# PyWavelets - Wavelet Transforms in Python¶

PyWavelets is open source wavelet transform software for Python. It combines a simple high level interface with low level C and Cython performance.

PyWavelets is very easy to use and get started with. Just install the package, open the Python interactive shell and type:

>>> import pywt
>>> cA, cD = pywt.dwt([1, 2, 3, 4], 'db1')


Voilà! Computing wavelet transforms has never been so simple :)

Here is a slightly more involved example of applying a digital wavelet transform to an image:

import numpy as np
import matplotlib.pyplot as plt

import pywt
import pywt.data

original = pywt.data.camera()

# Wavelet transform of image, and plot approximation and details
titles = ['Approximation', ' Horizontal detail',
'Vertical detail', 'Diagonal detail']
coeffs2 = pywt.dwt2(original, 'bior1.3')
LL, (LH, HL, HH) = coeffs2
fig = plt.figure(figsize=(12, 3))
for i, a in enumerate([LL, LH, HL, HH]):
ax = fig.add_subplot(1, 4, i + 1)
ax.imshow(a, interpolation="nearest", cmap=plt.cm.gray)
ax.set_title(titles[i], fontsize=10)
ax.set_xticks([])
ax.set_yticks([])

fig.tight_layout()
plt.show() ## Main features¶

The main features of PyWavelets are:

• 1D, 2D and nD Forward and Inverse Discrete Wavelet Transform (DWT and IDWT)
• 1D, 2D and nD Multilevel DWT and IDWT
• 1D, 2D and nD Stationary Wavelet Transform (Undecimated Wavelet Transform)
• 1D and 2D Wavelet Packet decomposition and reconstruction
• 1D Continuous Wavelet Transform
• Computing Approximations of wavelet and scaling functions
• Over 100 built-in wavelet filters and support for custom wavelets
• Single and double precision calculations
• Real and complex calculations
• Results compatible with Matlab Wavelet Toolbox (TM)

## Getting help¶

Use GitHub Issues, StackOverflow, or the PyWavelets discussions group to post your comments or questions.

PyWavelets is a free Open Source software released under the MIT license.

## Citing¶

If you use PyWavelets in a scientific publication, we would appreciate citations of the project:

Lee G, Gommers R, Wasilewski F, Wohlfahrt K, O’Leary A, Nahrstaedt H, and Contributors, “PyWavelets - Wavelet Transforms in Python”, 2006-, https://github.com/PyWavelets/pywt [Online; accessed 2018-MM-DD].

## Contents¶

### Installing¶

The latest release, including binary packages for Windows, macOS and Linux, is available for download from PyPI. You can also find source releases at the Releases Page.

You can install PyWavelets with:

pip install PyWavelets


Users of the Anaconda Python distribution may wish to obtain pre-built Windows, Intel Linux or macOS / OSX binaries from the main or conda-forge channel:

conda install pywavelets


Several Linux distributions have their own packages for PyWavelets, but these tend to be moderately out of date. Query your Linux package manager tool for python-pywavelets, python-wavelets, python-pywt or a similar package name.

#### Building from source¶

The most recent development version can be found on GitHub at https://github.com/PyWavelets/pywt.

The latest release, is available for download from PyPI or on the Releases Page.

If you want or need to install from source, you will need a working C compiler (any common one will work) and a recent version of Cython. Navigate to the PyWavelets source code directory (containing setup.py) and type:

pip install .


The requirements needed to build from source are:

• Python 2.7 or >=3.4
• Numpy >= 1.13.3
• Cython >= 0.23.5 (if installing from git, not from a PyPI source release)

To run all the tests for PyWavelets, you will also need to install the Matplotlib package.

Development guide section contains more information on building and installing from source code.

### API Reference¶

#### Wavelets¶

##### Wavelet families()¶
pywt.families(short=True)

Returns a list of available built-in wavelet families.

Currently the built-in families are:

• Haar (haar)
• Daubechies (db)
• Symlets (sym)
• Coiflets (coif)
• Biorthogonal (bior)
• Reverse biorthogonal (rbio)
• “Discrete” FIR approximation of Meyer wavelet (dmey)
• Gaussian wavelets (gaus)
• Mexican hat wavelet (mexh)
• Morlet wavelet (morl)
• Complex Gaussian wavelets (cgau)
• Shannon wavelets (shan)
• Frequency B-Spline wavelets (fbsp)
• Complex Morlet wavelets (cmor)
Parameters: short : bool, optional Use short names (default: True). families : list List of available wavelet families.

Examples

>>> import pywt
>>> pywt.families()
['haar', 'db', 'sym', 'coif', 'bior', 'rbio', 'dmey', 'gaus', 'mexh', 'morl', 'cgau', 'shan', 'fbsp', 'cmor']
>>> pywt.families(short=False)
['Haar', 'Daubechies', 'Symlets', 'Coiflets', 'Biorthogonal', 'Reverse biorthogonal', 'Discrete Meyer (FIR Approximation)', 'Gaussian', 'Mexican hat wavelet', 'Morlet wavelet', 'Complex Gaussian wavelets', 'Shannon wavelets', 'Frequency B-Spline wavelets', 'Complex Morlet wavelets']

##### Built-in wavelets - wavelist()¶
pywt.wavelist(family=None, kind='all')

Returns list of available wavelet names for the given family name.

Parameters: family : str, optional Short family name. If the family name is None (default) then names of all the built-in wavelets are returned. Otherwise the function returns names of wavelets that belong to the given family. Valid names are: 'haar', 'db', 'sym', 'coif', 'bior', 'rbio', 'dmey', 'gaus', 'mexh', 'morl', 'cgau', 'shan', 'fbsp', 'cmor'  kind : {‘all’, ‘continuous’, ‘discrete’}, optional Whether to return only wavelet names of discrete or continuous wavelets, or all wavelets. Default is 'all'. Ignored if family is specified. wavelist : list of str List of available wavelet names.

Examples

>>> import pywt
>>> pywt.wavelist('coif')
['coif1', 'coif2', 'coif3', 'coif4', 'coif5', 'coif6', 'coif7', ...
>>> pywt.wavelist(kind='continuous')
['cgau1', 'cgau2', 'cgau3', 'cgau4', 'cgau5', 'cgau6', 'cgau7', ...


Custom discrete wavelets are also supported through the Wavelet object constructor as described below.

##### Wavelet object¶
class pywt.Wavelet(name[, filter_bank=None])

Describes properties of a discrete wavelet identified by the specified wavelet name. For continuous wavelets see pywt.ContinuousWavelet instead. In order to use a built-in wavelet the name parameter must be a valid wavelet name from the pywt.wavelist() list.

Custom Wavelet objects can be created by passing a user-defined filters set with the filter_bank parameter.

Parameters: name – Wavelet name filter_bank – Use a user supplied filter bank instead of a built-in Wavelet.

The filter bank object can be a list of four filters coefficients or an object with filter_bank attribute, which returns a list of such filters in the following order:

[dec_lo, dec_hi, rec_lo, rec_hi]


Wavelet objects can also be used as a base filter banks. See section on using custom wavelets for more information.

Example:

>>> import pywt
>>> wavelet = pywt.Wavelet('db1')

name

Wavelet name.

short_name

Short wavelet name.

dec_lo

Decomposition filter values.

dec_hi

Decomposition filter values.

rec_lo

Reconstruction filter values.

rec_hi

Reconstruction filter values.

dec_len

Decomposition filter length.

rec_len

Reconstruction filter length.

filter_bank

Returns filters list for the current wavelet in the following order:

[dec_lo, dec_hi, rec_lo, rec_hi]

inverse_filter_bank

Returns list of reverse wavelet filters coefficients. The mapping from the filter_coeffs list is as follows:

[rec_lo[::-1], rec_hi[::-1], dec_lo[::-1], dec_hi[::-1]]

short_family_name

Wavelet short family name

family_name

Wavelet family name

orthogonal

Set if wavelet is orthogonal

biorthogonal

Set if wavelet is biorthogonal

symmetry

asymmetric, near symmetric, symmetric

vanishing_moments_psi

Number of vanishing moments for the wavelet function

vanishing_moments_phi

Number of vanishing moments for the scaling function

Example:

>>> def format_array(arr):
...     return "[%s]" % ", ".join(["%.14f" % x for x in arr])

>>> import pywt
>>> wavelet = pywt.Wavelet('db1')
>>> print(wavelet)
Wavelet db1
Family name:    Daubechies
Short name:     db
Filters length: 2
Orthogonal:     True
Biorthogonal:   True
Symmetry:       asymmetric
DWT:            True
CWT:            False
>>> print(format_array(wavelet.dec_lo), format_array(wavelet.dec_hi))
[0.70710678118655, 0.70710678118655] [-0.70710678118655, 0.70710678118655]
>>> print(format_array(wavelet.rec_lo), format_array(wavelet.rec_hi))
[0.70710678118655, 0.70710678118655] [0.70710678118655, -0.70710678118655]

###### Approximating wavelet and scaling functions - Wavelet.wavefun()¶
Wavelet.wavefun(level)

Changed in version 0.2: The time (space) localisation of approximation function points was added.

The wavefun() method can be used to calculate approximations of scaling function (phi) and wavelet function (psi) at the given level of refinement.

For orthogonal wavelets returns approximations of scaling function and wavelet function with corresponding x-grid coordinates:

[phi, psi, x] = wavelet.wavefun(level)


Example:

>>> import pywt
>>> wavelet = pywt.Wavelet('db2')
>>> phi, psi, x = wavelet.wavefun(level=5)


For other (biorthogonal but not orthogonal) wavelets returns approximations of scaling and wavelet function both for decomposition and reconstruction and corresponding x-grid coordinates:

[phi_d, psi_d, phi_r, psi_r, x] = wavelet.wavefun(level)


Example:

>>> import pywt
>>> wavelet = pywt.Wavelet('bior3.5')
>>> phi_d, psi_d, phi_r, psi_r, x = wavelet.wavefun(level=5)


You can find live examples of wavefun() usage and images of all the built-in wavelets on the Wavelet Properties Browser page. However, this website is no longer actively maintained and does not include every wavelet present in PyWavelets. The precision of the wavelet coefficients at that site is also lower than those included in PyWavelets.

##### Using custom wavelets¶

PyWavelets comes with a long list of the most popular wavelets built-in and ready to use. If you need to use a specific wavelet which is not included in the list it is very easy to do so. Just pass a list of four filters or an object with a filter_bank attribute as a filter_bank argument to the Wavelet constructor.

The filters list, either in a form of a simple Python list or returned via the filter_bank attribute, must be in the following order:

• lowpass decomposition filter
• highpass decomposition filter
• lowpass reconstruction filter
• highpass reconstruction filter

just as for the filter_bank attribute of the Wavelet class.

The Wavelet object created in this way is a standard Wavelet instance.

The following example illustrates the way of creating custom Wavelet objects from plain Python lists of filter coefficients and a filter bank-like object.

Example:

>>> import pywt, math
>>> c = math.sqrt(2)/2
>>> dec_lo, dec_hi, rec_lo, rec_hi = [c, c], [-c, c], [c, c], [c, -c]
>>> filter_bank = [dec_lo, dec_hi, rec_lo, rec_hi]
>>> myWavelet = pywt.Wavelet(name="myHaarWavelet", filter_bank=filter_bank)
>>>
>>> class HaarFilterBank(object):
...     @property
...     def filter_bank(self):
...         c = math.sqrt(2)/2
...         dec_lo, dec_hi, rec_lo, rec_hi = [c, c], [-c, c], [c, c], [c, -c]
...         return [dec_lo, dec_hi, rec_lo, rec_hi]
>>> filter_bank = HaarFilterBank()
>>> myOtherWavelet = pywt.Wavelet(name="myHaarWavelet", filter_bank=filter_bank)

##### ContinuousWavelet object¶
class pywt.ContinuousWavelet(name)

Describes properties of a continuous wavelet identified by the specified wavelet name. In order to use a built-in wavelet the name parameter must be a valid wavelet name from the pywt.wavelist() list.

Parameters: name – Wavelet name

Example:

>>> import pywt
>>> wavelet = pywt.ContinuousWavelet('gaus1')

name

Continuous Wavelet name.

short_family_name

Wavelet short family name

family_name

Wavelet family name

orthogonal

Set if wavelet is orthogonal

biorthogonal

Set if wavelet is biorthogonal

complex_cwt

Returns if wavelet is complex

lower_bound

Set the lower bound of the effective support

upper_bound

Set the upper bound of the effective support

center_frequency

Set the center frequency for the shan, fbsp and cmor wavelets

bandwidth_frequency

Set the bandwidth frequency for the shan, fbsp and cmor wavelets

fbsp_order

Set the order for the fbsp wavelet

symmetry

asymmetric, near symmetric, symmetric, anti-symmetric

Example:

>>> import pywt
>>> wavelet = pywt.ContinuousWavelet('gaus1')
>>> print(wavelet)
ContinuousWavelet gaus1
Family name:    Gaussian
Short name:     db
Symmetry:       anti-symmetric
DWT:            False
CWT:            True
Complex CWT:    False

###### Approximating wavelet functions - ContinuousWavelet.wavefun()¶
ContinuousWavelet.wavefun(level, length = None)

The wavefun() method can be used to calculate approximations of scaling function (psi) with grid (x). The vector length is set by length. The vector length can also be defined by 2**level if length is not set.

For complex_cwt wavelets returns a complex approximations of wavelet function with corresponding x-grid coordinates:

[psi, x] = wavelet.wavefun(level)


Example:

>>> import pywt
>>> wavelet = pywt.ContinuousWavelet('gaus1')
>>> psi, x = wavelet.wavefun(level=5)

###### Approximating wavelet functions - ContinuousWavelet.wavefun()¶
pywt.DiscreteContinuousWavelet(name[, filter_bank = None])
The DiscreteContinuousWavelet() returns a
Wavelet or a ContinuousWavelet object depending on the given name.

Example:

>>> import pywt
>>> wavelet = pywt.DiscreteContinuousWavelet('db1')
>>> print(wavelet)
Wavelet db1
Family name:    Daubechies
Short name:     db
Filters length: 2
Orthogonal:     True
Biorthogonal:   True
Symmetry:       asymmetric
DWT:            True
CWT:            False
>>> wavelet = pywt.DiscreteContinuousWavelet('gaus1')
>>> print(wavelet)
ContinuousWavelet gaus1
Family name:    Gaussian
Short name:     db
Symmetry:       anti-symmetric
DWT:            False
CWT:            True
Complex CWT:    False


#### Signal extension modes¶

Because the most common and practical way of representing digital signals in computer science is with finite arrays of values, some extrapolation of the input data has to be performed in order to extend the signal before computing the Discrete Wavelet Transform using the cascading filter banks algorithm.

Depending on the extrapolation method, significant artifacts at the signal’s borders can be introduced during that process, which in turn may lead to inaccurate computations of the DWT at the signal’s ends.

PyWavelets provides several methods of signal extrapolation that can be used to minimize this negative effect:

• zero - zero-padding - signal is extended by adding zero samples:

... 0  0 | x1 x2 ... xn | 0  0 ...

• constant - constant-padding - border values are replicated:

... x1 x1 | x1 x2 ... xn | xn xn ...

• symmetric - symmetric-padding - signal is extended by mirroring samples. This mode is also known as half-sample symmetric.:

... x2 x1 | x1 x2 ... xn | xn xn-1 ...

• reflect - reflect-padding - signal is extended by reflecting samples. This mode is also known as whole-sample symmetric.:

... x3 x2 | x1 x2 ... xn | xn-1 xn-2 ...

• periodic - periodic-padding - signal is treated as a periodic one:

... xn-1 xn | x1 x2 ... xn | x1 x2 ...

• smooth - smooth-padding - signal is extended according to the first derivatives calculated on the edges (straight line)
• antisymmetric - anti-symmetric padding - signal is extended by mirroring and negating samples. This mode is also known as half-sample anti-symmetric:

... -x2 -x1 | x1 x2 ... xn | -xn -xn-1 ...

• antireflect - anti-symmetric-reflect padding - signal is extended by reflecting anti-symmetrically about the edge samples. This mode is also known as whole-sample anti-symmetric:

... (2*x1 - x3) (2*x1 - x2) | x1 x2 ... xn | (2*xn - xn-1) (2*xn - xn-2) ...


DWT performed for these extension modes is slightly redundant, but ensures perfect reconstruction. To receive the smallest possible number of coefficients, computations can be performed with the periodization mode:

• periodization - periodization - is like periodic-padding but gives the smallest possible number of decomposition coefficients. IDWT must be performed with the same mode.

Example:

>>> import pywt
>>> print(pywt.Modes.modes)
['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect', 'antisymmetric', 'antireflect']


The following figure illustrates how a short signal (red) gets extended (black) outside of its original extent. Note that periodization first extends the signal to an even length prior to using periodic boundary conditions.

"""A visual illustration of the various signal extension modes supported in
PyWavelets. For efficiency, in the C routines the array is not actually
extended as is done here. This is just a demo for easier visual explanation of
the behavior of the various boundary modes.

In practice, which signal extension mode is beneficial will depend on the
signal characteristics.  For this particular signal, some modes such as
"periodic",  "antisymmetric" and "zeros" result in large discontinuities that
would lead to large amplitude boundary coefficients in the detail coefficients
of a discrete wavelet transform.
"""
import numpy as np
from matplotlib import pyplot as plt
from pywt._doc_utils import boundary_mode_subplot

# synthetic test signal
x = 5 - np.linspace(-1.9, 1.1, 9)**2

# Create a figure with one subplots per boundary mode
fig, axes = plt.subplots(3, 3, figsize=(10, 6))
axes = axes.ravel()
boundary_mode_subplot(x, 'symmetric', axes, symw=False)
boundary_mode_subplot(x, 'reflect', axes, symw=True)
boundary_mode_subplot(x, 'periodic', axes, symw=False)
boundary_mode_subplot(x, 'antisymmetric', axes, symw=False)
boundary_mode_subplot(x, 'antireflect', axes, symw=True)
boundary_mode_subplot(x, 'periodization', axes, symw=False)
boundary_mode_subplot(x, 'smooth', axes, symw=False)
boundary_mode_subplot(x, 'constant', axes, symw=False)
boundary_mode_subplot(x, 'zeros', axes, symw=False)
plt.show() Notice that you can use any of the following ways of passing wavelet and mode parameters:

>>> import pywt
>>> (a, d) = pywt.dwt([1,2,3,4,5,6], 'db2', 'smooth')
>>> (a, d) = pywt.dwt([1,2,3,4,5,6], pywt.Wavelet('db2'), pywt.Modes.smooth)


Note

Extending data in context of PyWavelets does not mean reallocation of the data in the computer’s physical memory and copying values, but rather computing the extra values only when they are needed. This feature saves extra memory and CPU resources and helps to avoid page swapping when handling relatively big data arrays on computers with low physical memory.

##### Naming Conventions¶

The correspondence between PyWavelets edge modes and the extension modes available in Matlab’s dwtmode and numpy’s pad are tabulated here for reference.

symmetric sym, symh symmetric
reflect symw reflect
smooth spd, sp1 N/A
constant sp0 edge
zero zpd constant, cval=0
periodic ppd wrap
periodization per N/A
antisymmetric asym, asymh N/A
antireflect asymw reflect, reflect_type=’odd’

#### Discrete Wavelet Transform (DWT)¶

Wavelet transform has recently become a very popular when it comes to analysis, de-noising and compression of signals and images. This section describes functions used to perform single- and multilevel Discrete Wavelet Transforms.

##### Single level dwt¶
pywt.dwt(data, wavelet, mode='symmetric', axis=-1)

Single level Discrete Wavelet Transform.

Parameters: data : array_like Input signal wavelet : Wavelet object or name Wavelet to use mode : str, optional Signal extension mode, see Modes axis: int, optional Axis over which to compute the DWT. If not given, the last axis is used. (cA, cD) : tuple Approximation and detail coefficients.

