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New Orthogonal Multiwavelet Filters Derived by Matrix Spectral Factorization

This paper introduces two new orthogonal multiwavelet filters with supercompact support, constructed via matrix spectral factorization of the CL multiwavelet filter, which demonstrate superior performance in image compression and signal denoising compared to existing multifilters.

Original authors: Vasil Kolev, Todor Cooklev, Fritz Keinert

Published 2026-08-13

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CC BY 4.0

Original authors: Vasil Kolev, Todor Cooklev, Fritz Keinert

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a giant, high-definition movie across the internet, but the connection is shaky and slow. To make it work, you have to chop the movie into tiny, manageable puzzle pieces, compress them, and send them one by one. The problem is, if you chop them up too roughly, the picture gets blurry or blocky when you put it back together. This is the daily struggle of digital signal processing: how to break complex information (like images or sounds) into simple parts without losing the important details.

To solve this, scientists use mathematical tools called "wavelets." Think of a wavelet not as a single wave, but as a tiny, self-contained wave packet that can zoom in on specific details. A "multiwavelet" is like having a whole team of these tiny waves working together, each with a special job. Some are great at spotting sharp edges (like the outline of a building), while others are perfect at smoothing out gentle curves (like a sunset). The goal is to find the perfect team of waves that can compress data efficiently while keeping the picture looking crisp and real. For a long time, researchers have been trying to design these teams so they are "orthogonal" (they don't step on each other's toes) and "supercompact" (they are very small and efficient, like a pocket-sized toolkit).

This paper introduces two new, super-efficient teams of multiwavelets, named New1 and New2. The authors, a team of mathematicians and engineers, didn't just guess these new designs; they built them using a clever mathematical recipe called "matrix spectral factorization." Imagine they took an existing, well-known team of waves (called the CL multiwavelet), analyzed its "DNA" (its mathematical product filter), and then used a fast, precise method (the Fast Bauer's method) to re-engineer it into two brand-new, even better versions.

The researchers found that these new wavelets are incredibly good at their jobs, though the results are nuanced. When they tested them on compressing images (making files smaller) and cleaning up noisy signals (removing static from a recording), New1 frequently achieved the best results for structural quality (SSIM and MS-SSIM) and often led in noise suppression. However, it wasn't a total sweep; in specific scenarios, such as compressing the "Bart Run" image at a ratio of 8:1 or the "Kiel" image at 4:1, the older Alpert filter actually produced the best clarity and detail. Similarly, the SA4 filter frequently outperformed New1 in standard sharpness metrics (PSNR and SSIM) for certain images and compression levels. It's as if they found a new set of lenses that make the picture look sharper in many situations, but sometimes the older lenses still hold their ground depending on the specific scene and how much you want to shrink the file. The paper suggests that these new tools could be very useful for anyone who needs to send high-quality images or sounds over limited bandwidth, proving that sometimes, a little mathematical magic can make a big difference in how we see the world.

Technical Summary: New Orthogonal Multiwavelet Filters Derived By Matrix Spectral Factorization

Problem Statement
The paper addresses the construction of orthogonal multiwavelets with "supercompact" support, defined as having a support interval of $[0, 1]$ . While supercompact multiwavelets offer significant advantages for applications on finite intervals and piecewise continuous signals (such as near shock waves or boundaries) by eliminating the need for special boundary treatments, existing designs often face trade-offs between orthogonality, symmetry, vanishing moments, and coding gain. Specifically, the authors aim to derive new supercompact orthogonal multiwavelets with exact recursion coefficients that potentially outperform existing filters (such as GHM, SA4, CL, Alpert, and Integer Haar) in terms of coding gain, smoothness, and performance in image compression and denoising.

Methodology
The core methodology involves Matrix Spectral Factorization (MSF) applied to the matrix product filter of the existing Chui-Lian (CL) orthogonal multiwavelet filter.

