Walsh-hadamard transform

Compressing images with a Hadamard transform.

A transform from the 1920s that needs nothing but additions and subtractions, turned into a complete image codec you can read in an afternoon. Every step of it is exact, so the same picture compresses to the same bytes on every machine, and seven file formats go in and come out. A DCT and a Haar transform are on board to race it against, and more than seven hundred tests keep all of it honest.

Contents

• Description
  • How it works, in one picture
• Contributing
  • List of contributors
• Acknowledgement
• Installation
  • Development
• How to run
  • Command line
  • Reading the PSNR figures
  • As a library
    • Skipping the .cim file
    • Looking at the vectors
    • Other transforms
    • Writing your own
  • Examples
• Requirements
• Development commands
  • Coverage
• Releasing
• File formats
  • Input and output
  • PNG
  • Pickled pixels
  • The .cim container

Description

From Wikipedia: The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on 2m real numbers (or complex numbers, although the Hadamard matrices themselves are purely real).

The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size 2 × 2 × ⋯ × 2 × 2. It decomposes an arbitrary input vector into a superposition of Walsh functions.

The transform is named for the French mathematician Jacques Hadamard, the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh.

The Hadamard transform is also used in data encryption, as well as many signal processing and data compression algorithms, such as JPEG XR and MPEG-4 AVC. In video compression applications, it is usually used in the form of the sum of absolute transformed differences. It is also a crucial part of Grover's algorithm and Shor's algorithm in quantum computing.

How it works, in one picture

Top: the eight Walsh functions of an 8-sample block are square waves whose only values are +1/√8 and -1/√8, ordered by how often they change sign (their sequency). Middle: two of them at a time, one across and one down, give the 64 patterns that any 8×8 block of pixels is a weighted mix of; the weights are the block's coefficients, and the codec keeps only the 16 in the outlined corner, the slow-changing patterns that carry most of a picture. Bottom: a real block from the Blue Marble sample goes through the codec. Its 64 coefficients are shown, the 16 that survive are rounded to integers, and the last panel is what those 16 numbers reconstruct: the shape is there and the fine detail is gone, which is the whole trade. Since the transform is exact, the coefficients in that figure are the ones in the .cim file, on any machine.

The figure is generated by examples/plot_transform.py from the sample image (needs the demo extra).

Contributing

See CONTRIBUTING.md for the development setup, the checks CI runs, and the conventions this codebase follows. Participation is covered by the Code of Conduct.

List of contributors

• :octocat: @oskar-j
• :octocat: @DYNOSuprovo

Acknowledgement

This code is partially based on the solution from ktisha/python2012

Installation

Requires Python 3.10 or newer.

pip install walsh

or, with uv:

uv add walsh          # into a project
uv tool install walsh # just the command line tool

The example script additionally needs matplotlib and Pillow, which are the demo extra: pip install "walsh[demo]".

Development

uv.lock is committed and CI installs from it with --locked, so a checkout reproduces exactly the environment CI uses:

uv sync --group dev --all-extras

--group dev brings in pytest, ruff and mypy; --all-extras adds the demo extra so examples/roundtrip.py runs too. Without uv:

pip install -e ".[demo]" -r requirements-dev.txt

How to run

Command line

walsh compress data/bmp/image.bmp data/cim/transformed.cim
walsh extract  data/cim/transformed.cim data/bmp/recreated.bmp

The format is taken from the filename suffix, so PPM works the same way, and a picture can be compressed from one format and restored as another:

walsh compress photo.ppm out.cim
walsh extract  out.cim restored.bmp     # PPM in, BMP out
walsh extract  out.cim restored.png     # or PNG out
walsh extract  out.cim restored.tif     # or TIFF out
walsh extract  out.cim restored.pam     # or PAM out
walsh extract  out.cim restored.npy     # or a bare NumPy array
walsh extract  out.cim restored.pkl     # or a pickle of rows of (r, g, b) tuples

To see what the codec does to a picture without keeping the .cim, name a picture as the output of compress. The picture is compressed and restored in memory, and what is written is its lossy reconstruction, byte for byte what the two commands above would have produced between them:

walsh compress photo.ppm photo_compressed.ppm
walsh compress photo.ppm photo_compressed.png    # or in any other format

The result is a picture like any other, as large as the original: it shows the compression, it is not the compressed data. See Skipping the .cim file.

