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
- dataarray_like
Input signal
- waveletWavelet object or name
Wavelet to use
- modestr, optional
Signal extension mode, see Modes.
- axis: int, optional
Axis over which to compute the DWT. If not given, the last axis is used.
- Returns
- (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
- waveletWavelet object or name string
Wavelet to use
- modestr, optional
Signal extension mode, see Modes.
- levelint, 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.
- Returns
- [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
- partstr
Coefficients type:
‘a’ - approximations reconstruction is performed
‘d’ - details reconstruction is performed
- dataarray_like
Input signal.
- waveletWavelet object or name
Wavelet to use
- modestr, optional
Signal extension mode, see Modes.
- levelint, optional
Decomposition level. Default is 1.
- Returns
- coeffsndarray
1-D array of coefficients.
See also
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_lenint
Input data length.
- filter_lenint, str or Wavelet
The wavelet filter length. Alternatively, the name of a discrete wavelet or a Wavelet object can be specified.
- Returns
- max_levelint
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
orwavedecn
.- Parameters
- shapesequence of ints
Input data shape.
- waveletWavelet 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
.- axessequence of ints, optional
Axes over which to compute the DWT. Axes may not be repeated.
- Returns
- levelint
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_lenint
Data length.
- filter_lenint
Filter length.
- modestr, optional
Signal extension mode, see Modes.
- Returns
- lenint
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.