Notes

Length of coefficients arrays depends on the selected mode. For all modes except periodization:

len(cA) == len(cD) == floor((len(data) + wavelet.dec_len - 1) / 2)

For periodization mode (“per”):

len(cA) == len(cD) == ceil(len(data) / 2)

Examples

>>> import pywt
>>> (cA, cD) = pywt.dwt([1, 2, 3, 4, 5, 6], 'db1')
>>> cA
array([ 2.12132034,  4.94974747,  7.77817459])
>>> cD
array([-0.70710678, -0.70710678, -0.70710678])


See the signal extension modes section for the list of available options and the dwt_coeff_len() function for information on getting the expected result length.

The transform can be performed over one axis of multi-dimensional data. By default this is the last axis. For multi-dimensional transforms see the 2D transforms section.

##### Multilevel decomposition using wavedec¶
pywt.wavedec(data, wavelet, mode='symmetric', level=None, axis=-1)

Multilevel 1D Discrete Wavelet Transform of data.

Parameters: data: array_like Input data wavelet : Wavelet object or name string Wavelet to use mode : str, optional Signal extension mode, see Modes (default: ‘symmetric’) level : int, optional Decomposition level (must be >= 0). If level is None (default) then it will be calculated using the dwt_max_level function. axis: int, optional Axis over which to compute the DWT. If not given, the last axis is used. [cA_n, cD_n, cD_n-1, …, cD2, cD1] : list Ordered list of coefficients arrays where n denotes the level of decomposition. The first element (cA_n) of the result is approximation coefficients array and the following elements (cD_n - cD_1) are details coefficients arrays.

Examples

>>> from pywt import wavedec
>>> coeffs = wavedec([1,2,3,4,5,6,7,8], 'db1', level=2)
>>> cA2, cD2, cD1 = coeffs
>>> cD1
array([-0.70710678, -0.70710678, -0.70710678, -0.70710678])
>>> cD2
array([-2., -2.])
>>> cA2
array([  5.,  13.])

##### Partial Discrete Wavelet Transform data decomposition downcoef¶
pywt.downcoef(part, data, wavelet, mode='symmetric', level=1)

Partial Discrete Wavelet Transform data decomposition.

Similar to pywt.dwt, but computes only one set of coefficients. Useful when you need only approximation or only details at the given level.

Parameters: part : str Coefficients type: ‘a’ - approximations reconstruction is performed ‘d’ - details reconstruction is performed data : array_like Input signal. wavelet : Wavelet object or name Wavelet to use mode : str, optional Signal extension mode, see Modes. Default is ‘symmetric’. level : int, optional Decomposition level. Default is 1. coeffs : ndarray 1-D array of coefficients.
##### Maximum decomposition level - dwt_max_level, dwtn_max_level¶
pywt.dwt_max_level(data_len, filter_len)

Compute the maximum useful level of decomposition.

Parameters: data_len : int Input data length. filter_len : int, str or Wavelet The wavelet filter length. Alternatively, the name of a discrete wavelet or a Wavelet object can be specified. max_level : int Maximum level.

Notes

The rational for the choice of levels is the maximum level where at least one coefficient in the output is uncorrupted by edge effects caused by signal extension. Put another way, decomposition stops when the signal becomes shorter than the FIR filter length for a given wavelet. This corresponds to:

$\mathtt{max\_level} = \left\lfloor\log_2\left(\mathtt{ \frac{data\_len}{filter\_len - 1}}\right)\right\rfloor$

Examples

>>> import pywt
>>> w = pywt.Wavelet('sym5')
>>> pywt.dwt_max_level(data_len=1000, filter_len=w.dec_len)
6
>>> pywt.dwt_max_level(1000, w)
6
>>> pywt.dwt_max_level(1000, 'sym5')
6

pywt.dwtn_max_level(shape, wavelet, axes=None)

Compute the maximum level of decomposition for n-dimensional data.

This returns the maximum number of levels of decomposition suitable for use with wavedec, wavedec2 or wavedecn.

Parameters: shape : sequence of ints Input data shape. wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. axes : sequence of ints, optional Axes over which to compute the DWT. Axes may not be repeated. level : int Maximum level.

Notes

The level returned is the smallest dwt_max_level over all axes.

Examples

>>> import pywt
>>> pywt.dwtn_max_level((64, 32), 'db2')
3

##### Result coefficients length - dwt_coeff_len¶
pywt.dwt_coeff_len(data_len, filter_len, mode='symmetric')

Returns length of dwt output for given data length, filter length and mode

Parameters: data_len : int Data length. filter_len : int Filter length. mode : str, optional (default: ‘symmetric’) Signal extension mode, see Modes len : int Length of dwt output.

Notes

For all modes except periodization:

len(cA) == len(cD) == floor((len(data) + wavelet.dec_len - 1) / 2)


for periodization mode (“per”):

len(cA) == len(cD) == ceil(len(data) / 2)


Based on the given input data length (data_len), wavelet decomposition filter length (filter_len) and signal extension mode, the dwt_coeff_len() function calculates the length of the resulting coefficients arrays that would be created while performing dwt() transform.

filter_len can be either an int or Wavelet object for convenience.

#### Inverse Discrete Wavelet Transform (IDWT)¶

##### Single level idwt¶
pywt.idwt(cA, cD, wavelet, mode='symmetric', axis=-1)

Single level Inverse Discrete Wavelet Transform.

Parameters: cA : array_like or None Approximation coefficients. If None, will be set to array of zeros with same shape as cD. cD : array_like or None Detail coefficients. If None, will be set to array of zeros with same shape as cA. wavelet : Wavelet object or name Wavelet to use mode : str, optional (default: ‘symmetric’) Signal extension mode, see Modes axis: int, optional Axis over which to compute the inverse DWT. If not given, the last axis is used. rec: array_like Single level reconstruction of signal from given coefficients.

Examples

>>> import pywt
>>> (cA, cD) = pywt.dwt([1,2,3,4,5,6], 'db2', 'smooth')
>>> pywt.idwt(cA, cD, 'db2', 'smooth')
array([ 1.,  2.,  3.,  4.,  5.,  6.])


One of the neat features of idwt is that one of the cA and cD arguments can be set to None. In that situation the reconstruction will be performed using only the other one. Mathematically speaking, this is equivalent to passing a zero-filled array as one of the arguments.

>>> (cA, cD) = pywt.dwt([1,2,3,4,5,6], 'db2', 'smooth')
>>> A = pywt.idwt(cA, None, 'db2', 'smooth')
>>> D = pywt.idwt(None, cD, 'db2', 'smooth')
>>> A + D
array([ 1.,  2.,  3.,  4.,  5.,  6.])

###### Multilevel reconstruction using waverec¶
pywt.waverec(coeffs, wavelet, mode='symmetric', axis=-1)

Multilevel 1D Inverse Discrete Wavelet Transform.

Parameters: coeffs : array_like Coefficients list [cAn, cDn, cDn-1, …, cD2, cD1] wavelet : Wavelet object or name string Wavelet to use mode : str, optional Signal extension mode, see Modes (default: ‘symmetric’) axis: int, optional Axis over which to compute the inverse DWT. If not given, the last axis is used.

Notes

It may sometimes be desired to run waverec with some sets of coefficients omitted. This can best be done by setting the corresponding arrays to zero arrays of matching shape and dtype. Explicitly removing list entries or setting them to None is not supported.

Specifically, to ignore detail coefficients at level 2, one could do:

coeffs[-2] == np.zeros_like(coeffs[-2])


Examples

>>> import pywt
>>> coeffs = pywt.wavedec([1,2,3,4,5,6,7,8], 'db1', level=2)
>>> pywt.waverec(coeffs, 'db1')
array([ 1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.])

###### Direct reconstruction with upcoef¶
pywt.upcoef(part, coeffs, wavelet, level=1, take=0)

Direct reconstruction from coefficients.

Parameters: part : str Coefficients type: * ‘a’ - approximations reconstruction is performed * ‘d’ - details reconstruction is performed coeffs : array_like Coefficients array to recontruct wavelet : Wavelet object or name Wavelet to use level : int, optional Multilevel reconstruction level. Default is 1. take : int, optional Take central part of length equal to ‘take’ from the result. Default is 0. rec : ndarray 1-D array with reconstructed data from coefficients.

Examples

>>> import pywt
>>> data = [1,2,3,4,5,6]
>>> (cA, cD) = pywt.dwt(data, 'db2', 'smooth')
>>> pywt.upcoef('a', cA, 'db2') + pywt.upcoef('d', cD, 'db2')
array([-0.25      , -0.4330127 ,  1.        ,  2.        ,  3.        ,
4.        ,  5.        ,  6.        ,  1.78589838, -1.03108891])
>>> n = len(data)
>>> pywt.upcoef('a', cA, 'db2', take=n) + pywt.upcoef('d', cD, 'db2', take=n)
array([ 1.,  2.,  3.,  4.,  5.,  6.])


#### Overview of multilevel wavelet decompositions¶

There are a number of different ways a wavelet decomposition can be performed for multiresolution analysis of n-dimensional data. Here we will review the three approaches currently implemented in PyWavelets. 2D cases are illustrated, but each of the approaches extends to the n-dimensional case in a straightforward manner.

##### Multilevel Discrete Wavelet Transform¶

The most common approach to the multilevel discrete wavelet transform involves further decomposition of only the approximation subband at each subsequent level. This is also sometimes referred to as the Mallat decomposition [Mall89]. In 2D, the discrete wavelet transform produces four sets of coefficients corresponding to the four possible compinations of the wavelet decomposition filters over the two separate axes. (In n-dimensions, there are 2**n sets of coefficients). For subsequent levels of decomposition, only the approximation coefficients (the lowpass subband) are further decomposed.

In PyWavelets, this decomposition is implemented for n-dimensional data by wavedecn() and the inverse by waverecn(). 1D and 2D versions of these routines also exist. It is illustrated in the figure below. The top row indicates the coefficient names as used by wavedec2() after each level of decomposition. The bottom row shows wavelet coefficients for the camerman image (with each subband independently normalized for easier visualization).

import numpy as np
import pywt
from matplotlib import pyplot as plt
from pywt._doc_utils import wavedec2_keys, draw_2d_wp_basis

x = pywt.data.camera().astype(np.float32)
shape = x.shape

max_lev = 3       # how many levels of decomposition to draw
label_levels = 3  # how many levels to explicitly label on the plots

fig, axes = plt.subplots(2, 4, figsize=[14, 8])
for level in range(0, max_lev + 1):
if level == 0:
# show the original image before decomposition
axes[0, 0].set_axis_off()
axes[1, 0].imshow(x, cmap=plt.cm.gray)
axes[1, 0].set_title('Image')
axes[1, 0].set_axis_off()
continue

# plot subband boundaries of a standard DWT basis
draw_2d_wp_basis(shape, wavedec2_keys(level), ax=axes[0, level],
label_levels=label_levels)
axes[0, level].set_title('{} level\ndecomposition'.format(level))

# compute the 2D DWT
c = pywt.wavedec2(x, 'db2', mode='periodization', level=level)
# normalize each coefficient array independently for better visibility
c /= np.abs(c).max()
for detail_level in range(level):
c[detail_level + 1] = [d/np.abs(d).max() for d in c[detail_level + 1]]
# show the normalized coefficients
arr, slices = pywt.coeffs_to_array(c)
axes[1, level].imshow(arr, cmap=plt.cm.gray)
axes[1, level].set_title('Coefficients\n({} level)'.format(level))
axes[1, level].set_axis_off()

plt.tight_layout()
plt.show() It can be seen that many of the coefficients are near zero (gray). This ability of the wavelet transform to sparsely represent natural images is a key property that makes it desirable in applications such as image compression and restoration.

##### Fully Seperable Discrete Wavelet Transform¶

An alternative decomposition results in first fully decomposing one axis of the data prior to moving onto each additional axis in turn. This is illustrated for the 2D case in the upper right panel of the figure below. This approach has a factor of two higher computational cost as compared to the Mallat approach, but has advantages in compactly representing anisotropic data. A demo of this is available).

This form of the DWT is also sometimes referred to as the tensor wavelet transform or the hyperbolic wavelet transform. In PyWavelets it is implemented for n-dimensional data by fswavedecn() and the inverse by fswaverecn().

##### Wavelet Packet Transform¶

Another possible choice is to apply additional levels of decomposition to all wavelet subbands from the first level as opposed to only the approximation subband. This is known as the wavelet packet transform and is illustrated in 2D in the lower left panel of the figure. It is also possible to only perform any subset of the decompositions, resulting in a wide number of potential wavelet packet bases. An arbitrary example is shown in the lower right panel of the figure below.

A further description is available in the wavelet packet documentation.

For the wavelet packets, the plots below use “natural” ordering for simplicity, but this does not directly match the “frequency” ordering for these wavelet packets. It is possible to rearrange the coefficients into frequency ordering (see the get_level method of WaveletPacket2D and [Wick94] for more details).

from itertools import product
import numpy as np
from matplotlib import pyplot as plt
from pywt._doc_utils import (wavedec_keys, wavedec2_keys, draw_2d_wp_basis,
draw_2d_fswavedecn_basis)

shape = (512, 512)

max_lev = 4       # how many levels of decomposition to draw
label_levels = 2  # how many levels to explicitly label on the plots

if False:
fig, axes = plt.subplots(1, 4, figsize=[16, 4])
axes = axes.ravel()
else:
fig, axes = plt.subplots(2, 2, figsize=[8, 8])
axes = axes.ravel()

# plot a 5-level standard DWT basis
draw_2d_wp_basis(shape, wavedec2_keys(max_lev), ax=axes,
label_levels=label_levels)
axes.set_title('wavedec2 ({} level)'.format(max_lev))

# plot for the fully separable case
draw_2d_fswavedecn_basis(shape, max_lev, ax=axes, label_levels=label_levels)
axes.set_title('fswavedecn ({} level)'.format(max_lev))

# get all keys corresponding to a full wavelet packet decomposition
wp_keys = list(product(['a', 'd', 'h', 'v'], repeat=max_lev))
draw_2d_wp_basis(shape, wp_keys, ax=axes)
axes.set_title('wavelet packet\n(full: {} level)'.format(max_lev))

# plot an example of a custom wavelet packet basis
'aavh', 'aavv', 'ad', 'ah', 'ava', 'avd', 'avh', 'avv', 'd', 'h',
'vaa', 'vad', 'vah', 'vav', 'vd', 'vh', 'vv']
draw_2d_wp_basis(shape, keys, ax=axes, label_levels=label_levels)
axes.set_title('wavelet packet\n(custom)'.format(max_lev))

plt.tight_layout()
plt.show() References

 [Mall89] Mallat, S.G. “A Theory for Multiresolution Signal Decomposition: The Wavelet Representation” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 2, no. 7. July 1989. DOI: 10.1109/34.192463
 [Wick94] Wickerhauser, M.V. “Adapted Wavelet Analysis from Theory to Software” Wellesley. Massachusetts: A K Peters. 1994.

#### 2D Forward and Inverse Discrete Wavelet Transform¶

##### Single level dwt2¶
pywt.dwt2(data, wavelet, mode='symmetric', axes=(-2, -1))

2D Discrete Wavelet Transform.

Parameters: data : array_like 2D array with input data wavelet : Wavelet object or name string, or 2-tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or 2-tuple of strings, optional Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple of modes specifying the mode to use on each axis in axes. axes : 2-tuple of ints, optional Axes over which to compute the DWT. Repeated elements mean the DWT will be performed multiple times along these axes. (cA, (cH, cV, cD)) : tuple Approximation, horizontal detail, vertical detail and diagonal detail coefficients respectively. Horizontal refers to array axis 0 (or axes for user-specified axes).

Examples

>>> import numpy as np
>>> import pywt
>>> data = np.ones((4,4), dtype=np.float64)
>>> coeffs = pywt.dwt2(data, 'haar')
>>> cA, (cH, cV, cD) = coeffs
>>> cA
array([[ 2.,  2.],
[ 2.,  2.]])
>>> cV
array([[ 0.,  0.],
[ 0.,  0.]])


The relation to the other common data layout where all the approximation and details coefficients are stored in one big 2D array is as follows:

                            -------------------
|        |        |
| cA(LL) | cH(LH) |
|        |        |
(cA, (cH, cV, cD))  <--->   -------------------
|        |        |
| cV(HL) | cD(HH) |
|        |        |
-------------------


PyWavelets does not follow this pattern because of pure practical reasons of simple access to particular type of the output coefficients.

##### Single level idwt2¶
pywt.idwt2(coeffs, wavelet, mode='symmetric', axes=(-2, -1))

2-D Inverse Discrete Wavelet Transform.

Reconstructs data from coefficient arrays.

Parameters: coeffs : tuple (cA, (cH, cV, cD)) A tuple with approximation coefficients and three details coefficients 2D arrays like from dwt2. If any of these components are set to None, it will be treated as zeros. wavelet : Wavelet object or name string, or 2-tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or 2-tuple of strings, optional Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple of modes specifying the mode to use on each axis in axes. axes : 2-tuple of ints, optional Axes over which to compute the IDWT. Repeated elements mean the IDWT will be performed multiple times along these axes.

Examples

>>> import numpy as np
>>> import pywt
>>> data = np.array([[1,2], [3,4]], dtype=np.float64)
>>> coeffs = pywt.dwt2(data, 'haar')
>>> pywt.idwt2(coeffs, 'haar')
array([[ 1.,  2.],
[ 3.,  4.]])

##### 2D multilevel decomposition using wavedec2¶
pywt.wavedec2(data, wavelet, mode='symmetric', level=None, axes=(-2, -1))

Multilevel 2D Discrete Wavelet Transform.

Parameters: data : ndarray 2D input data wavelet : Wavelet object or name string, or 2-tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or 2-tuple of str, optional Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple containing a mode to apply along each axis in axes. level : int, optional Decomposition level (must be >= 0). If level is None (default) then it will be calculated using the dwt_max_level function. axes : 2-tuple of ints, optional Axes over which to compute the DWT. Repeated elements are not allowed. [cAn, (cHn, cVn, cDn), … (cH1, cV1, cD1)] : list Coefficients list. For user-specified axes, cH* corresponds to axes while cV* corresponds to axes. The first element returned is the approximation coefficients for the nth level of decomposition. Remaining elements are tuples of detail coefficients in descending order of decomposition level. (i.e. cH1 are the horizontal detail coefficients at the first level)

Examples

>>> import pywt
>>> import numpy as np
>>> coeffs = pywt.wavedec2(np.ones((4,4)), 'db1')
>>> # Levels:
>>> len(coeffs)-1
2
>>> pywt.waverec2(coeffs, 'db1')
array([[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]])

##### 2D multilevel reconstruction using waverec2¶
pywt.waverec2(coeffs, wavelet, mode='symmetric', axes=(-2, -1))

Multilevel 2D Inverse Discrete Wavelet Transform.

coeffs : list or tuple
Coefficients list [cAn, (cHn, cVn, cDn), … (cH1, cV1, cD1)]
wavelet : Wavelet object or name string, or 2-tuple of wavelets
Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes.
mode : str or 2-tuple of str, optional
Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple containing a mode to apply along each axis in axes.
axes : 2-tuple of ints, optional
Axes over which to compute the IDWT. Repeated elements are not allowed.
Returns: 2D array of reconstructed data.

Notes

It may sometimes be desired to run waverec2 with some sets of coefficients omitted. This can best be done by setting the corresponding arrays to zero arrays of matching shape and dtype. Explicitly removing list or tuple entries or setting them to None is not supported.

Specifically, to ignore all detail coefficients at level 2, one could do:

coeffs[-2] == tuple([np.zeros_like(v) for v in coeffs[-2]])


Examples

>>> import pywt
>>> import numpy as np
>>> coeffs = pywt.wavedec2(np.ones((4,4)), 'db1')
>>> # Levels:
>>> len(coeffs)-1
2
>>> pywt.waverec2(coeffs, 'db1')
array([[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]])

##### 2D coordinate conventions¶

The labels for “horizontal” and “vertical” used by dwt2 and idwt2 follow the common mathematical convention that coordinate axis 0 is horizontal while axis 1 is vertical:

dwt2, idwt2 convention
----------------------

axis 1 ^
|
|
|
|--------->
axis 0


Note that this is different from another common convention used in computer graphics and image processing (e.g. by matplotlib’s imshow and functions in scikit-image). In those packages axis 0 is a vertical axis and axis 1 is horizontal as follows:

 imshow convention
-------------------
axis 1
|--------->
|
|
|
axis 0 v


#### nD Forward and Inverse Discrete Wavelet Transform¶

 _multilevel Multilevel 1D and 2D Discrete Wavelet Transform and Inverse Discrete Wavelet Transform.
##### Single level - dwtn¶
pywt.dwtn(data, wavelet, mode='symmetric', axes=None)

Single-level n-dimensional Discrete Wavelet Transform.

