asiryan / Weyl-Heisenberg-Toolbox

MATLAB Toolbox for digital signal processing using Weyl-Heisenberg bases.

Weyl-Heisenberg Toolbox

This MATLAB toolbox was developed to study the Weyl-Heisenberg bases and their properties in the field of signal processing and compression in time-frequency domain. The toolbox includes m-files that allow the synthesis and analysis of forming WH-functions and their corresponding orthogonal bases.

Installation

Add toolbox to your MATLAB path

addpath('[...] /matlab/toolbox_scripts')
addpath('[...] /matlab/toolbox_samples')

Introduction

We propose a new approach to raster image compression based on the use of an orthogonal WH basis specially optimized for the task of processing real images. To do this, we construct the two-dimensional real discrete Weyl-Heisenberg transform (DWHT), which has the property of orthogonality and the best time-frequency localization. We show that the DWHT provides better compression characteristics than other orthogonal bases - DCT and discrete Hartley transform (DHT). Moreover, we propose an image compression algorithm based on the JPEG ideas, but focused on DWHT.

Orthogonalization

It is well known that the gaussian function is ideally localized in the time-frequency domain. However, the Weyl-Heisenberg basis built on gaussian function (Gabor basis) is not orthogonal. However, the use of a standard orthogonalization procedure, for example, the Gram-Schmidt process, will lead to a significant deterioration in the time-frequency localization, and, as a consequence, such an orthogonal Weyl-Heisenberg basis will not be optimal. In old papers, an algorithm is proposed for constructing an Weyl-Heisenberg basis using the eigenvalue decomposition, but it is is not computationally efficient. We present fast algorithm for the synthesis of the forming WH-function based on discrete Zak transform.

This MATLAB toolbox includes all procedures of basis orthogonalizations. Run sample script to construct Gabor and Weyl-Heisenberg bases.

Number of frequency shifts M = 8
Number of time shifts L = 8
Beta-parameter = 1

Construction

This toolbox provides methods to construct Weyl-Heisenberg basis and its modifications.

Script	Orthogonal	Type	Dimension	Notation
gabor.m	no	complex	N x 2N	Gabor basis (rectangular)
gaborc.m	no	complex	N x N	Gabor basis (square)
gaborr.m	no	real	2N x 2N	Gabor basis (real)
weylhz.m	yes	complex	N x 2N	Weyl-Heisenberg basis (rectangular)
weylhzc.m	yes	complex	N x N	Weyl-Heisenberg basis (square)
weylhzr.m	yes	real	2N x 2N	Weyl-Heisenberg basis (real)
weylhzf.m	yes	complex	N x 2N	Weyl-Heisenberg basis (fast algorithm)
weylhzg.m	yes	complex	N x 2N	Weyl-Heisenberg basis (Gram-Schmidt)
weylhzh.m	yes	real	N x N	Weyl-Heisenberg basis (Hartley)

Compression

This toolbox includes a new approach to image compression based on the discrete Weyl-Heisenberg transform. Comparison of our technique and the original JPEG (DCT-based) algorithm for different values of the quantization coefficient R is presented below. Figures 1-2 show, as an example, the results of compressing the same image barbara.png (512×512 pixels) at R = 1, 4, 8 using DCT (JPEG) and DWHT, and Tables 1-2 show extended numerical results.

Figure 1. Compression using DCT (JPEG) for R = 1, 4, 8

Figure 2. Compression using DWHT (β = 2) for R = 1, 4, 8

Table 1. Compression using DCT (JPEG)

Quantization coefficient, R	1	2	4	8	16	32
Compression coefficient, K (%)	83.73	89.43	93.82	96.63	98.06	98.99
PSNR (dB)	32.53	29.30	26.26	23.92	22.01	19.06
Quality losses, E	438.58	660.72	980.48	1399.16	1601.70	3404.39

Table 2. Compression using DWHT (β = 2)

Quantization coefficient, R	1	2	4	8	16	32
Compression coefficient, K (%)	86.18	91.22	94.62	96.73	97.94	98.56
PSNR (dB)	34.19	31.19	28.27	25.57	23.48	21.91
Quality losses, E	326.86	494.31	737.01	1073.84	1451.38	1796.39

Other implementations

UMapx - Digital signal processing library for .NET Core and .NET Framework. Fast and optimized.

References

1. D. Petrov, V. Volchkov - Orthogonal well-localized Weyl-Heisenberg basis construction and optimization for multicarrier digital communication systems (2009).
2. V. Volchkov, V. Sannikov - Algebraic approach to the optimal synthesis of real signal Weyl-Heisenberg bases (2018).
3. H. Bolcskei, F. Hlawatsch - Discrete Zak Transforms, Polyphase Transforms, and Applications (1997).
4. A. Vahlin - Efficient Algorithms for Modulation and Demodulation in OFDM-Systems (2003).
5. V. Volchkov et al. - Synthesis of Real Weyl-Heisenberg Signal Frames with Desired Frequency-Time Localization (2019).
6. V. Asiryan et al. - Image Compression Using Discrete Weyl-Heisenberg Transform (2020).
7. V. Asiryan et al. - Weyl-Heisenberg Transform Capabilities in JPEG Compression Standard (2021).

License

Apache License v2.0