PyWavelets - Wavelet Transforms in Python

PyWavelets is free and Open Source wavelet transform software for the Python programming language. 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 :)

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 and 2D Stationary Wavelet Transform (Undecimated Wavelet Transform)
• 1D and 2D Wavelet Packet decomposition and reconstruction
• 1D Continuous Wavelet Transform
• Approximating wavelet and scaling functions
• Over seventy built-in wavelet filters and custom wavelets supported
• Single and double precision calculations
• Real and complex calculations
• Results compatible with Matlab Wavelet Toolbox (TM)

Requirements

PyWavelets is a package for the Python programming language. It requires:

• Python 2.6, 2.7 or >=3.3
• Numpy >= 1.9.1

Download

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

Latest release, including source and binary package for Windows, is available for download from the Python Package Index or on the Releases Page.

Install

In order to build PyWavelets from source, a working C compiler (GCC or MSVC) and a recent version of Cython is required.

• Install PyWavelets with pip install PyWavelets.
• To build and install from source, navigate to downloaded PyWavelets source code directory and type python setup.py install.

Prebuilt Windows binaries and source code packages are also available from Python Package Index.

Binary packages for several Linux distributors are maintained by Open Source community contributors. Query your Linux package manager tool for python-pywavelets, python-wavelets, python-pywt or similar package name.

See also

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

Documentation

Documentation with detailed examples and links to more resources is available online at http://pywavelets.readthedocs.org.

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

State of development & Contributing

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).

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!

Python 3

Python 3.x is fully supported from release v0.3.0 on.

Contact

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

License

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

Contents

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).

Returns:

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)

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.

Returns:

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 user 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 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.

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)

See also

You can find live examples of wavefun() usage and images of all the built-in wavelets on the Wavelet Properties Browser page.

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 continous 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:

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

• reflect - reflect-padding - signal is extended by reflecting samples:

  ... 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)

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']

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 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.

PyWavelets	Matlab	numpy.pad
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
N/A	asym, asymh	N/A
N/A	asymw	N/A

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.

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

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.

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:

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.

Returns:

coeffs : ndarray

1-D array of coefficients.

See also

upcoef

Maximum decomposition level - dwt_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

Wavelet filter length.

Returns:

max_level : int

Maximum level.

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

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

Returns:

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.

Returns:

rec: array_like

Single level reconstruction of signal from given coefficients.

Example:

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

Example:

>>> import pywt
>>> (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')
>>> print 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.

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.

Returns:

rec : ndarray

1-D array with reconstructed data from coefficients.

See also

downcoef

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.])

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

Wavelet to use

mode : str, optional

Signal extension mode, see Modes (default: ‘symmetric’)

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.

Returns:

(cA, (cH, cV, cD)) : tuple

Approximation, horizontal detail, vertical detail and diagonal detail coefficients respectively. Horizontal refers to array axis 0 (or axes[0] 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()

wavelet : Wavelet object or name string

Wavelet to use

mode : str, optional

Signal extension mode, see Modes (default: ‘symmetric’)

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

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.

axes : 2-tuple of ints, optional

Axes over which to compute the DWT. Repeated elements are not allowed.

Returns:

[cAn, (cHn, cVn, cDn), ... (cH1, cV1, cD1)] : list

Coefficients list. For user-specified axes, cH* corresponds to axes[0] while cV* corresponds to axes[1]. 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

Wavelet to use

mode : str, optional

Signal extension mode, see Modes (default: ‘symmetric’)

axes : 2-tuple of ints, optional

Axes over which to compute the IDWT. Repeated elements are not allowed.

Returns:2D array of reconstructed data.

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

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

Wavelet to use.

mode : str, optional

Signal extension mode, see Modes. Default is ‘symmetric’.

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.

Returns:

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': <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 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 zeroes.

wavelet : Wavelet object or name string

Wavelet to use

mode : str, optional

Signal extension mode used in the decomposition, see Modes (default: ‘symmetric’).

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.

Returns:

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

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.

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)).