Parameters: data : array_like n-dimensional array with input data. wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or tuple of string, optional Signal extension mode used in the decomposition, see Modes (default: ‘symmetric’). This can also be a tuple of modes specifying the mode to use on each axis in axes. axes : sequence of ints, optional Axes over which to compute the DWT. Repeated elements mean the DWT will be performed multiple times along these axes. A value of None (the default) selects all axes. Axes may be repeated, but information about the original size may be lost if it is not divisible by 2 ** nrepeats. The reconstruction will be larger, with additional values derived according to the mode parameter. pywt.wavedecn should be used for multilevel decomposition. coeffs : dict Results are arranged in a dictionary, where key specifies the transform type on each dimension and value is a n-dimensional coefficients array. For example, for a 2D case the result will look something like this: {'aa': # A(LL) - approx. on 1st dim, approx. on 2nd dim 'ad': # V(LH) - approx. on 1st dim, det. on 2nd dim 'da': # H(HL) - det. on 1st dim, approx. on 2nd dim 'dd': # D(HH) - det. on 1st dim, det. on 2nd dim }  For user-specified axes, the order of the characters in the dictionary keys map to the specified axes.
##### Single level - idwtn¶
pywt.idwtn(coeffs, wavelet, mode='symmetric', axes=None)

Single-level n-dimensional Inverse Discrete Wavelet Transform.

Parameters: coeffs: dict Dictionary as in output of dwtn. Missing or None items will be treated as zeros. wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or list of string, optional Signal extension mode used in the decomposition, see Modes (default: ‘symmetric’). This can also be a tuple of modes specifying the mode to use on each axis in axes. axes : sequence of ints, optional Axes over which to compute the IDWT. Repeated elements mean the IDWT will be performed multiple times along these axes. A value of None (the default) selects all axes. For the most accurate reconstruction, the axes should be provided in the same order as they were provided to dwtn. data: ndarray Original signal reconstructed from input data.
##### Multilevel decomposition - wavedecn¶
pywt.wavedecn(data, wavelet, mode='symmetric', level=None, axes=None)

Multilevel nD Discrete Wavelet Transform.

Parameters: data : ndarray nD input data wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or tuple of str, optional Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple containing a mode to apply along each axis in axes. level : int, optional Decomposition level (must be >= 0). If level is None (default) then it will be calculated using the dwt_max_level function. axes : sequence of ints, optional Axes over which to compute the DWT. Axes may not be repeated. The default is None, which means transform all axes (axes = range(data.ndim)). [cAn, {details_level_n}, … {details_level_1}] : list Coefficients list. Coefficients are listed in descending order of decomposition level. cAn are the approximation coefficients at level n. Each details_level_i element is a dictionary containing detail coefficients at level i of the decomposition. As a concrete example, a 3D decomposition would have the following set of keys in each details_level_i dictionary: {'aad', 'ada', 'daa', 'add', 'dad', 'dda', 'ddd'}  where the order of the characters in each key map to the specified axes.

Examples

>>> import numpy as np
>>> from pywt import wavedecn, waverecn
>>> coeffs = wavedecn(np.ones((4, 4, 4)), 'db1')
>>> # Levels:
>>> len(coeffs)-1
2
>>> waverecn(coeffs, 'db1')
array([[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]],
[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]],
[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]],
[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]]])

##### Multilevel reconstruction - waverecn¶
pywt.waverecn(coeffs, wavelet, mode='symmetric', axes=None)

Multilevel nD Inverse Discrete Wavelet Transform.

coeffs : array_like
Coefficients list [cAn, {details_level_n}, … {details_level_1}]
wavelet : Wavelet object or name string, or tuple of wavelets
Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes.
mode : str or tuple of str, optional
Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple containing a mode to apply along each axis in axes.
axes : sequence of ints, optional
Axes over which to compute the IDWT. Axes may not be repeated.
Returns: nD array of reconstructed data.

Notes

It may sometimes be desired to run waverecn with some sets of coefficients omitted. This can best be done by setting the corresponding arrays to zero arrays of matching shape and dtype. Explicitly removing list or dictionary entries or setting them to None is not supported.

Specifically, to ignore all detail coefficients at level 2, one could do:

coeffs[-2] = {k: np.zeros_like(v) for k, v in coeffs[-2].items()}


Examples

>>> import numpy as np
>>> from pywt import wavedecn, waverecn
>>> coeffs = wavedecn(np.ones((4, 4, 4)), 'db1')
>>> # Levels:
>>> len(coeffs)-1
2
>>> waverecn(coeffs, 'db1')
array([[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]],
[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]],
[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]],
[[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.],
[ 1.,  1.,  1.,  1.]]])

##### Multilevel fully separable decomposition - fswavedecn¶
pywt.fswavedecn(data, wavelet, mode='symmetric', levels=None, axes=None)

Fully Separable Wavelet Decomposition.

This is a variant of the multilevel discrete wavelet transform where all levels of decomposition are performed along a single axis prior to moving onto the next axis. Unlike in wavedecn, the number of levels of decomposition are not required to be the same along each axis which can be a benefit for anisotropic data.

Parameters: data: array_like Input data wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or tuple of str, optional Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple containing a mode to apply along each axis in axes. levels : int or sequence of ints, optional Decomposition levels along each axis (must be >= 0). If an integer is provided, the same number of levels are used for all axes. If levels is None (default), dwt_max_level will be used to compute the maximum number of levels possible for each axis. axes : sequence of ints, optional Axes over which to compute the transform. Axes may not be repeated. The default is to transform along all axes. fswavedecn_result : FswavedecnResult object Contains the wavelet coefficients, slice objects to allow obtaining the coefficients per detail or approximation level, and more. See FswavedecnResult for details.

fswaverecn
inverse of fswavedecn

Notes

This transformation has been variously referred to as the (fully) separable wavelet transform (e.g. refs , ), the tensor-product wavelet () or the hyperbolic wavelet transform (). It is well suited to data with anisotropic smoothness.

In  it was demonstrated that fully separable transform performs at least as well as the DWT for image compression. Computation time is a factor 2 larger than that for the DWT.

References

  (1, 2) PH Westerink. Subband Coding of Images. Ph.D. dissertation, Dept. Elect. Eng., Inf. Theory Group, Delft Univ. Technol., Delft, The Netherlands, 1989. (see Section 2.3) http://resolver.tudelft.nl/uuid:a4d195c3-1f89-4d66-913d-db9af0969509
  (1, 2, 3) CP Rosiene and TQ Nguyen. Tensor-product wavelet vs. Mallat decomposition: A comparative analysis, in Proc. IEEE Int. Symp. Circuits and Systems, Orlando, FL, Jun. 1999, pp. 431-434.
  (1, 2) V Velisavljevic, B Beferull-Lozano, M Vetterli and PL Dragotti. Directionlets: Anisotropic Multidirectional Representation With Separable Filtering. IEEE Transactions on Image Processing, Vol. 15, No. 7, July 2006.
  (1, 2) RA DeVore, SV Konyagin and VN Temlyakov. “Hyperbolic wavelet approximation,” Constr. Approx. 14 (1998), 1-26.

Examples

>>> from pywt import fswavedecn
>>> fs_result = fswavedecn(np.ones((32, 32)), 'sym2', levels=(1, 3))
>>> print(fs_result.detail_keys())
[(0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3)]
>>> approx_coeffs = fs_result.approx
>>> detail_1_2 = fs_result[(1, 2)]

##### Multilevel fully separable reconstruction - fswaverecn¶
pywt.fswaverecn(fswavedecn_result)

Fully Separable Inverse Wavelet Reconstruction.

Parameters: fswavedecn_result : FswavedecnResult object FswavedecnResult object from fswavedecn. reconstructed : ndarray Array of reconstructed data.

fswavedecn
inverse of fswaverecn

Notes

This transformation has been variously referred to as the (fully) separable wavelet transform (e.g. refs , ), the tensor-product wavelet () or the hyperbolic wavelet transform (). It is well suited to data with anisotropic smoothness.

In  it was demonstrated that the fully separable transform performs at least as well as the DWT for image compression. Computation time is a factor 2 larger than that for the DWT.

References

  (1, 2) PH Westerink. Subband Coding of Images. Ph.D. dissertation, Dept. Elect. Eng., Inf. Theory Group, Delft Univ. Technol., Delft, The Netherlands, 1989. (see Section 2.3) http://resolver.tudelft.nl/uuid:a4d195c3-1f89-4d66-913d-db9af0969509
  (1, 2, 3) CP Rosiene and TQ Nguyen. Tensor-product wavelet vs. Mallat decomposition: A comparative analysis, in Proc. IEEE Int. Symp. Circuits and Systems, Orlando, FL, Jun. 1999, pp. 431-434.
  (1, 2) V Velisavljevic, B Beferull-Lozano, M Vetterli and PL Dragotti. Directionlets: Anisotropic Multidirectional Representation With Separable Filtering. IEEE Transactions on Image Processing, Vol. 15, No. 7, July 2006.
  (1, 2) RA DeVore, SV Konyagin and VN Temlyakov. “Hyperbolic wavelet approximation,” Constr. Approx. 14 (1998), 1-26.
##### Multilevel fully separable reconstruction coeffs - FswavedecnResult¶
class pywt.FswavedecnResult(coeffs, coeff_slices, wavelets, mode_enums, axes)

Object representing fully separable wavelet transform coefficients.

Parameters: coeffs : ndarray The coefficient array. coeff_slices : list List of slices corresponding to each detail or approximation coefficient array. wavelets : list of pywt.DiscreteWavelet objects The wavelets used. Will be a list with length equal to len(axes). mode_enums : list of int The border modes used. Will be a list with length equal to len(axes). axes : tuple of int The set of axes over which the transform was performed. approx ndarray: The approximation coefficients. axes List of str: The axes the transform was performed along. coeff_slices List: List of coefficient slices. coeffs ndarray: All coefficients stacked into a single array. levels List of int: Levels of decomposition along each transformed axis. modes List of str: The border mode used along each transformed axis. ndim int: Number of data dimensions. ndim_transform int: Number of axes transformed. wavelet_names List of pywt.DiscreteWavelet: wavelet for each transformed axis. wavelets List of pywt.DiscreteWavelet: wavelet for each transformed axis.

Methods

 detail_keys() Return a list of all detail coefficient keys.

#### Handling DWT Coefficients¶

Convenience routines are available for converting the outputs of the multilevel dwt functions (wavedec, wavedec2 and wavedecn) to and from a single, concatenated coefficient array.

##### Concatenating all coefficients into a single n-d array¶
pywt.coeffs_to_array(coeffs, padding=0, axes=None)

Arrange a wavelet coefficient list from wavedecn into a single array.

Parameters: coeffs : array-like dictionary of wavelet coefficients as returned by pywt.wavedecn padding : float or None, optional If None, raise an error if the coefficients cannot be tightly packed. axes : sequence of ints, optional Axes over which the DWT that created coeffs was performed. The default value of None corresponds to all axes. coeff_arr : array-like Wavelet transform coefficient array. coeff_slices : list List of slices corresponding to each coefficient. As a 2D example, coeff_arr[coeff_slices[‘dd’]] would extract the first level detail coefficients from coeff_arr.

array_to_coeffs
the inverse of coeffs_to_array

Notes

Assume a 2D coefficient dictionary, c, from a two-level transform.

Then all 2D coefficients will be stacked into a single larger 2D array as follows:

+---------------+---------------+-------------------------------+
|               |               |                               |
|     c      |  c['da']   |                               |
|               |               |                               |
+---------------+---------------+           c['da']          |
|               |               |                               |
| c['ad']    |  c['dd']   |                               |
|               |               |                               |
+---------------+---------------+ ------------------------------+
|                               |                               |
|                               |                               |
|                               |                               |
|                               |                               |
|                               |                               |
|                               |                               |
+-------------------------------+-------------------------------+


Examples

>>> import pywt
>>> cam = pywt.data.camera()
>>> coeffs = pywt.wavedecn(cam, wavelet='db2', level=3)
>>> arr, coeff_slices = pywt.coeffs_to_array(coeffs)

##### Splitting concatenated coefficient array back into its components¶
pywt.array_to_coeffs(arr, coeff_slices, output_format='wavedecn')

Convert a combined array of coefficients back to a list compatible with waverecn.

Parameters: arr : array-like An array containing all wavelet coefficients. This should have been generated via coeffs_to_array. coeff_slices : list of tuples List of slices corresponding to each coefficient as obtained from array_to_coeffs. output_format : {‘wavedec’, ‘wavedec2’, ‘wavedecn’} Make the form of the coefficients compatible with this type of multilevel transform. coeffs: array-like Wavelet transform coefficient array.

coeffs_to_array
the inverse of array_to_coeffs

Notes

A single large array containing all coefficients will have subsets stored, into a waverecn list, c, as indicated below:

+---------------+---------------+-------------------------------+
|               |               |                               |
|     c      |  c['da']   |                               |
|               |               |                               |
+---------------+---------------+           c['da']          |
|               |               |                               |
| c['ad']    |  c['dd']   |                               |
|               |               |                               |
+---------------+---------------+ ------------------------------+
|                               |                               |
|                               |                               |
|                               |                               |
|                               |                               |
|                               |                               |
|                               |                               |
+-------------------------------+-------------------------------+


Examples

>>> import pywt
>>> from numpy.testing import assert_array_almost_equal
>>> cam = pywt.data.camera()
>>> coeffs = pywt.wavedecn(cam, wavelet='db2', level=3)
>>> arr, coeff_slices = pywt.coeffs_to_array(coeffs)
>>> coeffs_from_arr = pywt.array_to_coeffs(arr, coeff_slices)
>>> cam_recon = pywt.waverecn(coeffs_from_arr, wavelet='db2')
>>> assert_array_almost_equal(cam, cam_recon)

##### Raveling and unraveling coefficients to/from a 1D array¶
pywt.ravel_coeffs(coeffs, axes=None)

Ravel a set of multilevel wavelet coefficients into a single 1D array.

Parameters: coeffs : array-like A list of multilevel wavelet coefficients as returned by wavedec, wavedec2 or wavedecn. axes : sequence of ints, optional Axes over which the DWT that created coeffs was performed. The default value of None corresponds to all axes. coeff_arr : array-like Wavelet transform coefficient array. All coefficients have been concatenated into a single array. coeff_slices : list List of slices corresponding to each coefficient. As a 2D example, coeff_arr[coeff_slices['dd']] would extract the first level detail coefficients from coeff_arr. coeff_shapes : list List of shapes corresponding to each coefficient. For example, in 2D, coeff_shapes['dd'] would contain the original shape of the first level detail coefficients array.

unravel_coeffs
the inverse of ravel_coeffs

Examples

>>> import pywt
>>> cam = pywt.data.camera()
>>> coeffs = pywt.wavedecn(cam, wavelet='db2', level=3)
>>> arr, coeff_slices, coeff_shapes = pywt.ravel_coeffs(coeffs)

pywt.unravel_coeffs(arr, coeff_slices, coeff_shapes, output_format='wavedecn')

Unravel a raveled array of multilevel wavelet coefficients.

Parameters: arr : array-like An array containing all wavelet coefficients. This should have been generated by applying ravel_coeffs to the output of wavedec, wavedec2 or wavedecn. coeff_slices : list of tuples List of slices corresponding to each coefficient as obtained from ravel_coeffs. coeff_shapes : list of tuples List of shapes corresponding to each coefficient as obtained from ravel_coeffs. output_format : {‘wavedec’, ‘wavedec2’, ‘wavedecn’}, optional Make the form of the unraveled coefficients compatible with this type of multilevel transform. The default is ‘wavedecn’. coeffs: list List of wavelet transform coefficients. The specific format of the list elements is determined by output_format.

ravel_coeffs
the inverse of unravel_coeffs

Examples

>>> import pywt
>>> from numpy.testing import assert_array_almost_equal
>>> cam = pywt.data.camera()
>>> coeffs = pywt.wavedecn(cam, wavelet='db2', level=3)
>>> arr, coeff_slices, coeff_shapes = pywt.ravel_coeffs(coeffs)
>>> coeffs_from_arr = pywt.unravel_coeffs(arr, coeff_slices, coeff_shapes)
>>> cam_recon = pywt.waverecn(coeffs_from_arr, wavelet='db2')
>>> assert_array_almost_equal(cam, cam_recon)

##### Multilevel: Total size of all coefficients - wavedecn_size¶
pywt.wavedecn_size(shapes)

Compute the total number of wavedecn coefficients.

Parameters: shapes : list of coefficient shapes A set of coefficient shapes as returned by wavedecn_shapes. Alternatively, the user can specify a set of coefficients as returned by wavedecn. size : int The total number of coefficients.

Examples

>>> import pywt
>>> data_shape = (64, 32)
>>> shapes = pywt.wavedecn_shapes(data_shape, 'db2', mode='periodization')
>>> pywt.wavedecn_size(shapes)
2048
>>> coeffs = pywt.wavedecn(np.ones(data_shape), 'sym4', mode='symmetric')
>>> pywt.wavedecn_size(coeffs)
3087

##### Multilevel: n-d coefficient shapes - wavedecn_shapes¶
pywt.wavedecn_shapes(shape, wavelet, mode='symmetric', level=None, axes=None)

Subband shapes for a multilevel nD discrete wavelet transform.

Parameters: shape : sequence of ints The shape of the data to be transformed. wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple containing a wavelet to apply along each axis in axes. mode : str or tuple of str, optional Signal extension mode, see Modes (default: ‘symmetric’). This can also be a tuple containing a mode to apply along each axis in axes. level : int, optional Decomposition level (must be >= 0). If level is None (default) then it will be calculated using the dwt_max_level function. axes : sequence of ints, optional Axes over which to compute the DWT. Axes may not be repeated. The default is None, which means transform all axes (axes = range(data.ndim)). shapes : [cAn, {details_level_n}, … {details_level_1}] Coefficients shape list. Mirrors the output of wavedecn, except it contains only the shapes of the coefficient arrays rather than the arrays themselves.

Examples

>>> import pywt
>>> pywt.wavedecn_shapes((64, 32), wavelet='db2', level=3, axes=(0, ))
[(10, 32), {'d': (10, 32)}, {'d': (18, 32)}, {'d': (33, 32)}]


#### Stationary Wavelet Transform¶

Stationary Wavelet Transform (SWT), also known as Undecimated wavelet transform or Algorithme à trous is a translation-invariance modification of the Discrete Wavelet Transform that does not decimate coefficients at every transformation level.

##### Multilevel 1D swt¶
pywt.swt(data, wavelet, level=None, start_level=0, axis=-1)

Multilevel 1D stationary wavelet transform.

Parameters: data : Input signal wavelet : Wavelet to use (Wavelet object or name) level : int, optional The number of decomposition steps to perform. start_level : int, optional The level at which the decomposition will begin (it allows one to skip a given number of transform steps and compute coefficients starting from start_level) (default: 0) axis: int, optional Axis over which to compute the SWT. If not given, the last axis is used. coeffs : list List of approximation and details coefficients pairs in order similar to wavedec function: [(cAn, cDn), ..., (cA2, cD2), (cA1, cD1)]  where n equals input parameter level. If start_level = m is given, then the beginning m steps are skipped: [(cAm+n, cDm+n), ..., (cAm+1, cDm+1), (cAm, cDm)] 

Notes

The implementation here follows the “algorithm a-trous” and requires that the signal length along the transformed axis be a multiple of 2**level. If this is not the case, the user should pad up to an appropriate size using a function such as numpy.pad.

##### Multilevel 2D swt2¶
pywt.swt2(data, wavelet, level, start_level=0, axes=(-2, -1))

Multilevel 2D stationary wavelet transform.