Pickled pixels go in the same way, and a flat list of them, which does not carry its size, takes it from the command line:

walsh compress array.pkl  out.cim                            # a pickled NumPy array
walsh compress rows.pkl   out.cim                            # [[(r, g, b), ...], ...]
walsh compress pixels.pkl out.cim --width 400 --height 300   # [(r, g, b), ...]

A pickle is read through an allowlist and nothing in it is ever executed; see Pickled pixels.

compress accepts --packed-block-size (how many low-frequency coefficients per axis to keep -- lower is smaller and lossier), --y-block-size, --chroma-block-size, --coeff-removal, and --width with --height for input that cannot say how large it is. Add -v/-vv for progress logging, and see walsh compress -h for the full list.

Writes are atomic: output goes to a temporary file beside the destination and replaces it only on success, so a failed run leaves an existing file untouched. walsh also refuses to write over its own input, since both pipelines read the whole image before writing and would otherwise replace the original with a lossy reconstruction of itself.

The .cim container counts each channel's blocks in a 16-bit field, so at the default 8-pixel luma block an image must be under about 4.2 megapixels. Larger images are refused with a message naming the block size that would fit them: --y-block-size 16 roughly quadruples the ceiling, at some cost in detail.

--coeff-removal is the second, independent lossy knob: spectral coefficients smaller than the given magnitude are zeroed. It does not change the .cim file's size, because the format stores a fixed count of int16 values whether or not they are zero, but it makes the result far more compressible. On data/ppm/earth.ppm:

That table was measured at the default block sizes, 8 for luma and 16 for chroma, and the threshold is an absolute magnitude, so its effect depends on them. The surviving low-frequency coefficients grow with the block edge, and the same number prunes less at a larger one. The same --coeff-removal 25 on data/ppm/earth.ppm, with luma and chroma blocks set equal:

A threshold tuned against the first table and then combined with a larger --y-block-size has quietly stopped doing most of its work; retune it. Note also that it thresholds spectral coefficients, never the Hadamard matrix, whose entries all share one magnitude (0.35 at edge 8): a value at that scale is a no-op.

Reading the PSNR figures

PSNR is peak signal-to-noise ratio, the standard way to put a number on how much a lossy codec changed an image. It compares the reconstruction against the original pixel by pixel:

PSNR = 10 * log10(255**2 / MSE)

where MSE is the mean squared difference across every channel of every pixel, and 255 is the largest value an 8-bit channel can hold. It is measured in decibels, and higher is better: a perfect reconstruction has infinite PSNR, and every 3 dB gained means the mean squared error was halved.

Because the scale is logarithmic, small-looking differences matter. Going from 23 dB to 25 dB is not an 8% improvement, it is roughly a 37% reduction in error power. Equally, the near-identical 25.07 and 25.06 in the table above mean the first step of coefficient removal cost essentially nothing.

Rough expectations for 8-bit images, though they vary by content:

The figures here sit around 25 dB because the defaults are aggressive: each 8x8 luma block keeps 16 of its 64 coefficients and each 16x16 chroma block keeps 16 of 256. Raise --packed-block-size for a gentler setting.

One caveat worth knowing: PSNR measures arithmetic difference, not perceived quality. It is reproducible and easy to compare, which is why it is quoted here, but two images with the same PSNR can look noticeably different — it under- weights structured artefacts like block edges, which the eye picks out readily. Treat it as a consistent yardstick for comparing settings of this codec rather than an absolute measure of how good an image looks.

Exit codes follow the usual convention: 0 on success, 1 when the input cannot be processed (not a 24-bit BMP, truncated, unreadable), and 2 for a usage error such as a missing file or an unknown option.