Returns:

[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

Wavelet to use

mode : str, optional

Signal extension mode, see Modes (default: ‘symmetric’)

axes : sequence of ints, optional

Axes over which to compute the IDWT. Axes may not be repeated.

Returns:nD array of reconstructed data.

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.]]])

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 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.

Returns:

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[1][‘dd’]] would extract the first level detail coefficients from coeff_arr.

See also

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[0]      |  c[1]['da']   |                               |
|               |               |                               |
+---------------+---------------+           c[2]['da']          |
|               |               |                               |
| c[1]['ad']    |  c[1]['dd']   |                               |
|               |               |                               |
+---------------+---------------+ ------------------------------+
|                               |                               |
|                               |                               |
|                               |                               |
|          c[2]['ad']           |           c[2]['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.

Returns:

coeffs: array-like

Wavelet transform coefficient array.

See also

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[0]      |  c[1]['da']   |                               |
|               |               |                               |
+---------------+---------------+           c[2]['da']          |
|               |               |                               |
| c[1]['ad']    |  c[1]['dd']   |                               |
|               |               |                               |
+---------------+---------------+ ------------------------------+
|                               |                               |
|                               |                               |
|                               |                               |
|          c[2]['ad']           |           c[2]['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)

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.

Returns:

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)]

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

Wavelet to use

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.

Returns:

coeffs : list

Approximation and details coefficients:

[
    (cA_m,
        (cH_m, cV_m, cD_m)
    ),
    (cA_m+1,
        (cH_m+1, cV_m+1, cD_m+1)
    ),
    ...,
    (cA_m+level,
        (cH_m+level, cV_m+level, cD_m+level)
    )
]

where cA is approximation, cH is horizontal details, cV is vertical details, cD is diagonal details and m is start_level.

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

Wavelet to use.

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.

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 specified axes.

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.

Returns:

max_level : int

Maximum level of Stationary Wavelet Transform for data of given length.

Inverse Stationary Wavelet Transform

Inverse stationary wavelet transforms are provided for 1D and 2D data.

Note: These inverse transforms are not yet optimized for speed and only support a subset of the forward transform features. Specifically, there is not yet a general n-dimensional inverse transform and these routines do not yet have general axis/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

Returns:

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_1,
        (cH_1, cV_1, cD_1)
    ),
    (cA_2,
        (cH_2, cV_2, cD_2)
    ),
    ...,
    (cA_n
        (cH_n, cV_n, cD_n)
    )
]

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

Wavelet to use

Returns:

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.]])

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 - common interface for 1D and 2D nodes:
  • Node - data carrier node in a 1D decomposition tree
    • WaveletPacket - 1D decomposition tree root node
  • Node2D - data carrier node in a 2D decomposition tree
    • WaveletPacket2D - 2D decomposition tree root node

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.

Args:

arguments to pass to the func

Kwargs:

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.

Args:

arguments to pass to the func

Kwargs:

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()

See also

• dwt() for 1D Discrete Wavelet Transform output coefficients.

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()

See also

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.

Continous Wavelet Transform (CWT)

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

Single level - cwt

pywt.cwt(data, scales, wavelet)

One dimensional Continuous Wavelet Transform.

Parameters:

data : array_like

Input signal

scales : array_like

scales to use

wavelet : Wavelet object or name

Wavelet to use

sampling_period : float

Sampling period for frequencies output (optional)

Returns:

coefs : array_like

Continous 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 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() 

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, the data values where their absolute value is less than the value param are replaced with substitute. From the data values with absolute value greater or equal to the thresholding value, a quantity of (signum * value) is subtracted.

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.

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. Less data values pass untouched.

Parameters:

data : array_like

Numeric data.

value : scalar

Thresholding value.

mode : {‘soft’, ‘hard’, ‘greater’, ‘less’}

Decides the type of thresholding to be applied on input data. Default is ‘soft’.

substitute : float, optional

Substitute value (default: 0).

Returns:

output : array

Thresholded array.

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, '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. ])

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).