Parameters: data : array_like 2D array with input data wavelet : Wavelet object or name string, or 2-tuple of wavelets Wavelet to use. This can also be a tuple of wavelets to apply per axis in axes. level : int The number of decomposition steps to perform. start_level : int, optional The level at which the decomposition will start (default: 0) axes : 2-tuple of ints, optional Axes over which to compute the SWT. Repeated elements are not allowed. coeffs : list Approximation and details coefficients (for start_level = m): [ (cA_m+level, (cH_m+level, cV_m+level, cD_m+level) ), ..., (cA_m+1, (cH_m+1, cV_m+1, cD_m+1) ), (cA_m, (cH_m, cV_m, cD_m) ) ]  where cA is approximation, cH is horizontal details, cV is vertical details, cD is diagonal details and m is start_level.

Notes

The implementation here follows the “algorithm a-trous” and requires that the signal length along the transformed axes be a multiple of 2**level. If this is not the case, the user should pad up to an appropriate size using a function such as numpy.pad.

##### Multilevel n-dimensional swtn¶
pywt.swtn(data, wavelet, level, start_level=0, axes=None)

n-dimensional stationary wavelet transform.

Parameters: data : array_like n-dimensional array with input data. wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple of wavelets to apply per axis in axes. level : int The number of decomposition steps to perform. start_level : int, optional The level at which the decomposition will start (default: 0) axes : sequence of ints, optional Axes over which to compute the SWT. A value of None (the default) selects all axes. Axes may not be repeated. [{coeffs_level_n}, …, {coeffs_level_1}]: list of dict Results for each level are arranged in a dictionary, where the key specifies the transform type on each dimension and value is a n-dimensional coefficients array. For example, for a 2D case the result at a given level will look something like this: {'aa': # A(LL) - approx. on 1st dim, approx. on 2nd dim 'ad': # V(LH) - approx. on 1st dim, det. on 2nd dim 'da': # H(HL) - det. on 1st dim, approx. on 2nd dim 'dd': # D(HH) - det. on 1st dim, det. on 2nd dim }  For user-specified axes, the order of the characters in the dictionary keys map to the specified axes.

Notes

The implementation here follows the “algorithm a-trous” and requires that the signal length along the transformed axes be a multiple of 2**level. If this is not the case, the user should pad up to an appropriate size using a function such as numpy.pad.

##### Maximum decomposition level - swt_max_level¶
pywt.swt_max_level(input_len)

Calculates the maximum level of Stationary Wavelet Transform for data of given length.

Parameters: input_len : int Input data length. max_level : int Maximum level of Stationary Wavelet Transform for data of given length.

Notes

For the current implementation of the stationary wavelet transform, this corresponds to the number of times input_len is evenly divisible by two. In other words, for an n-level transform, the signal length must be a multiple of 2**n. numpy.pad can be used to pad a signal up to an appropriate length as needed.

#### Inverse Stationary Wavelet Transform¶

Inverse stationary wavelet transforms are provided.

Note: These inverse transforms are not yet optimized for speed. Only, the n-dimensional inverse transform currently has axes support.

##### Multilevel 1D iswt¶
pywt.iswt(coeffs, wavelet)

Multilevel 1D inverse discrete stationary wavelet transform.

Parameters: coeffs : array_like Coefficients list of tuples: [(cAn, cDn), ..., (cA2, cD2), (cA1, cD1)]  where cA is approximation, cD is details. Index 1 corresponds to start_level from pywt.swt. wavelet : Wavelet object or name string Wavelet to use 1D array of reconstructed data.

Examples

>>> import pywt
>>> coeffs = pywt.swt([1,2,3,4,5,6,7,8], 'db2', level=2)
>>> pywt.iswt(coeffs, 'db2')
array([ 1.,  2.,  3.,  4.,  5.,  6.,  7.,  8.])

##### Multilevel 2D iswt2¶
pywt.iswt2(coeffs, wavelet)

Multilevel 2D inverse discrete stationary wavelet transform.

Parameters: coeffs : list Approximation and details coefficients: [ (cA_n, (cH_n, cV_n, cD_n) ), ..., (cA_2, (cH_2, cV_2, cD_2) ), (cA_1, (cH_1, cV_1, cD_1) ) ]  where cA is approximation, cH is horizontal details, cV is vertical details, cD is diagonal details and n is the number of levels. Index 1 corresponds to start_level from pywt.swt2. wavelet : Wavelet object or name string, or 2-tuple of wavelets Wavelet to use. This can also be a 2-tuple of wavelets to apply per axis. 2D array of reconstructed data.

Examples

>>> import pywt
>>> coeffs = pywt.swt2([[1,2,3,4],[5,6,7,8],
...                     [9,10,11,12],[13,14,15,16]],
...                    'db1', level=2)
>>> pywt.iswt2(coeffs, 'db1')
array([[  1.,   2.,   3.,   4.],
[  5.,   6.,   7.,   8.],
[  9.,  10.,  11.,  12.],
[ 13.,  14.,  15.,  16.]])

##### Multilevel n-dimensional iswtn¶
pywt.iswtn(coeffs, wavelet, axes=None)

Multilevel nD inverse discrete stationary wavelet transform.

Parameters: coeffs : list [{coeffs_level_n}, …, {coeffs_level_1}]: list of dict wavelet : Wavelet object or name string, or tuple of wavelets Wavelet to use. This can also be a tuple of wavelets to apply per axis in axes. axes : sequence of ints, optional Axes over which to compute the inverse SWT. Axes may not be repeated. The default is None, which means transform all axes (axes = range(data.ndim)). nD array of reconstructed data.

Examples

>>> import pywt
>>> coeffs = pywt.swtn([[1,2,3,4],[5,6,7,8],
...                     [9,10,11,12],[13,14,15,16]],
...                    'db1', level=2)
>>> pywt.iswtn(coeffs, 'db1')
array([[  1.,   2.,   3.,   4.],
[  5.,   6.,   7.,   8.],
[  9.,  10.,  11.,  12.],
[ 13.,  14.,  15.,  16.]])


#### Wavelet Packets¶

New in version 0.2.

Version 0.2 of PyWavelets includes many new features and improvements. One of such new feature is a two-dimensional wavelet packet transform structure that is almost completely sharing programming interface with the one-dimensional tree structure.

In order to achieve this simplification, a new inheritance scheme was used in which a BaseNode base node class is a superclass for both Node and Node2D node classes.

The node classes are used as data wrappers and can be organized in trees (binary trees for 1D transform case and quad-trees for the 2D one). They are also superclasses to the WaveletPacket class and WaveletPacket2D class that are used as the decomposition tree roots and contain a couple additional methods.

The below diagram illustrates the inheritance tree:

##### BaseNode - a common interface of WaveletPacket and WaveletPacket2D¶
class pywt.BaseNode
class pywt.Node(BaseNode)
class pywt.WaveletPacket(Node)
class pywt.Node2D(BaseNode)
class pywt.WaveletPacket2D(Node2D)

Note

The BaseNode is a base class for Node and Node2D. It should not be used directly unless creating a new transformation type. It is included here to document the common interface of 1D and 2D node an wavelet packet transform classes.

__init__(parent, data, node_name)
Parameters: parent – parent node. If parent is None then the node is considered detached. data – data associated with the node. 1D or 2D numeric array, depending on the transform type. node_name – a name identifying the coefficients type. See Node.node_name and Node2D.node_name for information on the accepted subnodes names.
data

Data associated with the node. 1D or 2D numeric array (depends on the transform type).

parent

Parent node. Used in tree navigation. None for root node.

wavelet

Wavelet used for decomposition and reconstruction. Inherited from parent node.

mode

Signal extension mode for the dwt() (dwt2()) and idwt() (idwt2()) decomposition and reconstruction functions. Inherited from parent node.

level

Decomposition level of the current node. 0 for root (original data), 1 for the first decomposition level, etc.

path

Path string defining position of the node in the decomposition tree.

node_name

Node name describing data coefficients type of the current subnode.

See Node.node_name and Node2D.node_name.

maxlevel

Maximum allowed level of decomposition. Evaluated from parent or child nodes.

is_empty

Checks if data attribute is None.

has_any_subnode

Checks if node has any subnodes (is not a leaf node).

decompose()

Performs Discrete Wavelet Transform on the data and returns transform coefficients.

reconstruct([update=False])

Performs Inverse Discrete Wavelet Transform on subnodes coefficients and returns reconstructed data for the current level.

Parameters: update – If set, the data attribute will be updated with the reconstructed value.

Note

Descends to subnodes and recursively calls reconstruct() on them.

get_subnode(part[, decompose=True])

Returns subnode or None (see decomposition flag description).

Parameters: part – Subnode name decompose – If True and subnode does not exist, it will be created using coefficients from the DWT decomposition of the current node.
__getitem__(path)

Used to access nodes in the decomposition tree by string path.

Parameters: path – Path string composed from valid node names. See Node.node_name and Node2D.node_name for node naming convention.

Similar to get_subnode() method with decompose=True, but can access nodes on any level in the decomposition tree.

If node does not exist yet, it will be created by decomposition of its parent node.

__setitem__(path, data)

Used to set node or node’s data in the decomposition tree. Nodes are identified by string path.

Parameters: path – Path string composed from valid node names. See Node.node_name and Node2D.node_name for node naming convention. data – numeric array or BaseNode subclass.
__delitem__(path)

Used to delete node from the decomposition tree.

Parameters: path – Path string composed from valid node names. See Node.node_name and Node2D.node_name for node naming convention.
get_leaf_nodes([decompose=False])

Traverses through the decomposition tree and collects leaf nodes (nodes without any subnodes).

Parameters: decompose – If decompose is True, the method will try to decompose the tree up to the maximum level.
walk(self, func[, args=()[, kwargs={}[, decompose=True]]])

Traverses the decomposition tree and calls func(node, *args, **kwargs) on every node. If func returns True, descending to subnodes will continue.

Parameters: func – callable accepting BaseNode as the first param and optional positional and keyword arguments: func(node, *args, **kwargs)  decompose – If decompose is True (default), the method will also try to decompose the tree up to the maximum level. arguments to pass to the func keyword arguments to pass to the func
walk_depth(self, func[, args=()[, kwargs={}[, decompose=False]]])

Similar to walk() but traverses the tree in depth-first order.

Parameters: func – callable accepting BaseNode as the first param and optional positional and keyword arguments: func(node, *args, **kwargs)  decompose – If decompose is True, the method will also try to decompose the tree up to the maximum level. arguments to pass to the func keyword arguments to pass to the func
##### WaveletPacket and WaveletPacket tree Node¶
class pywt.Node(BaseNode)
class pywt.WaveletPacket(Node)
node_name

Node name describing data coefficients type of the current subnode.

For WaveletPacket case it is just as in dwt():
• a - approximation coefficients
• d - details coefficients
decompose()

class pywt.WaveletPacket(Node)
__init__(data, wavelet[, mode='symmetric'[, maxlevel=None]])
Parameters: data – data associated with the node. 1D numeric array. wavelet – Wavelet to use in the transform. This can be a name of the wavelet from the wavelist() list or a Wavelet object instance. mode – Signal extension mode for the dwt() and idwt() decomposition and reconstruction functions. maxlevel – Maximum allowed level of decomposition. If not specified it will be calculated based on the wavelet and data length using pywt.dwt_max_level().
get_level(level[, order="natural"[, decompose=True]])

Collects nodes from the given level of decomposition.

Parameters: level – Specifies decomposition level from which the nodes will be collected. order – Specifies nodes order - natural (natural) or frequency (freq). decompose – If set then the method will try to decompose the data up to the specified level.

If nodes at the given level are missing (i.e. the tree is partially decomposed) and the decompose is set to False, only existing nodes will be returned.

##### WaveletPacket2D and WaveletPacket2D tree Node2D¶
class pywt.Node2D(BaseNode)
class pywt.WaveletPacket2D(Node2D)
node_name
For WaveletPacket2D case it is just as in dwt2():
• a - approximation coefficients (LL)
• h - horizontal detail coefficients (LH)
• v - vertical detail coefficients (HL)
• d - diagonal detail coefficients (HH)
decompose()

dwt2() for 2D Discrete Wavelet Transform output coefficients.

expand_2d_path(self, path):
class pywt.WaveletPacket2D(Node2D)
__init__(data, wavelet[, mode='symmetric'[, maxlevel=None]])
Parameters: data – data associated with the node. 2D numeric array. wavelet – Wavelet to use in the transform. This can be a name of the wavelet from the wavelist() list or a Wavelet object instance. mode – Signal extension mode for the dwt() and idwt() decomposition and reconstruction functions. maxlevel – Maximum allowed level of decomposition. If not specified it will be calculated based on the wavelet and data length using pywt.dwt_max_level().
get_level(level[, order="natural"[, decompose=True]])

Collects nodes from the given level of decomposition.

Parameters: level – Specifies decomposition level from which the nodes will be collected. order – Specifies nodes order - natural (natural) or frequency (freq). decompose – If set then the method will try to decompose the data up to the specified level.

If nodes at the given level are missing (i.e. the tree is partially decomposed) and the decompose is set to False, only existing nodes will be returned.

#### Continuous Wavelet Transform (CWT)¶

This section describes functions used to perform single continuous wavelet transforms.

##### Single level - cwt¶
pywt.cwt(data, scales, wavelet)

One dimensional Continuous Wavelet Transform.

Parameters: data : array_like Input signal scales : array_like The wavelet scales to use. One can use f = scale2frequency(scale, wavelet)/sampling_period to determine what physical frequency, f. Here, f is in hertz when the sampling_period is given in seconds. wavelet : Wavelet object or name Wavelet to use sampling_period : float Sampling period for the frequencies output (optional). The values computed for coefs are independent of the choice of sampling_period (i.e. scales is not scaled by the sampling period). coefs : array_like Continuous wavelet transform of the input signal for the given scales and wavelet frequencies : array_like If the unit of sampling period are seconds and given, than frequencies are in hertz. Otherwise, a sampling period of 1 is assumed.

Notes

Size of coefficients arrays depends on the length of the input array and the length of given scales.

Examples

>>> import pywt
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> x = np.arange(512)
>>> y = np.sin(2*np.pi*x/32)
>>> coef, freqs=pywt.cwt(y,np.arange(1,129),'gaus1')
>>> plt.matshow(coef)
>>> plt.show()
----------
>>> import pywt
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> t = np.linspace(-1, 1, 200, endpoint=False)
>>> sig  = np.cos(2 * np.pi * 7 * t) + np.real(np.exp(-7*(t-0.4)**2)*np.exp(1j*2*np.pi*2*(t-0.4)))
>>> widths = np.arange(1, 31)
>>> cwtmatr, freqs = pywt.cwt(sig, widths, 'mexh')
>>> plt.imshow(cwtmatr, extent=[-1, 1, 1, 31], cmap='PRGn', aspect='auto',
...            vmax=abs(cwtmatr).max(), vmin=-abs(cwtmatr).max())
>>> plt.show()

##### Continuous Wavelet Families¶

A variety of continuous wavelets have been implemented. A list of the available wavelet names compatible with cwt can be obtained by:

wavlist = pywt.wavelist(kind='continuous')

###### Mexican Hat Wavelet¶

The mexican hat wavelet "mexh" is given by:

$\psi(t) = \frac{2}{\sqrt{3} \sqrt{\pi}} \exp^{-\frac{t^2}{2}} \left( 1 - t^2 \right)$

where the constant out front is a normalization factor so that the wavelet has unit energy.

###### Morlet Wavelet¶

The Morlet wavelet "morl" is given by:

$\psi(t) = \exp^{-\frac{t^2}{2}} \cos(5t)$
###### Complex Morlet Wavelets¶

The complex Morlet wavelet ("cmorB-C" with floating point values B, C) is given by:

$\psi(t) = \frac{1}{\sqrt{\pi B}} \exp^{-\frac{t^2}{B}} \exp^{\mathrm{j} 2\pi C t}$

where $$B$$ is the bandwidth and $$C$$ is the center frequency.

###### Gaussian Derivative Wavelets¶

The Gaussian wavelets ("gausP" where P is an integer between 1 and and 8) correspond to the Pth order derivatives of the function:

$\psi(t) = C \exp^{-t^2}$

where $$C$$ is an order-dependent normalization constant.

###### Complex Gaussian Derivative Wavelets¶

The complex Gaussian wavelets ("cgauP" where P is an integer between 1 and 8) correspond to the Pth order derivatives of the function:

$\psi(t) = C \exp^{-\mathrm{j} t}\exp^{-t^2}$

where $$C$$ is an order-dependent normalization constant.

###### Shannon Wavelets¶

The Shannon wavelets ("shanB-C" with floating point values B and C) correspond to the following wavelets:

$\psi(t) = \sqrt{B} \frac{\sin(\pi B t)}{\pi B t} \exp^{\mathrm{j}2 \pi C t}$

where $$B$$ is the bandwith and $$C$$ is the center frequency.

###### Freuqency B-Spline Wavelets¶

The frequency B-spline wavelets ("fpspM-B-C" with integer M and floating point B, C) correspond to the following wavelets:

$\psi(t) = \sqrt{B} \left[\frac{\sin(\pi B \frac{t}{M})}{\pi B \frac{t}{M}}\right]^M \exp^{2\mathrm{j} \pi C t}$

where $$M$$ is the spline order, $$B$$ is the bandwidth and $$C$$ is the center frequency.

##### Choosing the scales for cwt¶

For each of the wavelets described below, the implementation in PyWavelets evaluates the wavelet function for $$t$$ over the range [wavelet.lower_bound, wavelet.upper_bound] (with default range $$[-8, 8]$$). scale = 1 corresponds to the case where the extent of the wavelet is (wavelet.upper_bound - wavelet.lower_bound + 1) samples of the digital signal being analyzed. Larger scales correspond to stretching of the wavelet. For example, at scale=10 the wavelet is stretched by a factor of 10, making it sensitive to lower frequencies in the signal.

To relate a given scale to a specific signal frequency, the sampling period of the signal must be known. pywt.scale2frequency() can be used to convert a list of scales to their corresponding frequencies. The proper choice of scales depends on the chosen wavelet, so pywt.scale2frequency() should be used to get an idea of an appropriate range for the signal of interest.

For the cmor, fbsp and shan wavelets, the user can specify a specific a normalized center frequency. A value of 1.0 corresponds to 1/dt where dt is the sampling period. In other words, when analyzing a signal sampled at 100 Hz, a center frequency of 1.0 corresponds to ~100 Hz at scale = 1. This is above the Nyquist rate of 50 Hz, so for this particular wavelet, one would analyze a signal using scales >= 2.

>>> import numpy as np
>>> import pywt
>>> dt = 0.01  # 100 Hz sampling
>>> frequencies = pywt.scale2frequency('cmor1.5-1.0', [1, 2, 3, 4]) / dt
>>> frequencies
array([ 100.        ,   50.        ,   33.33333333,   25.        ])


The CWT in PyWavelets is applied to discrete data by convolution with samples of the integral of the wavelet. If scale is too low, this will result in a discrete filter that is inadequately sampled leading to aliasing as shown in the example below. Here the wavelet is 'cmor1.5-1.0'. The left column of the figure shows the discrete filters used in the convolution at various scales. The right column are the corresponding Fourier power spectra of each filter.. For scales 1 and 2 it can be seen that aliasing due to violation of the Nyquist limit occurs.

import numpy as np
import pywt
import matplotlib.pyplot as plt

wav = pywt.ContinuousWavelet('cmor1.5-1.0')

# print the range over which the wavelet will be evaluated
print("Continuous wavelet will be evaluated over the range [{}, {}]".format(
wav.lower_bound, wav.upper_bound))

width = wav.upper_bound - wav.lower_bound

scales = [1, 2, 3, 4, 10, 15]

max_len = int(np.max(scales)*width + 1)
t = np.arange(max_len)
fig, axes = plt.subplots(len(scales), 2, figsize=(12, 6))
for n, scale in enumerate(scales):

# The following code is adapted from the internals of cwt
int_psi, x = pywt.integrate_wavelet(wav, precision=10)
step = x - x
j = np.floor(
np.arange(scale * width + 1) / (scale * step))
if np.max(j) >= np.size(int_psi):
j = np.delete(j, np.where((j >= np.size(int_psi))))
j = j.astype(np.int)

# normalize int_psi for easier plotting
int_psi /= np.abs(int_psi).max()

# discrete samples of the integrated wavelet
filt = int_psi[j][::-1]

# The CWT consists of convolution of filt with the signal at this scale
# Here we plot this discrete convolution kernel at each scale.

nt = len(filt)
t = np.linspace(-nt//2, nt//2, nt)
axes[n, 0].plot(t, filt.real, t, filt.imag)
axes[n, 0].set_xlim([-max_len//2, max_len//2])
axes[n, 0].set_ylim([-1, 1])
axes[n, 0].text(50, 0.35, 'scale = {}'.format(scale))

f = np.linspace(-np.pi, np.pi, max_len)
filt_fft = np.fft.fftshift(np.fft.fft(filt, n=max_len))
filt_fft /= np.abs(filt_fft).max()
axes[n, 1].plot(f, np.abs(filt_fft)**2)
axes[n, 1].set_xlim([-np.pi, np.pi])
axes[n, 1].set_ylim([0, 1])
axes[n, 1].set_xticks([-np.pi, 0, np.pi])
axes[n, 1].set_xticklabels([r'$-\pi$', '0', r'$\pi$'])
axes[n, 1].grid(True, axis='x')
axes[n, 1].text(np.pi/2, 0.5, 'scale = {}'.format(scale))

axes[n, 0].set_xlabel('time (samples)')
axes[0, 0].legend(['real', 'imaginary'], loc='upper left')
axes[0, 1].legend(['Power'], loc='upper left')
axes[0, 0].set_title('filter')
axes[0, 1].set_title(r'|FFT(filter)|$^2$') #### Thresholding functions¶

The thresholding helper module implements the most popular signal thresholding functions.