As a library

from walsh import Codec

Codec().compress(input="data/bmp/image.bmp", output="out.cim").run()
Codec().extract(input="out.cim", output="back.bmp").run()

compress and extract say what to do and name the input and the output; nothing is read or written until run(). The settings go on the Codec itself, in the constructor or chained before run():

Codec(packed_block_size=2).with_coeff_removal(40).compress(
    input="photo.png", output="photo.cim"
).run()

Code written for 0.5.0 or earlier needs two small changes, listed under 0.5.1 in the changelog.

Skipping the .cim file

Give compress a picture as its output and the .cim never reaches the disk:

Codec().compress(input="data/ppm/earth.ppm", output="earth_compressed.ppm").run()

The picture goes through the whole codec in memory (colour conversion, transform, the crop to the kept coefficients, the rounding to the container's 16-bit integers, and back), and its lossy reconstruction is written. It is the same file, byte for byte, as compressing to a .cim and extracting that: the decoder is handed the very bytes the .cim would have held. Every setting applies, so it is also the short way to compare settings or transforms:

for name in ("walsh", "dct", "haar"):
    Codec(transform=name).compress(input="earth.ppm", output=f"earth_{name}.ppm").run()

What decides is the output's suffix. A picture suffix (.bmp, .png, .ppm and the rest of the table under File formats) writes the reconstruction; anything else, .cim by convention, writes the container.

The picture written need not be the format of the picture read. The codec works on pixels, which no format owns, so the output's suffix alone chooses the writer, as it does for extract:

Codec().compress(input="earth.ppm", output="earth_compressed.png").run()
Codec().compress(input="photo.png", output="photo_compressed.bmp").run()

All 36 pairs of the sample's formats are in the golden tests, each held to the checked-in reconstruction for its target, so where a picture came from leaves no trace in what is written.

Looking at the vectors

Codec goes from one file to another. Vectorizer stops in the middle, where the picture is a table of numbers, and hands you the table:

from walsh import Vectorizer

vectorizer = Vectorizer(transform="walsh").parse(file_name="data/png/earth.png").compute()

vectorizer.vectors  # int16, shape (3750, 16): one row per block
print(vectorizer.describe())
vectorizer.save(output_file_name="earth.cim")

picture       400 x 400
transform     WalshHadamardTransform
blocks        Y 8, Cb 16, Cr 16; 4 x 4 kept of each
vectors       3,750 of 16 (Y 2,500, Cb 625, Cr 625)
coefficients  60,000, 12.50% of the picture's samples; 48,122 non-zero (80.2%)
raw pixels    480,000 B
source file   301,514 B
compressed    120,026 B, 6.00 bits per pixel
reduction     75.0% smaller than the raw pixels, 60.2% smaller than the source file
PSNR          25.07 dB (mean squared error 202.13, largest error 143 of 255)

Each block of the picture becomes one vector: the coefficients the codec keeps of it, low frequencies first. Every channel keeps the same number per block, so they all fit one array, the luma blocks first, then Cb, then Cr. That is the order of the .cim file, and the array is int16 because the file is, so vectors.tobytes() is exactly the file after its 26-byte header, and save() writes the very .cim that Codec().compress() would.

Because the vectors are the state, changing them changes everything after them, which makes this a bench for experiments. Keep only each block's mean and see what that costs:

vectorizer.vectors[:, 1:] = 0
print(vectorizer.describe().psnr_db)  # 25.07 before, 19.37 now
vectorizer.save(output_file_name="earth_means.cim")

After load() there is no original picture to compare with, so describe() reports no PSNR; and since a .cim does not record its transform, reconstruct() inverts with whichever transform the Vectorizer was given. The constructor takes what Codec takes, plus coeff_removal=.