Returns:

[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).

Returns:

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.

Returns:

freq : scalar

Quadrature Mirror Filter

pywt.qmf(filt)

Returns the Quadrature Mirror Filter(QMF).

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.

Returns:

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).

Returns:

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)

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

Example: .. sourcecode:: python

>>> import pywt
>>> camera = pywt.data.camera()

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.

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:

Wavelet name, short_family_name and family_name:

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

• Decomposition (dec_len) and reconstruction (rec_len) filter lengths:

  >>> int(w.dec_len) # int() is for normalizing longs and ints for doctest
  6
  >>> int(w.rec_len)
  6
  

• Orthogonality (orthogonal) and biorthogonality (biorthogonal):

  >>> w.orthogonal
  True
  >>> w.biorthogonal
  True
  

• Symmetry (symmetry):

  >>> print(w.symmetry)
  asymmetric
  

• Number of vanishing moments for the scaling function phi (vanishing_moments_phi) and the wavelet function psi (vanishing_moments_psi) associated with the filters:

  >>> w.vanishing_moments_phi
  0
  >>> w.vanishing_moments_psi
  3
  

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)

See also

You can find live examples of wavefun() usage and images of all the built-in wavelets on the Wavelet Properties Browser page.

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 lenght 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)
ad

>>> 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')])
['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd']

or sorted based on the band frequency (freq):

>>> print([node.path for node in wp.get_level(3, 'freq')])
['aaa', 'aad', 'add', 'ada', 'dda', 'ddd', 'dad', 'daa']

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
>>> new_wp['ad'] = [-2., -2.]

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)])
['aa', 'ad', 'd']

>>> print([n.path for n in new_wp.get_leaf_nodes(True)])
['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd']

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.]
ad [-2. -2.]
da [-1. -1.]
dd [ 0.  0.]

>>> node = wp['ad']
>>> print(node)
ad: [-2. -2.]

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.]
ad [-2. -2.]
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

Start with preparing test data:

>>> 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=' ')
  aa ah av ad
  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
  ava avh avv avd ada adh adv add
  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
ava avh avv avd ada adh adv add
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

Development notes

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 official binary builds of Python 2.6 to 3.2, this will be VS 2008. Python 3.3 and 3.4 were compiled with VS 2010, and for Python 3.5 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 2.7.3 (default, Apr 10 2012, 23:31:26) [MSC v.1500 32 bit (Intel)] on win32
Python 3.2 (r32:88445, Feb 20 2011, 21:30:00) [MSC v.1500 64 bit (AMD64)] on win32

To get started first download, extract and install Microsoft Windows SDK for Windows 7 and .NET Framework 3.5 SP1 from http://www.microsoft.com/downloads/en/details.aspx?familyid=71DEB800-C591-4F97-A900-BEA146E4FAE1&displaylang=en.

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

See also http://wiki.cython.org/64BitCythonExtensionsOnWindows.

MinGW C/C++ compiler

MinGW distribution can be downloaded from http://sourceforge.net/projects/mingwbuilds/.

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.

See http://bugs.python.org/issue12641 bug report for more information on the issue.

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:

• Anaconda
• Enthought Canopy
• Python(x,y)

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 continous 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 2.7 and Python 3.4 use:

tox -e py27,py34

For more information see the Tox documentation.

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.

author lists can be generated via:

`python  ./util/authors.py vP.P.P..` (where vP.P.P is the previous release number)

and the lists of Issues closed and PRs merged via:

`python ./util/gh_lists.py vX.X.X` (script requires Python 2.X to run)

Tag the release and trigger the build of Windows wheels

Change ISRELEASED to TRUE in setup.py and commit.

Appveyor will build wheels for windows. 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. These should be downloaded so that they can later be uploaded to pypi.python.org.

Tag the release via:

`git tag -s vX.X.X`

Then push the vX.X.X tag to master.