##### Thresholding¶
pywt.threshold(data, value, mode='soft', substitute=0)

Thresholds the input data depending on the mode argument.

In soft thresholding , data values with absolute value less than param are replaced with substitute. Data values with absolute value greater or equal to the thresholding value are shrunk toward zero by value. In other words, the new value is data/np.abs(data) * np.maximum(np.abs(data) - value, 0).

In hard thresholding, the data values where their absolute value is less than the value param are replaced with substitute. Data values with absolute value greater or equal to the thresholding value stay untouched.

garrote corresponds to the Non-negative garrote threshold , . It is intermediate between hard and soft thresholding. It behaves like soft thresholding for small data values and approaches hard thresholding for large data values.

In greater thresholding, the data is replaced with substitute where data is below the thresholding value. Greater data values pass untouched.

In less thresholding, the data is replaced with substitute where data is above the thresholding value. Lesser data values pass untouched.

Both hard and soft thresholding also support complex-valued data.

Parameters: data : array_like Numeric data. value : scalar Thresholding value. mode : {‘soft’, ‘hard’, ‘garrote’, ‘greater’, ‘less’} Decides the type of thresholding to be applied on input data. Default is ‘soft’. substitute : float, optional Substitute value (default: 0). output : array Thresholded array.

References

  (1, 2) D.L. Donoho and I.M. Johnstone. Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika. Vol. 81, No. 3, pp.425-455, 1994. DOI:10.1093/biomet/81.3.425
  (1, 2) L. Breiman. Better Subset Regression Using the Nonnegative Garrote. Technometrics, Vol. 37, pp. 373-384, 1995. DOI:10.2307/1269730
  (1, 2) H-Y. Gao. Wavelet Shrinkage Denoising Using the Non-Negative Garrote. Journal of Computational and Graphical Statistics Vol. 7, No. 4, pp.469-488. 1998. DOI:10.1080/10618600.1998.10474789

Examples

>>> import numpy as np
>>> import pywt
>>> data = np.linspace(1, 4, 7)
>>> data
array([ 1. ,  1.5,  2. ,  2.5,  3. ,  3.5,  4. ])
>>> pywt.threshold(data, 2, 'soft')
array([ 0. ,  0. ,  0. ,  0.5,  1. ,  1.5,  2. ])
>>> pywt.threshold(data, 2, 'hard')
array([ 0. ,  0. ,  2. ,  2.5,  3. ,  3.5,  4. ])
>>> pywt.threshold(data, 2, 'garrote')
array([ 0.        ,  0.        ,  0.        ,  0.9       ,  1.66666667,
2.35714286,  3.        ])
>>> pywt.threshold(data, 2, 'greater')
array([ 0. ,  0. ,  2. ,  2.5,  3. ,  3.5,  4. ])
>>> pywt.threshold(data, 2, 'less')
array([ 1. ,  1.5,  2. ,  0. ,  0. ,  0. ,  0. ])

pywt.threshold_firm(data, value_low, value_high)

Firm threshold.

The approach is intermediate between soft and hard thresholding . It behaves the same as soft-thresholding for values below value_low and the same as hard-thresholding for values above thresh_high. For intermediate values, the thresholded value is in between that corresponding to soft or hard thresholding.

Parameters: data : array-like The data to threshold. This can be either real or complex-valued. value_low : float Any values smaller then value_low will be set to zero. value_high : float Any values larger than value_high will not be modified. val_new : array-like The values after firm thresholding at the specified thresholds.

Notes

This thresholding technique is also known as semi-soft thresholding .

For each value, x, in data. This function computes:

if np.abs(x) <= value_low:
return 0
elif np.abs(x) > value_high:
return x
elif value_low < np.abs(x) and np.abs(x) <= value_high:
return x * value_high * (1 - value_low/x)/(value_high - value_low)


firm is a continuous function (like soft thresholding), but is unbiased for large values (like hard thresholding).

If value_high == value_low this function becomes hard-thresholding. If value_high is infinity, this function becomes soft-thresholding.

References

  (1, 2) H.-Y. Gao and A.G. Bruce. Waveshrink with firm shrinkage. Statistica Sinica, Vol. 7, pp. 855-874, 1997.
  (1, 2) A. Bruce and H-Y. Gao. WaveShrink: Shrinkage Functions and Thresholds. Proc. SPIE 2569, Wavelet Applications in Signal and Image Processing III, 1995. DOI:10.1117/12.217582

The left panel of the figure below illustrates that non-negative Garotte thresholding is intermediate between soft and hard thresholding. Firm thresholding transitions between soft and hard thresholding behavior. It requires a pair of threshold values that define the width of the transition region.

import numpy as np
import matplotlib.pyplot as plt
import pywt

s = np.linspace(-4, 4, 1000)

s_soft = pywt.threshold(s, value=0.5, mode='soft')
s_hard = pywt.threshold(s, value=0.5, mode='hard')
s_garrote = pywt.threshold(s, value=0.5, mode='garrote')
s_firm1 = pywt.threshold_firm(s, value_low=0.5, value_high=1)
s_firm2 = pywt.threshold_firm(s, value_low=0.5, value_high=2)
s_firm3 = pywt.threshold_firm(s, value_low=0.5, value_high=4)

fig, ax = plt.subplots(1, 2, figsize=(10, 4))
ax.plot(s, s_soft)
ax.plot(s, s_hard)
ax.plot(s, s_garrote)
ax.legend(['soft (0.5)', 'hard (0.5)', 'non-neg. garrote (0.5)'])
ax.set_xlabel('input value')
ax.set_ylabel('thresholded value')

ax.plot(s, s_soft)
ax.plot(s, s_hard)
ax.plot(s, s_firm1)
ax.plot(s, s_firm2)
ax.plot(s, s_firm3)
ax.legend(['soft (0.5)', 'hard (0.5)', 'firm(0.5, 1)', 'firm(0.5, 2)',
'firm(0.5, 4)'])
ax.set_xlabel('input value')
ax.set_ylabel('thresholded value')
plt.show() #### Other functions¶

##### Integrating wavelet functions¶
pywt.integrate_wavelet(wavelet, precision=8)

Integrate psi wavelet function from -Inf to x using the rectangle integration method.

Parameters: wavelet : Wavelet instance or str Wavelet to integrate. If a string, should be the name of a wavelet. precision : int, optional Precision that will be used for wavelet function approximation computed with the wavefun(level=precision) Wavelet’s method (default: 8). [int_psi, x] : for orthogonal wavelets [int_psi_d, int_psi_r, x] : for other wavelets

Examples

>>> from pywt import Wavelet, integrate_wavelet
>>> wavelet1 = Wavelet('db2')
>>> [int_psi, x] = integrate_wavelet(wavelet1, precision=5)
>>> wavelet2 = Wavelet('bior1.3')
>>> [int_psi_d, int_psi_r, x] = integrate_wavelet(wavelet2, precision=5)


The result of the call depends on the wavelet argument:

• for orthogonal and continuous wavelets - an integral of the wavelet function specified on an x-grid:

[int_psi, x_grid] = integrate_wavelet(wavelet, precision)

• for other wavelets - integrals of decomposition and reconstruction wavelet functions and a corresponding x-grid:

[int_psi_d, int_psi_r, x_grid] = integrate_wavelet(wavelet, precision)

##### Central frequency of psi wavelet function¶
pywt.central_frequency(wavelet, precision=8)

Computes the central frequency of the psi wavelet function.

Parameters: wavelet : Wavelet instance, str or tuple Wavelet to integrate. If a string, should be the name of a wavelet. precision : int, optional Precision that will be used for wavelet function approximation computed with the wavefun(level=precision) Wavelet’s method (default: 8). scalar
pywt.scale2frequency(wavelet, scale, precision=8)
Parameters: wavelet : Wavelet instance or str Wavelet to integrate. If a string, should be the name of a wavelet. scale : scalar precision : int, optional Precision that will be used for wavelet function approximation computed with wavelet.wavefun(level=precision). Default is 8. freq : scalar
pywt.qmf(filt)

The magnitude response of QMF is mirror image about pi/2 of that of the input filter.

Parameters: filt : array_like Input filter for which QMF needs to be computed. qm_filter : ndarray Quadrature mirror of the input filter.
##### Orthogonal Filter Banks¶
pywt.orthogonal_filter_bank(scaling_filter)

Returns the orthogonal filter bank.

The orthogonal filter bank consists of the HPFs and LPFs at decomposition and reconstruction stage for the input scaling filter.

Parameters: scaling_filter : array_like Input scaling filter (father wavelet). orth_filt_bank : tuple of 4 ndarrays The orthogonal filter bank of the input scaling filter in the order : 1] Decomposition LPF 2] Decomposition HPF 3] Reconstruction LPF 4] Reconstruction HPF
##### Example Datasets¶

The following example datasets are available in the module pywt.data:

name description
ecg ECG waveform (1024 samples)
aero grayscale image (512x512)
ascent grayscale image (512x512)
camera grayscale image (512x512)
nino sea surface temperature (264 samples)
demo_signal various synthetic 1d test signals

Each can be loaded via a function of the same name.

pywt.data.demo_signal(name='Bumps', n=None)

Simple 1D wavelet test functions.

This function can generate a number of common 1D test signals used in papers by David Donoho and colleagues (e.g. ) as well as the wavelet book by Stéphane Mallat .

Parameters: name : {‘Blocks’, ‘Bumps’, ‘HeaviSine’, ‘Doppler’, …} The type of test signal to generate (name is case-insensitive). If name is set to ‘list’, a list of the avialable test functions is returned. n : int or None The length of the test signal. This should be provided for all test signals except ‘Gabor’ and ‘sineoneoverx’ which have a fixed length. f : np.ndarray Array of length n corresponding to the specified test signal type.

Notes

This function is a partial reimplementation of the MakeSignal function from the [Wavelab](https://statweb.stanford.edu/~wavelab/) toolbox. These test signals are provided with permission of Dr. Donoho to encourage reproducible research.

References

  (1, 2) D.L. Donoho and I.M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika, vol. 81, pp. 425–455, 1994.
  (1, 2) S. Mallat. A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press. 2009.

Example:

>>> import pywt
>>> camera = pywt.data.camera()
>>> doppler = pywt.data.demo_signal('doppler')
>>> available_signals = pywt.data.demo_signal('list')


### Usage examples¶

The following examples are used as doctest regression tests written using reST markup. They are included in the documentation since they contain various useful examples illustrating how to use and how not to use PyWavelets.

For more usage examples see the demo directory in the source package.

#### The Wavelet object¶

##### Wavelet families and builtin Wavelets names¶

Wavelet objects are really a handy carriers of a bunch of DWT-specific data like quadrature mirror filters and some general properties associated with them.

At first let’s go through the methods of creating a Wavelet object. The easiest and the most convenient way is to use builtin named Wavelets.

These wavelets are organized into groups called wavelet families. The most commonly used families are:

>>> import pywt
>>> pywt.families()
['haar', 'db', 'sym', 'coif', 'bior', 'rbio', 'dmey', 'gaus', 'mexh', 'morl', 'cgau', 'shan', 'fbsp', 'cmor']


The wavelist() function with family name passed as an argument is used to obtain the list of wavelet names in each family.

>>> for family in pywt.families():
...     print("%s family: " % family + ', '.join(pywt.wavelist(family)))
haar family: haar
db family: db1, db2, db3, db4, db5, db6, db7, db8, db9, db10, db11, db12, db13, db14, db15, db16, db17, db18, db19, db20, db21, db22, db23, db24, db25, db26, db27, db28, db29, db30, db31, db32, db33, db34, db35, db36, db37, db38
sym family: sym2, sym3, sym4, sym5, sym6, sym7, sym8, sym9, sym10, sym11, sym12, sym13, sym14, sym15, sym16, sym17, sym18, sym19, sym20
coif family: coif1, coif2, coif3, coif4, coif5, coif6, coif7, coif8, coif9, coif10, coif11, coif12, coif13, coif14, coif15, coif16, coif17
bior family: bior1.1, bior1.3, bior1.5, bior2.2, bior2.4, bior2.6, bior2.8, bior3.1, bior3.3, bior3.5, bior3.7, bior3.9, bior4.4, bior5.5, bior6.8
rbio family: rbio1.1, rbio1.3, rbio1.5, rbio2.2, rbio2.4, rbio2.6, rbio2.8, rbio3.1, rbio3.3, rbio3.5, rbio3.7, rbio3.9, rbio4.4, rbio5.5, rbio6.8
dmey family: dmey
gaus family: gaus1, gaus2, gaus3, gaus4, gaus5, gaus6, gaus7, gaus8
mexh family: mexh
morl family: morl
cgau family: cgau1, cgau2, cgau3, cgau4, cgau5, cgau6, cgau7, cgau8
shan family: shan
fbsp family: fbsp
cmor family: cmor


To get the full list of builtin wavelets’ names just use the wavelist() with no argument.

##### Creating Wavelet objects¶

Now when we know all the names let’s finally create a Wavelet object:

>>> w = pywt.Wavelet('db3')


So.. that’s it.

##### Wavelet properties¶

But what can we do with Wavelet objects? Well, they carry some interesting information.

First, let’s try printing a Wavelet object. This shows a brief information about its name, its family name and some properties like orthogonality and symmetry.

>>> print(w)
Wavelet db3
Family name:    Daubechies
Short name:     db
Filters length: 6
Orthogonal:     True
Biorthogonal:   True
Symmetry:       asymmetric
DWT:            True
CWT:            False


But the most important information are the wavelet filters coefficients, which are used in Discrete Wavelet Transform. These coefficients can be obtained via the dec_lo, Wavelet.dec_hi, rec_lo and rec_hi attributes, which corresponds to lowpass and highpass decomposition filters and lowpass and highpass reconstruction filters respectively:

>>> def print_array(arr):
...     print("[%s]" % ", ".join(["%.14f" % x for x in arr]))


Another way to get the filters data is to use the filter_bank attribute, which returns all four filters in a tuple:

>>> w.filter_bank == (w.dec_lo, w.dec_hi, w.rec_lo, w.rec_hi)
True


Other Wavelet’s properties are:

>>> print(w.name)
db3
>>> print(w.short_family_name)
db
>>> print(w.family_name)
Daubechies


Now when we know a bit about the builtin Wavelets, let’s see how to create custom Wavelets objects. These can be done in two ways:

1. Passing the filter bank object that implements the filter_bank attribute. The attribute must return four filters coefficients.

>>> class MyHaarFilterBank(object):
...     @property
...     def filter_bank(self):
...         from math import sqrt
...         return ([sqrt(2)/2, sqrt(2)/2], [-sqrt(2)/2, sqrt(2)/2],
...                 [sqrt(2)/2, sqrt(2)/2], [sqrt(2)/2, -sqrt(2)/2])

>>> my_wavelet = pywt.Wavelet('My Haar Wavelet', filter_bank=MyHaarFilterBank())

2. Passing the filters coefficients directly as the filter_bank parameter.

>>> from math import sqrt
>>> my_filter_bank = ([sqrt(2)/2, sqrt(2)/2], [-sqrt(2)/2, sqrt(2)/2],
...                   [sqrt(2)/2, sqrt(2)/2], [sqrt(2)/2, -sqrt(2)/2])
>>> my_wavelet = pywt.Wavelet('My Haar Wavelet', filter_bank=my_filter_bank)


Note that such custom wavelets will not have all the properties set to correct values:

>>> print(my_wavelet)
Wavelet My Haar Wavelet
Family name:
Short name:
Filters length: 2
Orthogonal:     False
Biorthogonal:   False
Symmetry:       unknown
DWT:            True
CWT:            False


You can however set a couple of them on your own:

>>> my_wavelet.orthogonal = True
>>> my_wavelet.biorthogonal = True

>>> print(my_wavelet)
Wavelet My Haar Wavelet
Family name:
Short name:
Filters length: 2
Orthogonal:     True
Biorthogonal:   True
Symmetry:       unknown
DWT:            True
CWT:            False

##### And now… the wavefun!¶

We all know that the fun with wavelets is in wavelet functions. Now what would be this package without a tool to compute wavelet and scaling functions approximations?

This is the purpose of the wavefun() method, which is used to approximate scaling function (phi) and wavelet function (psi) at the given level of refinement, based on the filters coefficients.

The number of returned values varies depending on the wavelet’s orthogonality property. For orthogonal wavelets the result is tuple with scaling function, wavelet function and xgrid coordinates.

>>> w = pywt.Wavelet('sym3')
>>> w.orthogonal
True
>>> (phi, psi, x) = w.wavefun(level=5)


For biorthogonal (non-orthogonal) wavelets different scaling and wavelet functions are used for decomposition and reconstruction, and thus five elements are returned: decomposition scaling and wavelet functions approximations, reconstruction scaling and wavelet functions approximations, and the xgrid.

>>> w = pywt.Wavelet('bior1.3')
>>> w.orthogonal
False
>>> (phi_d, psi_d, phi_r, psi_r, x) = w.wavefun(level=5)


You can find live examples of wavefun() usage and images of all the built-in wavelets on the Wavelet Properties Browser page. However, this website is no longer actively maintained and does not include every wavelet present in PyWavelets. The precision of the wavelet coefficients at that site is also lower than those included in PyWavelets.

#### Signal Extension Modes¶

Import pywt first

>>> import pywt

>>> def format_array(a):
...     """Consistent array representation across different systems"""
...     import numpy
...     a = numpy.where(numpy.abs(a) < 1e-5, 0, a)
...     return numpy.array2string(a, precision=5, separator=' ', suppress_small=True)


List of available signal extension modes:

>>> print(pywt.Modes.modes)
['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect']


Invalid mode name should rise a ValueError:

>>> pywt.dwt([1,2,3,4], 'db2', 'invalid')
Traceback (most recent call last):
...
ValueError: Unknown mode name 'invalid'.


You can also refer to modes via Modes class attributes:

>>> x = [1, 2, 1, 5, -1, 8, 4, 6]
>>> for mode_name in ['zero', 'constant', 'symmetric', 'reflect', 'periodic', 'smooth', 'periodization']:
...     mode = getattr(pywt.Modes, mode_name)
...     cA, cD = pywt.dwt(x, 'db2', mode)
...     print("Mode: %d (%s)" % (mode, mode_name))
Mode: 0 (zero)
Mode: 2 (constant)
Mode: 1 (symmetric)
Mode: 6 (reflect)
Mode: 4 (periodic)
Mode: 3 (smooth)
Mode: 5 (periodization)


The default mode is symmetric:

>>> cA, cD = pywt.dwt(x, 'db2')
>>> print(cA)
[ 1.76776695  1.73309178  3.40612438  6.32928585  7.77817459]
>>> print(cD)
[-0.61237244 -2.15599552 -5.95034847 -1.21545369  1.22474487]
>>> print(pywt.idwt(cA, cD, 'db2'))
[ 1.  2.  1.  5. -1.  8.  4.  6.]