Other transforms

Codec takes the block transform as a keyword, so another transform can reuse the whole pipeline — the colour conversion, the padding, the crop to the low-frequency corner, the container — with only the transform swapped. Three ship with the package and are selected by name, in any case:

from walsh import Codec

Codec(transform="dct").compress(input="data/ppm/earth.ppm", output="dct.cim").run()
Codec(transform="dct").extract(input="dct.cim", output="back.ppm").run()

An unknown name is a ValueError that lists the known ones. "dct" and "haar" are ordinary floating-point matrix products, accurate to rounding; only "walsh" carries the bit-exactness guarantee.

The .cim does not record which transform wrote it. A file written with anything but the default must be extracted by a Codec given the same transform. The walsh command never takes one, and will decode such a file without complaint into a degraded picture. This keyword is for experiments, not for files you hand to someone else.

examples/compare_transforms.py runs them over the Blue Marble sample. The byte count depends on the geometry alone, so each row is a like-for-like comparison of how much picture a transform packs into its first few coefficients:

The DCT wins throughout, which is why JPEG uses it; Walsh-Hadamard needs no multiplications and is exact. Haar ties Walsh-Hadamard wherever the kept size is a power of two, and that is mathematics rather than coincidence: the first 2, 4 or 8 Walsh functions and the first 2, 4 or 8 Haar functions span the same piecewise-constant subspace, so the two projections are the same picture.

Writing your own

The last column of that table is not in the package. Anything that subclasses Transform can be passed as an instance, and for a separable orthonormal transform MatrixTransform needs only the matrix:

import numpy as np
from walsh import MatrixTransform, Codec


class Hartley(MatrixTransform):
    """cas(2*pi*i*k/n) / sqrt(n), with cas = cos + sin."""

    def matrix(self, size):
        angle = 2 * np.pi * np.outer(np.arange(size), np.arange(size)) / size
        return (np.cos(angle) + np.sin(angle)) / np.sqrt(size)


Codec(transform=Hartley()).compress(input="data/ppm/earth.ppm", output="hartley.cim").run()

It trails the others for an instructive reason: the codec keeps the top-left corner of each spectrum, which assumes rows rise in frequency, and a Hartley matrix puts half of its low frequencies in its last rows. A transform that is not a matrix product subclasses Transform directly and implements transform and inverse_transform for one square block; Codec calls transform_stack and inverse_transform_stack, whose defaults loop over the blocks, so override those when a whole (count, edge, edge) stack can go through in one operation. with_coeff_removal is applied by the codec, so it works for any transform.

Examples

examples/roundtrip.py compresses the sample image, restores it, and plots both images with their histograms side by side (needs the demo extra):

python examples/roundtrip.py

examples/compare_transforms.py prints the table above for any image, and needs numpy only:

python examples/compare_transforms.py [image]

Requirements

The package needs numpy and click -- every format is parsed by hand with struct, PNG included, whose compression is the standard library's zlib, and click powers the command line interface. matplotlib and Pillow are needed only by the example script, and are declared as the demo extra. Versions are pinned in pyproject.toml; requirements.txt, requirements-demo.txt and requirements-dev.txt mirror them for plain pip install -r workflows, and tests/project/test_requirements_mirror.py fails if the two ever disagree. On pip 25.1 or newer, pip install -e ".[demo]" --group dev reads the same groups straight from pyproject.toml and needs no mirror at all.

Development commands

uv run pytest                          # test suite
uv run pytest --cov --cov-report=term-missing   # with coverage
uv run ruff check .                    # lint
uv run ruff format .                   # format
uv run mypy                            # strict type check
uv build                               # sdist + wheel into dist/

CI runs exactly these on every pull request, plus the test suite against Python 3.10 through 3.14.

Coverage

Coverage is measured with branch coverage on, and CI enforces a floor of 90% on every supported Python version. A pull request that drops below it fails the test jobs, which are required checks on master — so the badge above states what is actually guaranteed rather than a number that could drift.

Coverage is opt-in locally (--cov) so a plain pytest stays fast; CI always passes it.

Releasing

The version in pyproject.toml is the single source of truth. To cut a release, bump it, add the matching ## [x.y.z] section to CHANGELOG.md, and merge to master. The release workflow then tags v<version>, creates a GitHub Release with those notes, and publishes the sdist and wheel to PyPI using Trusted Publishing — no API token is stored in this repository.

