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, trim_approx=False, norm=False)¶
Multilevel 1D stationary wavelet transform.
- Parameters
- data
Input signal
- wavelet
Wavelet to use (Wavelet object or name)
- levelint, optional
The number of decomposition steps to perform.
- start_levelint, 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.
- trim_approxbool, optional
If True, approximation coefficients at the final level are retained.
- normbool, optional
If True, transform is normalized so that the energy of the coefficients will be equal to the energy of
data
. In other words,np.linalg.norm(data.ravel())
will equal the norm of the concatenated transform coefficients whentrim_approx
is True.
- Returns
- coeffslist
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)]
If
trim_approx
isTrue
, then the output list is exactly as inpywt.wavedec
, where the first coefficient in the list is the approximation coefficient at the final level and the rest are the detail coefficients:[cAn, cDn, ..., cD2, cD1]
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 asnumpy.pad
.A primary benefit of this transform in comparison to its decimated counterpart (
pywt.wavedecn
), is that it is shift-invariant. This comes at cost of redundancy in the transform (the size of the output coefficients is larger than the input).When the following three conditions are true:
The wavelet is orthogonal
swt
is called withnorm=True
swt
is called withtrim_approx=True
the transform has the following additional properties that may be desirable in applications:
energy is conserved
variance is partitioned across scales
When used with
norm=True
, this transform is closely related to the multiple-overlap DWT (MODWT) as popularized for time-series analysis, although the underlying implementation is slightly different from the one published in [1]. Specifically, the implementation used here requires a signal that is a multiple of2**level
in length.References
- 1
DB Percival and AT Walden. Wavelet Methods for Time Series Analysis. Cambridge University Press, 2000.
Multilevel 2D swt2
¶
- pywt.swt2(data, wavelet, level, start_level=0, axes=(-2, -1), trim_approx=False, norm=False)¶
Multilevel 2D stationary wavelet transform.
- Parameters
- dataarray_like
2D array with input data
- waveletWavelet 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
.- levelint
The number of decomposition steps to perform.
- start_levelint, optional
The level at which the decomposition will start (default: 0)
- axes2-tuple of ints, optional
Axes over which to compute the SWT. Repeated elements are not allowed.
- trim_approxbool, optional
If True, approximation coefficients at the final level are retained.
- normbool, optional
If True, transform is normalized so that the energy of the coefficients will be equal to the energy of
data
. In other words,np.linalg.norm(data.ravel())
will equal the norm of the concatenated transform coefficients whentrim_approx
is True.
- Returns
- coeffslist
Approximation and details coefficients (for
start_level = m
). Iftrim_approx
isFalse
, approximation coefficients are retained for all levels:[ (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
.If
trim_approx
isTrue
, approximation coefficients are only retained at the final level of decomposition. This matches the format used bypywt.wavedec2
:[ cA_m+level, (cH_m+level, cV_m+level, cD_m+level), ..., (cH_m+1, cV_m+1, cD_m+1), (cH_m, cV_m, cD_m), ]
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 asnumpy.pad
.A primary benefit of this transform in comparison to its decimated counterpart (
pywt.wavedecn
), is that it is shift-invariant. This comes at cost of redundancy in the transform (the size of the output coefficients is larger than the input).When the following three conditions are true:
The wavelet is orthogonal
swt2
is called withnorm=True
swt2
is called withtrim_approx=True
the transform has the following additional properties that may be desirable in applications:
energy is conserved
variance is partitioned across scales
Multilevel n-dimensional swtn
¶
- pywt.swtn(data, wavelet, level, start_level=0, axes=None, trim_approx=False, norm=False)¶
n-dimensional stationary wavelet transform.
- Parameters
- dataarray_like
n-dimensional array with input data.
- waveletWavelet 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
.- levelint
The number of decomposition steps to perform.
- start_levelint, optional
The level at which the decomposition will start (default: 0)
- axessequence of ints, optional
Axes over which to compute the SWT. A value of
None
(the default) selects all axes. Axes may not be repeated.- trim_approxbool, optional
If True, approximation coefficients at the final level are retained.
- normbool, optional
If True, transform is normalized so that the energy of the coefficients will be equal to the energy of
data
. In other words,np.linalg.norm(data.ravel())
will equal the norm of the concatenated transform coefficients whentrim_approx
is True.
- Returns
- [{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': <coeffs> # A(LL) - approx. on 1st dim, approx. on 2nd dim 'ad': <coeffs> # V(LH) - approx. on 1st dim, det. on 2nd dim 'da': <coeffs> # H(HL) - det. on 1st dim, approx. on 2nd dim 'dd': <coeffs> # 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 specifiedaxes
.If
trim_approx
isTrue
, the first element of the list contains the array of approximation coefficients from the final level of decomposition, while the remaining coefficient dictionaries contain only detail coefficients. This matches the behavior of pywt.wavedecn.
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 asnumpy.pad
.A primary benefit of this transform in comparison to its decimated counterpart (
pywt.wavedecn
), is that it is shift-invariant. This comes at cost of redundancy in the transform (the size of the output coefficients is larger than the input).When the following three conditions are true:
The wavelet is orthogonal
swtn
is called withnorm=True
swtn
is called withtrim_approx=True
the transform has the following additional properties that may be desirable in applications:
energy is conserved
variance is partitioned across scales
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_lenint
Input data length.
- Returns
- max_levelint
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 of2**n
.numpy.pad
can be used to pad a signal up to an appropriate length as needed.