And using a keyword argument:

>>> cA, cD = pywt.dwt(x, 'db2', mode='symmetric')
>>> print(cA)
[ 1.76776695  1.73309178  3.40612438  6.32928585  7.77817459]
>>> print(cD)
[-0.61237244 -2.15599552 -5.95034847 -1.21545369  1.22474487]
>>> print(pywt.idwt(cA, cD, 'db2'))
[ 1.  2.  1.  5. -1.  8.  4.  6.]


#### DWT and IDWT¶

##### Discrete Wavelet Transform¶

Let’s do a Discrete Wavelet Transform of a sample data x using the db2 wavelet. It’s simple..

>>> import pywt
>>> x = [3, 7, 1, 1, -2, 5, 4, 6]
>>> cA, cD = pywt.dwt(x, 'db2')


And the approximation and details coefficients are in cA and cD respectively:

>>> print(cA)
[ 5.65685425  7.39923721  0.22414387  3.33677403  7.77817459]
>>> print(cD)
[-2.44948974 -1.60368225 -4.44140056 -0.41361256  1.22474487]

##### Inverse Discrete Wavelet Transform¶

Now let’s do an opposite operation - Inverse Discrete Wavelet Transform:

>>> print(pywt.idwt(cA, cD, 'db2'))
[ 3.  7.  1.  1. -2.  5.  4.  6.]


Voilà! That’s it!

##### More Examples¶

Now let’s experiment with the dwt() some more. For example let’s pass a Wavelet object instead of the wavelet name and specify signal extension mode (the default is symmetric) for the border effect handling:

>>> w = pywt.Wavelet('sym3')
>>> cA, cD = pywt.dwt(x, wavelet=w, mode='constant')
>>> print(cA)
[ 4.38354585  3.80302657  7.31813271 -0.58565539  4.09727044  7.81994027]
>>> print(cD)
[-1.33068221 -2.78795192 -3.16825651 -0.67715519 -0.09722957 -0.07045258]


Note that the output coefficients arrays length depends not only on the input data length but also on the :class:Wavelet type (particularly on its filters length that are used in the transformation).

To find out what will be the output data size use the dwt_coeff_len() function:

>>> # int() is for normalizing Python integers and long integers for documentation tests
>>> int(pywt.dwt_coeff_len(data_len=len(x), filter_len=w.dec_len, mode='symmetric'))
6
>>> int(pywt.dwt_coeff_len(len(x), w, 'symmetric'))
6
>>> len(cA)
6


Looks fine. (And if you expected that the output length would be a half of the input data length, well, that’s the trade-off that allows for the perfect reconstruction…).

The third argument of the dwt_coeff_len() is the already mentioned signal extension mode (please refer to the PyWavelets’ documentation for the modes description). Currently there are six extension modes available:

>>> pywt.Modes.modes
['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect']


As you see in the above example, the periodization (periodization) mode is slightly different from the others. It’s aim when doing the DWT transform is to output coefficients arrays that are half of the length of the input data.

Knowing that, you should never mix the periodization mode with other modes when doing DWT and IDWT. Otherwise, it will produce invalid results:

>>> x
[3, 7, 1, 1, -2, 5, 4, 6]
>>> cA, cD = pywt.dwt(x, wavelet=w, mode='periodization')
>>> print(pywt.idwt(cA, cD, 'sym3', 'symmetric')) # invalid mode
[ 1.  1. -2.  5.]
>>> print(pywt.idwt(cA, cD, 'sym3', 'periodization'))
[ 3.  7.  1.  1. -2.  5.  4.  6.]

##### Tips & tricks¶
###### Passing None instead of coefficients data to idwt()¶

Now some tips & tricks. Passing None as one of the coefficient arrays parameters is similar to passing a zero-filled array. The results are simply the same:

>>> print(pywt.idwt([1,2,0,1], None, 'db2', 'symmetric'))
[ 1.19006969  1.54362308  0.44828774 -0.25881905  0.48296291  0.8365163 ]

>>> print(pywt.idwt([1, 2, 0, 1], [0, 0, 0, 0], 'db2', 'symmetric'))
[ 1.19006969  1.54362308  0.44828774 -0.25881905  0.48296291  0.8365163 ]

>>> print(pywt.idwt(None, [1, 2, 0, 1], 'db2', 'symmetric'))
[ 0.57769726 -0.93125065  1.67303261 -0.96592583 -0.12940952 -0.22414387]

>>> print(pywt.idwt([0, 0, 0, 0], [1, 2, 0, 1], 'db2', 'symmetric'))
[ 0.57769726 -0.93125065  1.67303261 -0.96592583 -0.12940952 -0.22414387]


Remember that only one argument at a time can be None:

>>> print(pywt.idwt(None, None, 'db2', 'symmetric'))
Traceback (most recent call last):
...
ValueError: At least one coefficient parameter must be specified.

###### Coefficients data size in idwt¶

When doing the IDWT transform, usually the coefficient arrays must have the same size.

>>> print(pywt.idwt([1, 2, 3, 4, 5], [1, 2, 3, 4], 'db2', 'symmetric'))
Traceback (most recent call last):
...
ValueError: Coefficients arrays must have the same size.


Not every coefficient array can be used in IDWT. In the following example the idwt() will fail because the input arrays are invalid - they couldn’t be created as a result of DWT, because the minimal output length for dwt using db4 wavelet and the symmetric mode is 4, not 3:

>>> pywt.idwt([1,2,4], [4,1,3], 'db4', 'symmetric')
Traceback (most recent call last):
...
ValueError: Invalid coefficient arrays length for specified wavelet. Wavelet and mode must be the same as used for decomposition.

>>> int(pywt.dwt_coeff_len(1, pywt.Wavelet('db4').dec_len, 'symmetric'))
4


#### Multilevel DWT, IDWT and SWT¶

##### Multilevel DWT decomposition¶
>>> import pywt
>>> x = [3, 7, 1, 1, -2, 5, 4, 6]
>>> db1 = pywt.Wavelet('db1')
>>> cA3, cD3, cD2, cD1 = pywt.wavedec(x, db1)
>>> print(cA3)
[ 8.83883476]
>>> print(cD3)
[-0.35355339]
>>> print(cD2)
[ 4.  -3.5]
>>> print(cD1)
[-2.82842712  0.         -4.94974747 -1.41421356]

>>> pywt.dwt_max_level(len(x), db1)
3

>>> cA2, cD2, cD1 = pywt.wavedec(x, db1, mode='constant', level=2)

##### Multilevel IDWT reconstruction¶
>>> coeffs = pywt.wavedec(x, db1)
>>> print(pywt.waverec(coeffs, db1))
[ 3.  7.  1.  1. -2.  5.  4.  6.]

##### Multilevel SWT decomposition¶
>>> x = [3, 7, 1, 3, -2, 6, 4, 6]
>>> (cA2, cD2), (cA1, cD1) = pywt.swt(x, db1, level=2)
>>> print(cA1)
[ 7.07106781  5.65685425  2.82842712  0.70710678  2.82842712  7.07106781
7.07106781  6.36396103]
>>> print(cD1)
[-2.82842712  4.24264069 -1.41421356  3.53553391 -5.65685425  1.41421356
-1.41421356  2.12132034]
>>> print(cA2)
[  7.    4.5   4.    5.5   7.    9.5  10.    8.5]
>>> print(cD2)
[ 3.   3.5  0.  -4.5 -3.   0.5  0.   0.5]

>>> [(cA2, cD2)] = pywt.swt(cA1, db1, level=1, start_level=1)
>>> print(cA2)
[  7.    4.5   4.    5.5   7.    9.5  10.    8.5]
>>> print(cD2)
[ 3.   3.5  0.  -4.5 -3.   0.5  0.   0.5]

>>> coeffs = pywt.swt(x, db1)
>>> len(coeffs)
3
>>> pywt.swt_max_level(len(x))
3

>>> from __future__ import print_function


#### Wavelet Packets¶

##### Import pywt¶
>>> import pywt

>>> def format_array(a):
...     """Consistent array representation across different systems"""
...     import numpy
...     a = numpy.where(numpy.abs(a) < 1e-5, 0, a)
...     return numpy.array2string(a, precision=5, separator=' ', suppress_small=True)

##### Create Wavelet Packet structure¶

Ok, let’s create a sample WaveletPacket:

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')


The input data and decomposition coefficients are stored in the WaveletPacket.data attribute:

>>> print(wp.data)
[1, 2, 3, 4, 5, 6, 7, 8]


Nodes are identified by paths. For the root node the path is '' and the decomposition level is 0.

>>> print(repr(wp.path))
''
>>> print(wp.level)
0


The maxlevel, if not given as param in the constructor, is automatically computed:

>>> print(wp['ad'].maxlevel)
3

##### Traversing WP tree:¶
###### Accessing subnodes:¶
>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')


First check what is the maximum level of decomposition:

>>> print(wp.maxlevel)
3


and try accessing subnodes of the WP tree:

• 1st level:

>>> print(wp['a'].data)
[  2.12132034   4.94974747   7.77817459  10.60660172]
>>> print(wp['a'].path)
a

• 2nd level:

>>> print(wp['aa'].data)
[  5.  13.]
>>> print(wp['aa'].path)
aa

• 3rd level:

>>> print(wp['aaa'].data)
[ 12.72792206]
>>> print(wp['aaa'].path)
aaa


Ups, we have reached the maximum level of decomposition and got an IndexError:

>>> print(wp['aaaa'].data)
Traceback (most recent call last):
...
IndexError: Path length is out of range.


Now try some invalid path:

>>> print(wp['ac'])
Traceback (most recent call last):
...
ValueError: Subnode name must be in ['a', 'd'], not 'c'.


which just yielded a ValueError.

###### Accessing Node’s attributes:¶

WaveletPacket object is a tree data structure, which evaluates to a set of Node objects. WaveletPacket is just a special subclass of the Node class (which in turn inherits from the BaseNode).

Tree nodes can be accessed using the obj[x] (Node.__getitem__()) operator. Each tree node has a set of attributes: data, path, node_name, parent, level, maxlevel and mode.

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

>>> print(wp['ad'].data)
[-2. -2.]

>>> print(wp['ad'].path)

>>> print(wp['ad'].node_name)
d

>>> print(wp['ad'].parent.path)
a

>>> print(wp['ad'].level)
2

>>> print(wp['ad'].maxlevel)
3

>>> print(wp['ad'].mode)
symmetric

###### Collecting nodes¶
>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')


We can get all nodes on the particular level either in natural order:

>>> print([node.path for node in wp.get_level(3, 'natural')])


or sorted based on the band frequency (freq):

>>> print([node.path for node in wp.get_level(3, 'freq')])


Note that WaveletPacket.get_level() also performs automatic decomposition until it reaches the specified level.

##### Reconstructing data from Wavelet Packets:¶
>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')


Now create a new Wavelet Packet and set its nodes with some data.

>>> new_wp = pywt.WaveletPacket(data=None, wavelet='db1', mode='symmetric')

>>> new_wp['aa'] = wp['aa'].data


For convenience, Node.data gets automatically extracted from the Node object:

>>> new_wp['d'] = wp['d']


And reconstruct the data from the aa, ad and d packets.

>>> print(new_wp.reconstruct(update=False))
[ 1.  2.  3.  4.  5.  6.  7.  8.]


If the update param in the reconstruct method is set to False, the node’s data will not be updated.

>>> print(new_wp.data)
None


Otherwise, the data attribute will be set to the reconstructed value.

>>> print(new_wp.reconstruct(update=True))
[ 1.  2.  3.  4.  5.  6.  7.  8.]
>>> print(new_wp.data)
[ 1.  2.  3.  4.  5.  6.  7.  8.]

>>> print([n.path for n in new_wp.get_leaf_nodes(False)])

>>> print([n.path for n in new_wp.get_leaf_nodes(True)])

##### Removing nodes from Wavelet Packet tree:¶

Let’s create a sample data:

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')


First, start with a tree decomposition at level 2. Leaf nodes in the tree are:

>>> dummy = wp.get_level(2)
>>> for n in wp.get_leaf_nodes(False):
...     print(n.path, format_array(n.data))
aa [  5.  13.]
da [-1. -1.]
dd [ 0.  0.]

>>> node = wp['ad']
>>> print(node)


To remove a node from the WP tree, use Python’s del obj[x] (Node.__delitem__):

>>> del wp['ad']


The leaf nodes that left in the tree are:

>>> for n in wp.get_leaf_nodes():
...     print(n.path, format_array(n.data))
aa [  5.  13.]
da [-1. -1.]
dd [ 0.  0.]


And the reconstruction is:

>>> print(wp.reconstruct())
[ 2.  3.  2.  3.  6.  7.  6.  7.]


Now restore the deleted node value.

>>> wp['ad'].data = node.data


Printing leaf nodes and tree reconstruction confirms the original state of the tree:

>>> for n in wp.get_leaf_nodes(False):
...     print(n.path, format_array(n.data))
aa [  5.  13.]
da [-1. -1.]
dd [ 0.  0.]

>>> print(wp.reconstruct())
[ 1.  2.  3.  4.  5.  6.  7.  8.]

##### Lazy evaluation:¶

Note

This section is for demonstration of pywt internals purposes only. Do not rely on the attribute access to nodes as presented in this example.

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

1. At first the wp’s attribute a is None

>>> print(wp.a)
None


Remember that you should not rely on the attribute access.

2. At first attempt to access the node it is computed via decomposition of its parent node (the wp object itself).

>>> print(wp['a'])
a: [  2.12132034   4.94974747   7.77817459  10.60660172]

3. Now the wp.a is set to the newly created node:

>>> print(wp.a)
a: [  2.12132034   4.94974747   7.77817459  10.60660172]


And so is wp.d:

>>> print(wp.d)
d: [-0.70710678 -0.70710678 -0.70710678 -0.70710678]


#### 2D Wavelet Packets¶

##### Import pywt¶
>>> from __future__ import print_function
>>> import pywt
>>> import numpy

##### Create 2D Wavelet Packet structure¶

>>> x = numpy.array([[1, 2, 3, 4, 5, 6, 7, 8]] * 8, 'd')
>>> print(x)
[[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]]


Now create a 2D Wavelet Packet object:

>>> wp = pywt.WaveletPacket2D(data=x, wavelet='db1', mode='symmetric')


The input data and decomposition coefficients are stored in the WaveletPacket2D.data attribute:

>>> print(wp.data)
[[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]]


Nodes are identified by paths. For the root node the path is '' and the decomposition level is 0.

>>> print(repr(wp.path))
''
>>> print(wp.level)
0


The WaveletPacket2D.maxlevel, if not given in the constructor, is automatically computed based on the data size:

>>> print(wp.maxlevel)
3

##### Traversing WP tree:¶

Wavelet Packet nodes are arranged in a tree. Each node in a WP tree is uniquely identified and addressed by a path string.

In the 1D WaveletPacket case nodes were accessed using 'a' (approximation) and 'd' (details) path names (each node has two 1D children).

Because now we deal with a bit more complex structure (each node has four children), we have four basic path names based on the dwt 2D output convention to address the WP2D structure:

• a - LL, low-low coefficients
• h - LH, low-high coefficients
• v - HL, high-low coefficients
• d - HH, high-high coefficients

In other words, subnode naming corresponds to the dwt2() function output naming convention (as wavelet packet transform is based on the dwt2 transform):

                            -------------------
|        |        |
| cA(LL) | cH(LH) |
|        |        |
(cA, (cH, cV, cD))  <--->   -------------------
|        |        |
| cV(HL) | cD(HH) |
|        |        |
-------------------

(fig.1: DWT 2D output and interpretation)


Knowing what the nodes names are, we can now access them using the indexing operator obj[x] (WaveletPacket2D.__getitem__()):

>>> print(wp['a'].data)
[[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]]
>>> print(wp['h'].data)
[[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]]
>>> print(wp['v'].data)
[[-1. -1. -1. -1.]
[-1. -1. -1. -1.]
[-1. -1. -1. -1.]
[-1. -1. -1. -1.]]
>>> print(wp['d'].data)
[[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]]


Similarly, a subnode of a subnode can be accessed by:

>>> print(wp['aa'].data)
[[ 10.  26.]
[ 10.  26.]]


Indexing base WaveletPacket2D (as well as 1D WaveletPacket) using compound path is just the same as indexing WP subnode:

>>> node = wp['a']
>>> print(node['a'].data)
[[ 10.  26.]
[ 10.  26.]]
>>> print(wp['a']['a'].data is wp['aa'].data)
True


Following down the decomposition path:

>>> print(wp['aaa'].data)
[[ 36.]]
>>> print(wp['aaaa'].data)
Traceback (most recent call last):
...
IndexError: Path length is out of range.


Ups, we have reached the maximum level of decomposition for the 'aaaa' path, which btw. was:

>>> print(wp.maxlevel)
3


Now try some invalid path:

>>> print(wp['f'])
Traceback (most recent call last):
...
ValueError: Subnode name must be in ['a', 'h', 'v', 'd'], not 'f'.

###### Accessing Node2D’s attributes:¶

WaveletPacket2D is a tree data structure, which evaluates to a set of Node2D objects. WaveletPacket2D is just a special subclass of the Node2D class (which in turn inherits from a BaseNode, just like with Node and WaveletPacket for the 1D case.).

>>> print(wp['av'].data)
[[-4. -4.]
[-4. -4.]]

>>> print(wp['av'].path)
av

>>> print(wp['av'].node_name)
v

>>> print(wp['av'].parent.path)
a

>>> print(wp['av'].parent.data)
[[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]]

>>> print(wp['av'].level)
2

>>> print(wp['av'].maxlevel)
3

>>> print(wp['av'].mode)
symmetric

###### Collecting nodes¶

We can get all nodes on the particular level using the WaveletPacket2D.get_level() method:

• 0 level - the root wp node:

>>> len(wp.get_level(0))
1
>>> print([node.path for node in wp.get_level(0)])
['']

• 1st level of decomposition:

>>> len(wp.get_level(1))
4
>>> print([node.path for node in wp.get_level(1)])
['a', 'h', 'v', 'd']

• 2nd level of decomposition:

>>> len(wp.get_level(2))
16
>>> paths = [node.path for node in wp.get_level(2)]
>>> for i, path in enumerate(paths):
...     if (i+1) % 4 == 0:
...         print(path)
...     else:
...         print(path, end=' ')
ha hh hv hd
va vh vv vd
da dh dv dd

• 3rd level of decomposition:

>>> print(len(wp.get_level(3)))
64
>>> paths = [node.path for node in wp.get_level(3)]
>>> for i, path in enumerate(paths):
...     if (i+1) % 8 == 0:
...         print(path)
...     else:
...         print(path, end=' ')
aaa aah aav aad aha ahh ahv ahd
haa hah hav had hha hhh hhv hhd
hva hvh hvv hvd hda hdh hdv hdd
vaa vah vav vad vha vhh vhv vhd
vva vvh vvv vvd vda vdh vdv vdd
daa dah dav dad dha dhh dhv dhd
dva dvh dvv dvd dda ddh ddv ddd


Note that WaveletPacket2D.get_level() performs automatic decomposition until it reaches the given level.

##### Reconstructing data from Wavelet Packets:¶

Let’s create a new empty 2D Wavelet Packet structure and set its nodes values with known data from the previous examples:

>>> new_wp = pywt.WaveletPacket2D(data=None, wavelet='db1', mode='symmetric')

>>> new_wp['vh'] = wp['vh'].data # [[0.0, 0.0], [0.0, 0.0]]
>>> new_wp['vv'] = wp['vh'].data # [[0.0, 0.0], [0.0, 0.0]]
>>> new_wp['vd'] = [[0.0, 0.0], [0.0, 0.0]]

>>> new_wp['a'] = [[3.0, 7.0, 11.0, 15.0], [3.0, 7.0, 11.0, 15.0],
...                [3.0, 7.0, 11.0, 15.0], [3.0, 7.0, 11.0, 15.0]]
>>> new_wp['d'] = [[0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0],
...                [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]]


For convenience, Node2D.data gets automatically extracted from the base Node2D object:

>>> new_wp['h'] = wp['h'] # all zeros


Note: just remember to not assign to the node.data parameter directly (todo).