Merges that do not change the version are a no-op, since PyPI permanently refuses to accept the same version twice.

File formats

Input and output

Every reader presents the same in-memory view -- RGB pixels, top row first -- whatever the file itself stores. BMP is the awkward one on both counts, storing blue-green-red samples in bottom-up rows, and BMPImage converts in each direction. That shared contract is what makes cross-format conversion work.

PNG

PNG is the one compressed format here, and the only one most people have pictures in. It costs the package nothing: a PNG's pixel rows sit in a zlib stream, the inflater is zlib in the standard library, and everything around it (chunks, CRCs, the five row filters) is read by hand, so there is still no image library behind walsh. The compression is lossless, so a PNG reaches the transform as exactly the pixels a PPM of the same picture would: data/png/earth.png, written by libpng, compresses to a .cim byte-identical to the one from data/ppm/earth.ppm.

walsh compress photo.png out.cim
walsh extract  out.cim restored.png

Three of the five filters predict a byte from the pixel to its left, which was itself predicted, so a row cannot be undone in one step and the textbook decoder is a loop over every byte. Here the image is undone one anti-diagonal at a time instead: a pixel needs only its left, upper and upper-left neighbours, all of which lie on the two diagonals before its own, so each diagonal is a single array operation. A 2000x2000 PNG from libpng loads in about half a second, where a byte loop in Python takes five to twelve.

Transparency is refused, not flattened. Dropping an alpha channel means choosing a background to put behind it, and nothing in the file says which, so an RGBA file is read only when every pixel is opaque, and the message counts the pixels that are not:

$ walsh compress logo.png out.cim
Error: PNG has real transparency: alpha is below 255 in 1840 of 65536 pixels; only RGBA that is opaque throughout is supported

The files written are 8-bit RGB with a filter chosen per row, the same choice libpng makes, which keeps them 14% smaller on the samples here and half the size on a smooth picture. Their bytes are not reproducible from one machine to the next, because DEFLATE output may differ between zlib builds. Their pixels are, and that is how data/png/recreated.png is pinned.

A small file cannot cost much memory: inflation stops at the size the header declares, and nothing is allocated from the header, only from what the stream delivers.

Pickled pixels

pickle.load and numpy.load(allow_pickle=True) run the program a pickle contains, with the power to import any module and call anything in it, so opening an untrusted pickle is running untrusted code. This package never does that. It reads the same files through an allowlist: lists, tuples, dicts, numbers and bytes need no lookups at all, and the only names a file may refer to are the handful NumPy's own pickles use to rebuild an array. Anything else is refused by name before it is called:

$ walsh compress evil.pkl out.cim
Error: unsupported pickle: it refers to posix.system; only lists, tuples, integers and NumPy arrays are read, and nothing in a pickle is ever executed

What a .pkl or .pickle may hold:

Lists and tuples are interchangeable at every level. Samples must be integers in 0-255, Python's or NumPy's; floats, booleans, out-of-range values and ragged rows are refused by name. A declared size is never silently dropped: input that carries its own size, in any format, must match it.

from walsh import Codec

Codec().with_input_size(400, 300).compress(input="pixels.pkl", output="out.cim").run()

An object array saved by numpy.save, which only numpy.load(allow_pickle=True) opens, is a pickle inside a .npy and is read the same way. The writer produces rows of (r, g, b) tuples of plain int at protocol 4: any Python loads that without NumPy, and its bytes do not depend on which NumPy wrote it.

The .cim container

compress writes a .cim file: an atypical, project-specific container, so most commercial tools will not be able to read it. It stores the image dimensions, three block-layout descriptions (Y, Cb, Cr), and the retained Walsh-Hadamard coefficients as little-endian int16.

Note. .cim files written by 0.1.x are not compatible with 0.2.0. The in-memory pixel contract changed, so an old file extracted with 0.2.0 comes back with red and blue swapped and vertically flipped. Re-compress from the source image instead.