Create the source distribution

Remove untracked files and directories via 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/*`

Upload the documentation to pypi

The documentation on readthedocs (http://pywavelets.readthedocs.org) will have been automatically generated, but the documentation linked from pypi.python.org (http://pythonhosted.org/PyWavelets/) will need to be manually updated. This can be done by building the documentation locally and zipping ‘doc/build/html’ so that index.html is at the top level of the archive. This archive can then be uploaded in the “files” section of the PyWavelets page on pypi.

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:

• https://groups.google.com/forum/#!forum/pywavelets (pywavelets@googlegroups.com)
• python-announce-list@python.org
• scipy-user@scipy.org

Prepare for continued development

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

Prepare new release note files for theupcoming release:

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

Resources

Code

The GitHub repository is now the main code repository.

If you are using the Mercurial repository at Bitbucket, please switch to Git/GitHub and follow for development updates.

Questions and bug reports

Use GitHub Issues or PyWavelets discussions group to post questions and open tickets.

Wavelet Properties Browser

Browse properties and graphs of wavelets included in PyWavelets on wavelets.pybytes.com.

Articles

• Denoising: wavelet thresholding
• Wavelet Regression in Python

PyWavelets

Release Notes

PyWavelets 0.3.0 Release Notes

Contents

• PyWavelets 0.3.0 Release Notes
  • New features
    • Test suite
    • n-D Inverse Discrete Wavelet Transform
    • Thresholding
  • Backwards incompatible changes
  • Other changes
  • Authors
    • Issues closed for v0.3.0
    • Pull requests for v0.3.0

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 +
• David Menéndez Hurtado +
• 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
• #43: MAINT: update six.py to not use lazy loading.
• #49: Taking on API Issues
• #50: Add idwtn
• #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

Contents

• PyWavelets 0.4.0 Release Notes
  • New features
    • 1D and 2D inverse stationary wavelet transforms
    • Faster 2D and nD wavelet transforms
    • Complex floating point support
    • nD implementation of the multilevel DWT and IDWT
    • Wavelet transforms can be applied along a specific axis/axes
    • Example Datasets
  • Deprecated features
  • Backwards incompatible changes
  • Bugs Fixed
  • Other changes
  • Authors
    • Issues closed for v0.4.0
    • Pull requests for v0.4.0

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
• #144: Add importable images
• #145: DOC: Document MSVC versions

PyWavelets 0.5.0 Release Notes

Contents

• PyWavelets 0.5.0 Release Notes
  • New features
    • 1D Continous Wavelet Transforms
    • New discrete wavelets
    • New extension mode: reflect
    • Multilevel DWT Coefficient Handling
    • More C function calls release the GIL
    • Multilevel wavelet transforms along specific axes
    • Faster multilevel stationary wavelet transforms
  • Deprecated features
  • Backwards incompatible changes
  • Bugs Fixed
  • Other changes
  • Authors
    • Issues closed for v0.5.0
    • Pull requests for v0.5.0

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 discrete wavelets added (additional Debauchies and Coiflet wavelets)
• 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.

Deprecated features

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
• David Menéndez Hurtado
• 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
• #167: Add coverage (codecov.io)
• #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
• #207: minor documentation updates
• #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
• #223: add FutureWarning about swt2 coefficient order
• #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 0.5.1 Release Notes

PyWavelets 0.5.1 is a bug-fix release with no new features compared to 0.5.0

Bugs Fixed

In release 0.5.0 the wrong edge mode was used for the following three deprecated modes: ppd, sp1, and per. All deprecated edge mode names are now correctly converted to the corresponding new names.

One-dimensional discrete wavelet transforms did not properly respect the axis argument for complex-valued data. Prior to this release, the last axis was always transformed for arrays with complex dtype. This fix affects dwt, idwt, wavedec, waverec.

Authors

• Gregory R. Lee

Issues closed for v0.5.1

• #245: Keyword “per” for dwt extension mode

Pull requests for v0.5.1

• #244: FIX: dwt, idwt with complex data now pass axis argument properly
• #246: fix bug in deprecated mode name conversion

Indices and tables

• Index
• Search Page