And reconstruct the data from the a, d, vh, vv, vd and h packets (Note that va node was not set and the WP tree is “not complete” - the va branch will be treated as zero-array):

>>> print(new_wp.reconstruct(update=False))
[[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]]


Now set the va node with the known values and do the reconstruction again:

>>> new_wp['va'] = wp['va'].data # [[-2.0, -2.0], [-2.0, -2.0]]
>>> print(new_wp.reconstruct(update=False))
[[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]]


which is just the same as the base sample data x.

Of course we can go the other way and remove nodes from the tree. If we delete the va node, again, we get the “not complete” tree from one of the previous examples:

>>> del new_wp['va']
>>> print(new_wp.reconstruct(update=False))
[[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]]


Just restore the node before next examples.

>>> new_wp['va'] = wp['va'].data


If the update param in the WaveletPacket2D.reconstruct() method is set to False, the node’s Node2D.data attribute will not be updated.

>>> print(new_wp.data)
None


Otherwise, the WaveletPacket2D.data attribute will be set to the reconstructed value.

>>> print(new_wp.reconstruct(update=True))
[[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]]
>>> print(new_wp.data)
[[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]
[ 1.  2.  3.  4.  5.  6.  7.  8.]]


Since we have an interesting WP structure built, it is a good occasion to present the WaveletPacket2D.get_leaf_nodes() method, which collects non-zero leaf nodes from the WP tree:

>>> print([n.path for n in new_wp.get_leaf_nodes()])
['a', 'h', 'va', 'vh', 'vv', 'vd', 'd']


Passing the decompose = True parameter to the method will force the WP object to do a full decomposition up to the maximum level of decomposition:

>>> paths = [n.path for n in new_wp.get_leaf_nodes(decompose=True)]
>>> len(paths)
64
>>> for i, path in enumerate(paths):
...     if (i+1) % 8 == 0:
...         print(path)
...     else:
...         try:
...             print(path, end=' ')
...         except:
...             print(path, end=' ')
aaa aah aav aad aha ahh ahv ahd
haa hah hav had hha hhh hhv hhd
hva hvh hvv hvd hda hdh hdv hdd
vaa vah vav vad vha vhh vhv vhd
vva vvh vvv vvd vda vdh vdv vdd
daa dah dav dad dha dhh dhv dhd
dva dvh dvv dvd dda ddh ddv ddd

##### Lazy evaluation:¶

Note

This section is for demonstration of pywt internals purposes only. Do not rely on the attribute access to nodes as presented in this example.

>>> x = numpy.array([[1, 2, 3, 4, 5, 6, 7, 8]] * 8)
>>> wp = pywt.WaveletPacket2D(data=x, wavelet='db1', mode='symmetric')

1. At first the wp’s attribute a is None

>>> print(wp.a)
None


Remember that you should not rely on the attribute access.

2. During the first attempt to access the node it is computed via decomposition of its parent node (the wp object itself).

>>> print(wp['a'])
a: [[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]]

3. Now the a is set to the newly created node:

>>> print(wp.a)
a: [[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]
[  3.   7.  11.  15.]]


And so is wp.d:

>>> print(wp.d)
d: [[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]
[ 0.  0.  0.  0.]]


#### Gotchas¶

PyWavelets utilizes NumPy under the hood. That’s why handling the data containing None values can be surprising. None values are converted to ‘not a number’ (numpy.NaN) values:

>>> import numpy, pywt
>>> x = [None, None]
>>> mode = 'symmetric'
>>> wavelet = 'db1'
>>> cA, cD = pywt.dwt(x, wavelet, mode)
>>> numpy.all(numpy.isnan(cA))
True
>>> numpy.all(numpy.isnan(cD))
True
>>> rec = pywt.idwt(cA, cD, wavelet, mode)
>>> numpy.all(numpy.isnan(rec))
True


### Contributing¶

All contributions including bug reports, bug fixes, new feature implementations and documentation improvements are welcome. Moreover, developers with an interest in PyWavelets are very welcome to join the development team! Please see our guidelines for pull requests for more information.

Contributors are expected to behave in a productive and respectful manner in accordance with our community guidelines.

#### History¶

PyWavelets started in 2006 as an academic project for a masters thesis on Analysis and Classification of Medical Signals using Wavelet Transforms and was maintained until 2012 by its original developer. In 2013 maintenance was taken over in a new repo) by a larger development team - a move supported by the original developer. The repo move doesn’t mean that this is a fork - the package continues to be developed under the name “PyWavelets”, and released on PyPI and Github (see this issue for the discussion where that was decided).

### Development guide¶

This section contains information on building and installing PyWavelets from source code as well as instructions for preparing the build environment on Windows and Linux.

#### Preparing Windows build environment¶

To start developing PyWavelets code on Windows you will have to install a C compiler and prepare the build environment.

##### Installing Windows SDK C/C++ compiler¶

Depending on your Python version, a different version of the Microsoft Visual C++ compiler will be required to build extensions. The same compiler that was used to build Python itself should be used.

For Python 3.5, 3.6 and 3.7 it will be MSVC 2015.

The MSVC version should be printed when starting a Python REPL, and can be checked against the note below:

Note

For reference:

• the MSC v.1500 in the Python version string is Microsoft Visual C++ 2008 (Microsoft Visual Studio 9.0 with msvcr90.dll runtime)
• MSC v.1600 is MSVC 2010 (10.0 with msvcr100.dll runtime)
• MSC v.1700 is MSVC 2012 (11.0)
• MSC v.1800 is MSVC 2013 (12.0)
• MSC v.1900 is MSVC 2015 (14.0)
Python 3.5.5 (default, Feb 13 2018, 06:15:35) [MSC v.1900 64 bit (AMD64)] on win32


There are several ISO images on the site, so just grab the one that is suitable for your platform:

• GRMSDK_EN_DVD.iso for 32-bit x86 platform
• GRMSDKX_EN_DVD.iso for 64-bit AMD64 platform (AMD64 is the codename for 64-bit CPU architecture, not the processor manufacturer)

After installing the SDK and before compiling the extension you have to configure some environment variables.

For 32-bit build execute the util/setenv_build32.bat script in the cmd window:

rem Configure the environment for 32-bit builds.
rem Use "vcvars32.bat" for a 32-bit build.
"C:\Program Files (x86)\Microsoft Visual Studio 9.0\VC\bin\vcvars32.bat"
rem Convince setup.py to use the SDK tools.
set MSSdk=1
setenv /x86 /release
set DISTUTILS_USE_SDK=1


For 64-bit use util/setenv_build64.bat:

rem Configure the environment for 64-bit builds.
rem Use "vcvars32.bat" for a 32-bit build.
"C:\Program Files (x86)\Microsoft Visual Studio 9.0\VC\bin\vcvars64.bat"
rem Convince setup.py to use the SDK tools.
set MSSdk=1
setenv /x64 /release
set DISTUTILS_USE_SDK=1

##### MinGW C/C++ compiler¶

In order to change the settings and use MinGW as the default compiler, edit or create a Distutils configuration file c:\Python2*\Lib\distutils\distutils.cfg and place the following entry in it:

[build]
compiler = mingw32


You can also take a look at Cython’s “Installing MinGW on Windows” page at http://wiki.cython.org/InstallingOnWindows for more info.

Note

Python 2.7/3.2 distutils package is incompatible with the current version (4.7+) of MinGW (MinGW dropped the -mno-cygwin flag, which is still passed by distutils).

To use MinGW to compile Python extensions you have to patch the distutils/cygwinccompiler.py library module and remove every occurrence of -mno-cygwin.

##### Next steps¶

After completing these steps continue with Installing build dependencies.

#### Preparing Linux build environment¶

There is a good chance that you already have a working build environment. Just skip steps that you don’t need to execute.

##### Installing basic build tools¶

Note that the example below uses aptitude package manager, which is specific to Debian and Ubuntu Linux distributions. Use your favourite package manager to install these packages on your OS.

aptitude install build-essential gcc python-dev git-core

##### Next steps¶

After completing these steps continue with Installing build dependencies.

#### Installing build dependencies¶

##### Setting up Python virtual environment¶

A good practice is to create a separate Python virtual environment for each project. If you don’t have virtualenv yet, install and activate it using:

curl -O https://raw.github.com/pypa/virtualenv/master/virtualenv.py
python virtualenv.py <name_of_the_venv>
. <name_of_the_venv>/bin/activate

##### Installing Cython¶

Use pip (http://pypi.python.org/pypi/pip) to install Cython:

pip install Cython>=0.16

##### Installing numpy¶

Use pip to install numpy:

pip install numpy


Numpy can also be obtained via scientific python distributions such as:

Note

You can find binaries for 64-bit Windows on http://www.lfd.uci.edu/~gohlke/pythonlibs/.

##### Installing Sphinx¶

Sphinx is a documentation tool that converts reStructuredText files into nicely looking html documentation. Install it with:

pip install Sphinx


numpydoc is used to format the API docmentation appropriately. Install it via:

pip install numpydoc


#### Building and installing PyWavelets¶

##### Installing from source code¶

Go to https://github.com/PyWavelets/pywt GitHub project page, fork and clone the repository or use the upstream repository to get the source code:

git clone https://github.com/PyWavelets/pywt.git PyWavelets


Activate your Python virtual environment, go to the cloned source directory and type the following commands to build and install the package:

python setup.py build
python setup.py install


To verify the installation run the following command:

python setup.py test


To build docs:

cd doc
make html

##### Installing a development version¶

You can also install directly from the source repository:

pip install -e git+https://github.com/PyWavelets/pywt.git#egg=PyWavelets


or:

pip install PyWavelets==dev

##### Installing a regular release from PyPi¶

A regular release can be installed with pip or easy_install:

pip install PyWavelets


#### Testing¶

##### Continous integration with Travis-CI¶

The project is using Travis-CI service for continuous integration and testing.

Current build status is: If you are submitting a patch or pull request please make sure it does not break the build.

##### Running tests locally¶

Tests are implemented with nose, so use one of:

\$ nosetests pywt

>>> pywt.test()


Note doctests require Matplotlib in addition to the usual dependencies.

##### Running tests with Tox¶

There’s also a config file for running tests with Tox (pip install tox). To for example run tests for Python 3.5 and 3.6 use:

tox -e py35,py36


#### Guidelines for Releasing PyWavelets¶

The following are guidelines for preparing a release of PyWavelets. The notation vX.X.X in the commands below would be replaced by the actual release number.

##### Updating the release notes¶

Prior to the release, make sure the release notes are up to date. The author lists can be generated via:

python  ./util/authors.py vP.P.P..


where vP.P.P is the previous release number.

The lists of issues closed and PRs merged can be generated via (script requires Python 2.X to run):

python ./util/gh_lists.py vX.X.X

##### Tag the release¶

Change ISRELEASED to True in setup.py and commit.

Tag the release via:

git tag -s vX.X.X


Then push the vX.X.X tag to the PyWavelets GitHub repo.

Note that while Appveyor will build wheels for Windows, it is preferred to get those wheels from the step below. Instructions for grabbing Appveyor wheels manually here for reference only: if the commit with ISRELEASED=True is submitted as a PR, the wheels can be downloaded from Appveyor once it has run on the PR. They can be found under the “Artifacts” tab in the Appveyor interface.

##### Build Windows, OS X and Linux wheels and upload to PyPI¶

Push a commit with the new tag and updates of dependency versions where needed to https://github.com/MacPython/pywavelets-wheels. The wheels will be produced automatically and uploaded to http://wheels.scipy.org/. From there they can be uploaded to PyPI automatically with wheel-uploader.

See the README on https://github.com/MacPython/pywavelets-wheels for more details.

##### Create the source distribution¶

Remove untracked files and directories with git clean. Warning: this will delete files & directories that are not under version control so you may want to do a dry run first by adding -n, so you can see what will be removed:

git clean -xfdn


Then run without -n:

git clean -xfd


Create the source distribution files via:

python setup.py sdist --formats=gztar,zip

##### Upload the release to PyPI¶

The binary Windows wheels downloaded from Appveyor (see above) should also be placed into the /dist subfolder along with the sdist archives.

The wheels and source distributions created above can all be securely uploaded to pypi.python.org using twine:

twine upload -s dist/*


Note that the documentation on ReadTheDocs (http://pywavelets.readthedocs.org) will have been automatically generated, so no actions need to be taken for documentation.

##### Update conda-forge¶

Send a PR with the new version number and sha256 hash of the source release to https://github.com/conda-forge/pywavelets-feedstock.

##### Create the release on GitHub¶

On the project’s GitHub page, click the releases tab and then press the “Draft a new release” button to create a release from the appropriate tag.

##### Announcing the release¶

Send release announcements to:

##### Prepare for continued development¶

Increment the version number in setup.py and change ISRELEASED to False.

Prepare new release note files for the upcoming release:

git add doc/release/X.X.X-notes.rst


And add release.X.X.X to the list in doc/source/releasenotes.rst

#### Something not working?¶

If these instructions are not clear or you need help setting up your development environment, go ahead and ask on the PyWavelets discussion group at http://groups.google.com/group/pywavelets or open a ticket on GitHub.

### Release Notes¶

#### PyWavelets 0.3.0 Release Notes¶

PyWavelets 0.3.0 is the first release of the package in 3 years. It is the result of a significant effort of a growing development team to modernize the package, to provide Python 3.x support and to make a start with providing new features as well as improved performance. A 0.4.0 release will follow shortly, and will contain more significant new features as well as changes/deprecations to streamline the API.

This release requires Python 2.6, 2.7 or 3.3-3.5 and NumPy 1.6.2 or greater.

Highlights of this release include:

• Support for Python 3.x (>=3.3)
• Added a test suite (based on nose, coverage up to 61% so far)
• Maintenance work: C style complying to the Numpy style guide, improved templating system, more complete docstrings, pep8/pyflakes compliance, and more.
##### New features¶
###### Test suite¶

The test suite can be run with nosetests pywt or with:

>>> import pywt
>>> pywt.test()

###### n-D Inverse Discrete Wavelet Transform¶

The function pywt.idwtn, which provides n-dimensional inverse DWT, has been added. It complements idwt, idwt2 and dwtn.

###### Thresholding¶

The function pywt.threshold has been added. It unifies the four thresholding functions that are still provided in the pywt.thresholding namespace.

##### Backwards incompatible changes¶

None in this release.

##### Other changes¶

Development has moved to a new repo. Everyone with an interest in wavelets is welcome to contribute!

Building wheels, building with python setup.py develop and many other standard ways to build and install PyWavelets are supported now.

##### Authors¶
• Ankit Agrawal +
• François Boulogne +
• Ralf Gommers +
• Gregory R. Lee +
• David McInnis +
• Helder Oliveira +
• Filip Wasilewski
• Kai Wohlfahrt +

A total of 9 people contributed to this release. People with a “+” by their names contributed a patch for the first time. This list of names is automatically generated, and may not be fully complete.

###### Issues closed for v0.3.0¶
• #3: Remove numerix compat layer
• #4: Add single code base Python 3 support
• #5: PEP8 issues
• #6: Migrate tests to nose
• #7: Expand test coverage without Matlab to a reasonable level
• #8: Replace custom C templates by Numpy’s templating system
• #9: Replace Cython templates by fused types
• #10: Replace use of __array_interface__ with Cython’s memoryviews
• #11: Format existing docstrings in numpydoc format.
• #12: Complete docstrings, they’re quite sparse right now
• #13: Reorganize source tree
• #24: doc/source/regression should be moved
• #27: Broken test: test_swt_decomposition
• #28: Install issue, no module tools.six
• #29: wp.update fails after removal of nodes
• #32: wp.update fails on 2D
• #34: Wavelet string attributes shouldn’t be bytes in Python 3
• #35: Re-enable float32 support
• #36: wavelet instance vs string
• #40: Test with Numpy 1.8rc1
• #45: demos should be updated and integrated in docs
• #60: Moving pywt forward faster
• #61: issues to address in moving towards 0.3.0
• #71: BUG: _pywt.downcoef always returns level=1 result
###### Pull requests for v0.3.0¶
• #1: travis: check all branches + fix URL
• #17: [DOC] doctrings for multilevel functions
• #18: DOC: format -> functions.py
• #20: MAINT: remove unnecessary zero() copy()
• #21: Doc wavelet_packets
• #22: Minor doc fixes
• #25: TEST: remove useless functions and use numpy instead
• #26: Merge most recent work
• #30: Adding test for wp.rst
• #41: Change to Numpy templating system
• #49: Taking on API Issues
• #53: readme updated with info related to Py3 version
• #63: Remove six
• #65: Thresholding
• #70: MAINT: PEP8 fixes
• #72: BUG: fix _downcoef for level > 1
• #73: MAINT: documentation and metadata update for repo fork
• #74: STY: fix pep8/pyflakes issues
• #77: MAINT: raise ValueError if data given to dwt or idwt is not 1D…

#### PyWavelets 0.4.0 Release Notes¶

PyWavelets 0.4.0 is the culmination of 6 months of work. In addition to several new features, some changes and deprecations have been made to streamline the API.

This release requires Python 2.6, 2.7 or 3.3-3.5 and NumPy 1.6.2 or greater.

Highlights of this release include:

• 1D and 2D inverse stationary wavelet transforms
• Substantially faster 2D and nD discrete wavelet transforms
• Complex number support
• nD versions of the multilevel DWT and IDWT
##### New features¶
###### 1D and 2D inverse stationary wavelet transforms¶

1D (iswt) and 2D (iswt2) inverse stationary wavelet transforms were added. These currently only support even length inputs.

###### Faster 2D and nD wavelet transforms¶

The multidimensional DWT and IDWT code was refactored and is now an order of magnitude faster than in previous releases. The following functions benefit: dwt2, idwt2, dwtn, idwtn.

###### Complex floating point support¶

64 and 128-bit complex data types are now supported by all wavelet transforms.

###### nD implementation of the multilevel DWT and IDWT¶

The existing 1D and 2D multilevel transforms were supplemented with an nD implementation.

###### Wavelet transforms can be applied along a specific axis/axes¶

All wavelet transform functions now support explicit specification of the axis or axes upon which to perform the transform.

###### Example Datasets¶

Two additional 2D grayscale images were added (camera, ascent). The previously existing 1D ECG data (ecg) and the 2D aerial image (aero) used in the demos can also now be imported via functions defined in pywt.data (e.g. camera = pywt.data.camera())

##### Deprecated features¶

A number of functions have been renamed, the old names are deprecated and will be removed in a future release:

• intwave, renamed to integrate_wavelet
• centrfrq, renamed to central_frequency
• scal2frq, renamed to scale2frequency
• orthfilt, renamed to orthogonal_filter_bank

Integration of general signals (i.e. not wavelets) with integrate_wavelet is deprecated.

The MODES object and its attributes are deprecated. The new name is Modes, and the attribute names are expanded:

• zpd, renamed to zero
• cpd, renamed to constant
• sp1, renamed to smooth
• sym, renamed to symmetric
• ppd, renamed to periodic
• per, renamed to periodization
##### Backwards incompatible changes¶

idwt no longer takes a correct_size parameter. As a consequence, idwt2 inputs must match exactly in length. For multilevel transforms, where arrays differing in size by one element may be produced, use the waverec functions from the multilevel module instead.

##### Bugs Fixed¶

float32 inputs were not always respected. All transforms now return float32 outputs when called using float32 inputs.

Incorrect detail coefficients were returned by downcoef when level > 1.

##### Other changes¶

Much of the API documentation is now autogenerated from the corresponding function docstrings. The numpydoc sphinx extension is now needed to build the documentation.

##### Authors¶
• Thomas Arildsen +
• François Boulogne
• Ralf Gommers
• Gregory R. Lee
• Michael Marino +
• Aaron O’Leary +
• Daniele Tricoli +
• Kai Wohlfahrt

A total of 8 people contributed to this release. People with a “+” by their names contributed a patch for the first time. This list of names is automatically generated, and may not be fully complete.

