Pytorch Wavelet Toolbox (ptwt)

Welcome to the PyTorch wavelet toolbox. This package implements:

• the fast wavelet transform (fwt) via wavedec and its inverse by providing the waverec function,
• the two-dimensional fwt is called wavedec2 the synthesis counterpart waverec2,
• wavedec3 and waverec3 cover the three-dimensional analysis and synthesis case,
• MatrixWavedec and MatrixWaverec provide sparse-matrix-based fast wavelet transforms with boundary filters,
• 2d sparse-matrix transforms with separable & non-separable boundary filters are available (experimental),
• cwt computes a one-dimensional continuous forward transform,
• single and two-dimensional wavelet packet forward and backward transforms are available via the WaveletPacket and WaveletPacket2D objects,
• finally, this package provides adaptive wavelet support (experimental).

This toolbox supports pywt-wavelets. Complete documentation is available: https://pytorch-wavelet-toolbox.readthedocs.io/

Installation

Install the toolbox via pip or clone this repository. In order to use pip, type:

$ pip install ptwt

You can remove it later by typing pip uninstall ptwt.

Example usage:

Single dimensional transform

One way to compute fast wavelet transforms is to rely on padding and convolution. Consider the following example:

import torch
import numpy as np
import pywt
import ptwt  # use "from src import ptwt" for a cloned the repo

# generate an input of even length.
data = np.array([0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0])
data_torch = torch.from_numpy(data.astype(np.float32))
wavelet = pywt.Wavelet('haar')

# compare the forward fwt coefficients
print(pywt.wavedec(data, wavelet, mode='zero', level=2))
print(ptwt.wavedec(data_torch, wavelet, mode='zero', level=2))

# invert the fwt.
print(ptwt.waverec(ptwt.wavedec(data_torch, wavelet, mode='zero'),
                   wavelet))

The functions wavedec and waverec compute the 1d-fwt and its inverse. Internally both rely on conv1d, and its transposed counterpart conv_transpose1d from the torch.nn.functional module. This toolbox supports discrete wavelets see also pywt.wavelist(kind='discrete'). I have tested Daubechies-Wavelets db-x and symlets sym-x, which are usually a good starting point.

Two-dimensional transform

Analog to the 1d-case wavedec2 and waverec2 rely on conv2d, and its transposed counterpart conv_transpose2d. To test an example run:

import ptwt, pywt, torch
import numpy as np
import scipy.misc

face = np.transpose(scipy.misc.face(),
                        [2, 0, 1]).astype(np.float64)
pytorch_face = torch.tensor(face)
coefficients = ptwt.wavedec2(pytorch_face, pywt.Wavelet("haar"),
                                level=2, mode="constant")
reconstruction = ptwt.waverec2(coefficients, pywt.Wavelet("haar"))
np.max(np.abs(face - reconstruction.squeeze(1).numpy()))

Boundary Wavelets with Sparse-Matrices

In addition to convolution and padding approaches, sparse-matrix-based code with boundary wavelet support is available. In contrast to padding, boundary wavelets do not add extra pixels at the edges. Internally, boundary wavelet support relies on torch.sparse.mm. Generate 1d sparse matrix forward and backward transforms with the MatrixWavedec and MatrixWaverec classes. Reconsidering the 1d case, try:

import torch
import numpy as np
import pywt
import ptwt  # use "from src import ptwt" for a cloned the repo

# generate an input of even length.
data = np.array([0, 1, 2, 3, 4, 5, 6, 7, 7, 6, 5, 4, 3, 2, 1, 0])
data_torch = torch.from_numpy(data.astype(np.float32))
# forward
matrix_wavedec = ptwt.MatrixWavedec(pywt.Wavelet("haar"), level=2)
coeff = matrix_wavedec(data_torch)
print(coeff)
# backward
matrix_waverec = ptwt.MatrixWaverec(pywt.Wavelet("haar"))
rec = matrix_waverec(coeff)
print(rec)

The process for the 2d transforms MatrixWavedec2, MatrixWaverec2 works similarly. By default, a non-separable transformation is used. To use a separable transformation, pass separable=True to MatrixWavedec2 and MatrixWaverec2. Separable transformations use a 1d transformation along both axes, which might be faster since fewer matrix entries have to be orthogonalized.

Adaptive Wavelets

Experimental code to train an adaptive wavelet layer in PyTorch is available in the examples folder. In addition to static wavelets from pywt,

• Adaptive product-filters
• and optimizable orthogonal-wavelets are supported.

See https://github.com/v0lta/PyTorch-Wavelet-Toolbox/tree/main/examples/network_compression/ for a complete implementation.

Testing

The tests folder contains multiple tests to allow independent verification of this toolbox. After cloning the repository, and moving into the main directory, and installing nox with pip install nox run:

$ nox --session test

Citation

If you use this work in a scientific context please cite:

@phdthesis{handle:20.500.11811/9245,
  urn: https://nbn-resolving.org/urn:nbn:de:hbz:5-63361,
  author = {{Moritz Wolter}},
  title = {Frequency Domain Methods in Recurrent Neural Networks for Sequential Data Processing},
  school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
  year = 2021,
  month = jul,
  url = {https://hdl.handle.net/20.500.11811/9245}
}

If you use the boundary wavelet support please additionally cite:

@thesis{Blanke2021,
  author = {Felix Blanke},
  title = {{Randbehandlung bei Wavelets für Faltungsnetzwerke}},
  type = {Bachelor's Thesis},
  annote = {Gbachelor},
  year = {2021},
  school = {Institut f\"ur Numerische Simulation, Universit\"at Bonn}
}