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 also supports discrete wavelets see pywt.wavelist(kind='discrete'). I have tested Daubechies-Wavelets db-x and symlets sym-x, 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, torch
from scipy import datasets

data = torch.tensor(datasets.face(), dtype=torch.float64)
# permute [H, W, C] -> [C, H, W]
data = data.permute(2, 0, 1)
coefficients = ptwt.wavedec2(face, "haar", level=2, mode="constant")
reconstruction = ptwt.waverec2(coefficients, "haar")
torch.max(torch.abs(face - reconstruction))

Speed tests

Speed tests comparing our tools to related libraries are available.

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 pywt
import ptwt  # use "from src import ptwt" for a cloned the repo

# generate an input of even length.
data = torch.arange(16, dtype=torch.float32)
# forward
matrix_wavedec = ptwt.MatrixWavedec(haar, level=2)
coeff = matrix_wavedec(data)
print(coeff)
# backward
matrix_waverec = ptwt.MatrixWaverec("haar")
rec = matrix_waverec(coeff)
print(rec)

The process for the 2d transforms MatrixWavedec2, MatrixWaverec2 works similarly. By default, a separable transformation is used. To use a non-separable transformation, pass separable=False 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. The GitHub workflow executes a subset of all tests for efficiency reasons. After cloning the repository, moving into the main directory, and installing nox with pip install nox run

nox --session test

for all existing tests.

Citation

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

@article{JMLR:v25:23-0636,
  author  = {Moritz Wolter and Felix Blanke and Jochen Garcke and Charles Tapley Hoyt},
  title   = {ptwt - The PyTorch Wavelet Toolbox},
  journal = {Journal of Machine Learning Research},
  year    = {2024},
  volume  = {25},
  number  = {80},
  pages   = {1--7},
  url     = {http://jmlr.org/papers/v25/23-0636.html}
}