###### Issues closed for v0.4.0¶
• #46: Independent test comparison
• #95: Simplify Matlab tests
• #97: BUG: erroneous detail coefficients returned by downcoef with…
• #140: demo/dwt_signal_decomposition.py : TypeError: object of type…
• #141: Documentation needs update: ImportError: cannot import name ‘multilevel’
###### Pull requests for v0.4.0¶
• #55: [RFC] Api changes
• #59: Refactor convolution.c.src
• #64: MAINT: make LH, HL variable names in idwt2 consistent with dwt2
• #67: ENH: add wavedecn and waverecn functions
• #68: ENH: Faster dwtn and idwtn
• #88: DOC minor edit about possible naming
• #93: Added implementation of iswt and iswt2
• #98: fix downcoef detail coefficients for level > 1
• #99: complex support in all dwt and idwt related functions
• #100: replace mlabwrap with python-matlab-bridge in Matlab tests
• #102: Replace some .src expansion with macros
• #104: Faster idwtn/dwtn
• #106: make sure transforms respect float32 dtype
• #109: DOC: fix broken link in sidebar for html docs.
• #112: Complex fix
• #113: TST: don’t build .exe installers on Appveyor anymore, only wheels.
• #116: [RFC] ENH: Add axis argument to dwt
• #117: MAINT: remove deprecated for loop syntax from Cython code
• #121: Fix typo
• #123: MAINT: remove some unused imports
• #124: switch travis from python 3.5-dev to 3.5
• #130: Add axis argument to multidim
• #138: WIP: Documentation updates for v0.4.0
• #139: Autogenerate function API docs
• #142: fix broken docstring examples in _multilevel.py
• #143: handle None properly in waverec
• #145: DOC: Document MSVC versions

#### PyWavelets 0.5.0 Release Notes¶

PyWavelets is a Python toolbox implementing both discrete and continuous wavelet transforms (mathematical time-frequency transforms) with a wide range of built-in wavelets. C/Cython are used for the low-level routines, enabling high performance. Key Features of PyWavelets are:

• 1D, 2D and nD Forward and Inverse Discrete Wavelet Transform (DWT and IDWT)
• 1D, 2D and nD Multilevel DWT and IDWT
• 1D and 2D Forward and Inverse Stationary Wavelet Transform
• 1D and 2D Wavelet Packet decomposition and reconstruction
• 1D Continuous Wavelet Transform
• When multiple valid implementations are available, we have chosen to maintain consistency with MATLAB™’s Wavelet Toolbox.

PyWavelets 0.5.0 is the culmination of 1 year of work. In addition to several new features, substantial refactoring of the underlying C and Cython code have been made.

This release requires Python 2.6, 2.7 or 3.3-3.5 and NumPy 1.9.1 or greater. This will be the final release supporting Python 2.6 and 3.3.

Highlights of this release include:

• 1D continuous wavelet transforms
• new ‘reflect’ extension mode for discrete wavelet transforms
• faster performance for multilevel forward stationary wavelet transforms (SWT)
• n-dimensional support added to forward SWT
• routines to convert multilevel DWT coefficients to and from a single array
• axis support for multilevel DWT
• substantial refactoring/reorganization of the underlying C and Cython code
##### New features¶
###### 1D Continous Wavelet Transforms¶

A wide range of continous wavelets are now available. These include the following:

• Gaussian wavelets (gaus1…gaus8)
• Mexican hat wavelet (mexh)
• Morlet wavelet (morl)
• Complex Gaussian wavelets (cgau1…cgau8)
• Shannon wavelet (shan)
• Frequency B-Spline wavelet (fbsp)
• Complex Morlet wavelet (cmor)

Also, see the new CWT-related demo: demo/cwt_analysis.py

###### New discrete wavelets¶

Additional Debauchies wavelets (db20…db38) and Coiflets (coif6…coif17) have been added.

###### New extension mode: reflect¶

Discrete wavelet transforms support a new extension mode, reflect. This mode pads an array symmetrically, but without repeating the edge value. As an example:

 pad      array      pad
4 3 2 | 1 2 3 4 5 | 4 3 2


This differs from symmetric, which repeats the values at the boundaries:

 pad      array      pad
3 2 1 | 1 2 3 4 5 | 5 4 3

###### Multilevel DWT Coefficient Handling¶

New routines to convert the coefficients returned by multilevel DWT routines to and from a single n-dimensional array have been added. pywt.coeffs_to_array concatenates the output of wavedec, wavedec2 or wavedecn into a single numpy array. pywt.array_to_coeffs can be used to transform back from a single coefficient array to a format appropriate for waverec, waverec2 or waverecn.

###### More C function calls release the GIL¶

Cython code calling the wavelet filtering routines (DWT and SWT) now releases the global interpreter lock (GIL) where possible. A potential use case is in speeding up the batch computation of several large DWTs using multi-threading (e.g. via concurrent.futures).

###### Multilevel wavelet transforms along specific axes¶

The axis specific transform support introduced in the prior release was extended to the multilevel DWT transforms. All wavedec* and waverec* routines have a new axis (1D) or axes (2D, nD) keyword argument. If unspecified the default behaviour is to transform all axes of the input.

###### Faster multilevel stationary wavelet transforms¶

Stationary wavelet transforms are now faster when the number of levels is greater than one. The improvement can be very large (multiple orders of magnitude) for transforms with a large number of levels.

##### Backwards incompatible changes¶

A FutureWarning was added to swt2 and iswt2 to warn about a pending backwards incompatible change to the order of the coefficients in the list returned by these routines. The actual change will not occur until the next release. Transform coefficients will be returned in descending rather than ascending order. This change is being made for consistency with all other existing multi-level transforms in PyWavelets.

##### Bugs Fixed¶

demo/image_blender.py was updated to support the new api of Pillow 3.x

A bug related to size of assumed size_t on some platforms/compilers (e.g. Windows with mingw64) was fixed.

Fix to memory leak in (i)dwt_axis

Fix to a performance regression in idwt and iswt that was introduced in v0.4.0.

Fixed a bug in dwtn and idwtn for data with complex dtype when axes != None.

##### Other changes¶

The minimum supported numpy version has been increased to 1.9.1.

Test coverage (including for the Cython and C code) via Codecov was added and the overall test coverage has been improved.

A substantial overhaul of the C extension code has been performed. Custom templating is no longer used. The intention is to make this code easier to maintain and expand in the future.

The Cython code has been split out into a multiple files to hopefully make relevant portions of the wrappers easier to find for future developers.

setup.py now relies on setuptools in all cases (rather than distutils).

##### Authors¶
• Jonathan Dan +
• Ralf Gommers
• Gregory R. Lee
• Holger Nahrstaedt +
• Daniel M. Pelt +
• Alexandre Saint +
• Scott Sievert +
• Kai Wohlfahrt
• Frank Yu +

A total of 10 people contributed to this release. People with a “+” by their names contributed a patch for the first time. This list of names is automatically generated, and may not be fully complete.

###### Issues closed for v0.5.0¶
• #48: Continous wavelet transform?
• #127: Reorganize _pywt
• #160: Appveyor failing on recent PRs
• #163: Set up coveralls
• #166: Wavelet coefficients to single array (and vice versa?)
• #177: Fail to install pywt due to the use of index_t which conflict with the defination in /usr/include/sys/types.h on smartos sysmte(open solaris like system)
• #180: Memory leak
• #187: ‘reflect’ signal extension mode
• #189: bump minimum numpy version?
• #191: Upgrade removed Pillow methods
• #196: building in-place for development.
• #200: swt implementation is considerably slower than MATLAB
• #209: broken doctests
• #210: Run doctests in CI setup
• #211: Typo in iswt documentation
• #217: blank_discrete_wavelet does not properly intiailize some properties
• #231: I can’t compile pywt
###### Pull requests for v0.5.0¶
• #148: Reorganize C v2
• #161: Remove numpy distutils
• #162: fix: iswt/idwt performance regression
• #164: Improved coefficients for db and coif
• #168: convert transform coefficients to and from a single n-dimensional array
• #169: Remove templating
• #170: :Always install new pip on Appveyor
• #172: Adding of missing wavelets from the matlab list
• #178: use Index_t instead of index_t
• #179: add axis/axes support to multilevel discrete wavelet transforms
• #181: Fix memory leak
• #182: improve test coverage for _multidim.py and _multilevel.py
• #183: improve coverage for _dwt.py
• #184: fix corner case in coeffs_to_array
• #188: Drop GIL in c_wt calls
• #190: bump minimum numpy to 1.9
• #192: Upgrade to Pillow>=3 api
• #193: ENH: add ‘reflect’ extension mode
• #197: BLD: fix “python setup.py develop”. Closes gh-196
• #198: Choose clz* based on SIZE_MAX
• #201: speedup multi-level swt
• #205: fix dwtn/idwtn with axes != None and complex data
• #206: DOC: correct typo in iswt docstring
• #208: document coeff_to_array and array_to_coeff
• #214: FIX: update several doctests to reflect the new wavelets added
• #218: FIX: initialize all properties of a blank discrete wavelet
• #219: document coordinate conventions for 2D DWT routines.
• #220: Run doctests on TravisCI
• #221: Documentation for cwt and ContinuousWavelet
• #222: consistent use of double backticks in docs
• #224: n-dimensional stationary wavelet transform (swtn) and axis support in swt, swt2
• #225: BUG: fix breakage on 32-bit Python.
• #226: DOC: update Copyright statements.
• #227: ENH: add kind keyword to wavelist()
• #228: MAINT: avoid using a builtin as variable name in qmf().
• #229: DOC: add swtn, iswt, iswt2 to the API documentation
• #230: add demo of batch processing via concurrent.futures
• #234: ENH: coeffs_to_array supports axes argument as recently added to wavedec*
• #236: BLD: raise an ImportError if Cython should be installed but isn’t.

#### PyWavelets 1.0.0 Release Notes¶

We are very pleased to announce the release of PyWavelets 1.0. We view this version number as a milestone in the project’s now more than a decade long history. It reflects that PyWavelets has stabilized over the past few years, and is now a mature package which a lot of other important packages depend on. A listing of those package won’t be complete, but some we are aware of are:

• scikit-image - image processing in Python
• imagehash - perceptual image hashing
• tomopy - Tomographic Reconstruction in Python
• SpikeSort - Spike sorting library implemented in Python/NumPy/PyTables
• ODL - operator discretization library

This release requires Python 2.7 or >=3.5 and NumPy 1.9.1 or greater. The 1.0 release will be the last release supporting Python 2.7. It will be a Long Term Support (LTS) release, meaning that we will backport critical bug fixes to 1.0.x for as long as Python itself does so (i.e. until 1 Jan 2020).

##### New features¶
###### New 1D test signals¶

Many common synthetic 1D test signals have been implemented in the new function pywt.data.demo_signals to encourage reproducible research. To get a list of the available signals, call pywt.data.demo_signals('list'). These signals have been validated to match the test signals of the same name from the Wavelab toolbox (with the kind permission of Dr. David Donoho).

###### C99 complex support¶

The Cython modules and underlying C library can now be built with C99 complex support when supported by the compiler. Doing so improves performance when running wavelet transforms on complex-valued data. On POSIX systems (Linux, Mac OS X), C99 complex support is enabled by default at build time. The user can set the environment variable USE_C99_COMPLEX to 0 or 1 to manually disable or enable C99 support at compile time.

###### complex-valued CWT¶

The continuous wavelet transform, cwt, now also accepts complex-valued data.

###### More flexible specification of some continuous wavelets¶

The continous wavelets "cmor", "shan" and "fbsp" now let the user specify attributes such as their center frequency and bandwidth that were previously fixed. See more on this in the section on deprecated features.

###### Fully Separable Discrete Wavelet Transfrom¶

A new variant of the multilevel n-dimensional DWT has been implemented. It is known as the fully separable wavelet transform (FSWT). The functions fswavedecn fswaverecn correspond to the forward and inverse transforms, respectively. This differs from the existing wavedecn and waverecn in dimensions >= 2 in that all levels of decomposition are performed along a single axis prior to moving on to the next.

###### New thresholding methods¶

pywt.threshold now supports non-negative Garotte thresholding (mode='garotte'). There is also a new function pywt.threshold_firm that implements firm (semi-soft) thresholding. Both of the these new thresholding methods are intermediate between soft and hard thresholding.

###### New anti-symmetric boundary modes¶

Two new boundary handling modes for the discrete wavelet transforms have been implemented. These correspond to whole-sample and half-sample anti-symmetric boundary conditions (antisymmetric and antireflect).

###### New functions to ravel and unravel wavedecn coefficients¶

The function ravel_coeffs can be used to ravel all coefficients from wavedec, wavedec2 or wavedecn into a single 1D array. Unraveling back into a list of individual n-dimensional coefficients can be performed by unravel_coeffs.

###### New functions to determine multilevel DWT coefficient shapes and sizes¶

The new function wavedecn_size outputs the total number of coefficients that will be produced by a wavedecn decomposition. The function wavedecn_shapes returns full shape information for all coefficient arrays produced by wavedecn. These functions provide the size/shape information without having to explicitly compute a transform.

##### Deprecated features¶

The continous wavelets with names "cmor", "shan" and "fbsp" should now be modified to include formerly hard-coded attributes such as their center frequency and bandwidth. Use of the bare names “cmor”. “shan” and “fbsp” is now deprecated. For “cmor” (and “shan”), the form of the wavelet name is now “cmorB-C” (“shanB-C”) where B and C are floats representing the bandwidth frequency and center frequency. For “fbsp” the form should now incorporate three floats as in “fbspM-B-C” where M is the spline order and B and C are the bandwidth and center frequencies.

##### Backwards incompatible changes¶

Python 2.6, 3.3 and 3.4 are no longer supported.

The order of coefficients returned by swt2 and input to iswt2 have been reversed so that the decomposition levels are now returned in descending rather than ascending order. This makes these 2D stationary wavelet functions consistent with all of the other multilevel discrete transforms in PyWavelets.

For wavedec, wavedec2 and wavedecn, the ability for the user to specify a level that is greater than the value returned by dwt_max_level has been restored. A UserWarning is raised instead of a ValueError in this case.

##### Bugs Fixed¶

Assigning new data to the Node or Node2D no longer forces a cast to float64 when the data is one of the other dtypes supported by the dwt (float32, complex64, complex128).

Calling pywt.threshold with mode='soft' now works properly for complex-valued inputs.

A segfault when running multiple swt2 or swtn transforms concurrently has been fixed.

Several instances of deprecated numpy multi-indexing that caused warnings in numpy >=1.15 have been resolved.

The 2d inverse stationary wavelet transform, iswt2, now supports non-square inputs (an unnecessary check for square inputs was removed).

Wavelet packets no longer convert float32 to float64 upon assignment to nodes.

Doctests have been updated to also work with NumPy >= 1.14,

Indexing conventions have been updated to avoid FutureWarnings in NumPy 1.15.

##### Other changes¶

Python 3.7 is now officially supported.

##### Authors¶
• 0-tree +
• Jacopo Antonello +
• Matthew Brett +
• Saket Choudhary +
• Michael V. DePalatis +
• Daniel Goertzen +
• Ralf Gommers
• Mark Harfouche +
• John Kirkham +
• Dawid Laszuk +
• Gregory R. Lee
• Michel Pelletier +
• Balint Reczey +
• SylvainLan +
• Daniele Tricoli
• Kai Wohlfahrt

A total of 16 people contributed to this release. People with a “+” by their names contributed a patch for the first time. This list of names is automatically generated, and may not be fully complete.

###### Issues closed for v1.0.0¶

The following 15 issues were closed for this release.

• #405: New warning appearing
• #397: Make pip install work if numpy is not yet installed
• #396: Allow more levels in wavedec
• #386: Improve documentation for cwt
• #396: Allow more levels in wavedec
• #368: Bug in ISWT2 for non-rectangular arrays
• #357: reconstruction from array_to_coeff and waverec
• #352: FYI: PyWavelet does not correctly declare setup.py dependency…
• #338: upcoef - TypeError: No matching signature found
• #335: Build issue: PyWavelets does not install from sdist
• #333: user-friendly error messages regarding discrete vs. continuous…
• #326: Allow complex dtype of input
• #316: Test fail in some architectures
• #312: Documentation should suggest using the default conda channel
• #308: incorporate bandwidths into CWT wavelet names for families cmor,…
• #306: dwt_max_levels not enough documentation
• #302: Can’t remove cA and then reconstruct
• #290: idwtn should treat coefficients set to None as zeros
• #288: RuntimeErrors and segfaults from swt2() in threaded environments
###### Pull requests for v1.0.0¶

A total of 53 pull requests were merged for this release.

• #248: DOC: sync PyWavelets main descriptions.
• #249: Add pyqtgraph demo for plotting wavelets
• #254: DOC: fix rendering of wavelist docstring
• #255: ENH: improve iswt performance
• #256: ENH: add iswtn (n-dimensional inverse SWT)
• #260: TST: test OS X build on TravisCI. Closes gh-75.
• #262: avoid some compiler warnings
• #263: MAINT: better exception message for Wavelet(‘continuous_familyname’)
• #265: MAINT: fix more compiler warnings
• #269: allow string input in dwt_max_level
• #270: DOC: update ISWT documentation
• #272: allow separate wavelet/mode for each axis in routines based on…
• #273: fix non-integer index error
• #275: ENH: use single precision routines for half-precision inputs
• #276: update wp_scalogram demo work with matplotlib 2.0
• #285: Fix spelling typo
• #286: MAINT: Package the license file
• #291: idwtn should allow coefficients to be set as None
• #292: MAINT: ensure tests are included in wheels
• #294: FIX: shape adjustment in waverec should not assume a transform…
• #299: DOC: update outdated scipy-user email address
• #300: ENH: compiling with C99 support (non-MSVC only)
• #303: DOC: better document how to handle omitted coefficients in multilevel…
• #309: Document how max levels are determined for multilevel DWT and…
• #310: parse CWT wavelet names for parameters
• #314: TST: Explicity align data records in test_byte_offset()
• #317: TST: specify rtol and atol for assert_allclose calls in test_swt_decomposition
• #320: Suggest using default conda channel to install
• #321: BLD: add pyproject.toml file (PEP 518 support).
• #322: support soft thresholding of complex valued data
• #331: Rename to CONTRIBUTING.rst
• #337: provide a more helpful error message for wrong wavelet type
• #339: check for wrong number of dimensions in upcoef and downcoef
• #340: DOC: fix broken link to Airspeed Velocity documentation
• #344: force legacy numpy repr for doctests
• #349: test case for CWT with complex input
• #350: better document the size requirements for swt/swt2/swtn
• #351: Add two new antisymmetric edge modes
• #353: DOC: add citation info to the front page of the docs.
• #354: add firm (semi-soft) and non-negative garotte thresholding
• #355: swt(): inference of level=None to depend on axis
• #356: fix: default level in wavedec2 and wavedecn can be too conservative
• #360: fix Continuous spelling
• #361: AttributeError when using coeffs_to_array
• #362: Fix spelling of continuous globally
• #364: DOC: Explicitly print wavelet name for invalid wavelets
• #367: fix segfault related to parallel SWT
• #369: remove iswt2’s restriction on non-square inputs
• #376: add common 1d synthetic signals
• #377: minor update to demo_signals
• #378: numpy: 1.15 multiindexing warning. targetted fix
• #380: BLD: fix doc build on ReadTheDocs, need matplotlib for plots…
• #381: Fix corner case for small scales in CWT
• #382: avoid FutureWarnings related to multiindexing in Numpy1.15
• #384: swap swt2 coefficient order (and remove FutureWarnings)
• #387: improve CWT docs
• #390: MAINT: update Python version support. Closes gh-385.
• #391: fix broken link in documentation
• #392: do not force float64 dtype on assignment to Node, Node2D
• #398: MAINT: update .gitignore for files generated during build.
• #401: Fix failing numpy 1.9.3 build on Travis CI
• #403: Change ValueError to UserWarning when level is > dwt_max_level
• #404: BLD: fix ReadTheDocs build. Outdated NumPy gave a conflict with MPL.
• #410: DOC: rewrite docs front page

#### PyWavelets 1.1.0 Release Notes¶

We are very pleased to announce the release of PyWavelets 1.1.

This release requires Python >=3.5 and has dropped Python 2.7 support.