Matrix Product Filter Construction: The authors first derive the matrix product filter $P(z) = H(z)H(z)^*$ from the known CL multiscaling function coefficients. This product filter is a positive semidefinite polynomial matrix satisfying the half-band condition.
Spectral Factorization via Fast Bauer's Method: To recover the new multiscaling filter $H(z)$ $H (z)$ from $P(z)$ $P (z)$ , the paper employs the Fast Bauer's method.
- The classical Bauer's method, which relies on the Cholesky decomposition of large block Toeplitz matrices, is noted to converge very slowly when the product filter possesses multiple zeros on the unit circle (a requirement for regularity in supercompact filters).
- The authors utilize the Fast Bauer's method, which iteratively solves a non-linear matrix equation (NME): $X = P_0 - P_1^* X^{-1} P_1$ .
- For the supercompact case ( $m=1$ ), this equation is solved using a Computer Algebra System (CAS) to obtain exact symbolic coefficients rather than numerical approximations. This avoids the precision loss associated with large-scale numerical iterations required by the classical method.
Wavelet Derivation: Once the new multiscaling functions (New1 and New2) are determined, the corresponding multiwavelet functions are derived using the QR decomposition (or the orthogonal vectors method) to ensure orthogonality and the desired symmetry/antisymmetry properties.

Key Contributions

Derivation of New Filters: The paper successfully derives two new orthogonal supercompact multiwavelet filters, designated New1 and New2, with exact recursion coefficients.
Exact Coefficients: By solving the spectral factorization equation symbolically, the authors provide exact mathematical expressions for the filter coefficients, avoiding the numerical errors inherent in iterative methods for high-multiplicity zeros.
Performance Comparison: The new filters are rigorously compared against established multifilters (GHM, SA4, CL, Alpert, Integer Haar) across several metrics:
- Coding Gain (CG) and Smoothness: New1 and New2 demonstrate higher coding gains and better Sobolev smoothness than CL and Integer Haar, though slightly lower than SA4 and GHM.
- Image Compression: Extensive testing on grayscale and color images (Bart Run, Cameraman, Kiel, House, Cara, Voit, Texture 1, Texture 4, Brick Wall, Subwater) at various compression ratios (CR) and decomposition levels. Performance is measured using MSE, PSNR, SSIM, and MS-SSIM.
- Signal Denoising: Evaluation on 1D signals (Piece-Polynomial, Cusp, HeaviSine) with additive white Gaussian noise (AWGN) and 2D image denoising using soft and hard thresholding.
- Edge Detection: Application to subband-based edge detection using the Canny method.

Results

Image Compression: The comparative analysis indicates that New1 (and to a lesser extent New2) frequently achieves superior or comparable performance to existing filters. Specifically, New1 often yields the best PSNR, SSIM, and MS-SSIM values, particularly at higher compression ratios and deeper decomposition levels. The paper notes that while Integer Haar or Alpert may sometimes achieve lower MSE, New1 often provides better structural similarity (SSIM/MS-SSIM), which correlates more closely with human visual perception.
Denoising: In 1D denoising, New1 and New2 demonstrate minimal Maximum Absolute Error (MAE) for various test signals and noise levels, outperforming GHM, SA4, CL, Alpert, and Integer Haar. In 2D image denoising, New1 consistently achieves high PSNR, SSIM, and MS-SSIM values across different noise variances ( $\sigma=10, 20$ ) and thresholding methods.
Edge Detection: In subband-based edge detection, the contour lines generated by the SA4 filter show that it is an excellent detector, performing better than the other six transformations tested (including the new filters, GHM, CL, Alpert, and Integer Haar).

Significance and Claims
The paper claims that the proposed New1 and New2 multifilters offer a robust alternative for applications requiring supercompact support. The primary significance lies in the successful application of the Fast Bauer's method to obtain exact coefficients for filters with multiple zeros on the unit circle, a task where classical methods struggle with convergence and precision.

The authors conclude that these new filters provide:

Effective Noise Suppression: Superior performance in denoising 1D and 2D signals compared to well-known orthogonal and biorthogonal multifilters.
Enhanced Image Quality: Better human visual measures (HVPs), SSIM, and MS-SSIM in image compression and denoising applications, suggesting improved preservation of image structure and detail.
Utility in Edge Detection: While the new filters were applied to subband-based edge detection, the results indicate that the SA4 filter remains the superior detector among those tested.

The paper maintains a modest tone, stating that the new filters "can provide" better performance in specific contexts and "can be useful" in the mentioned applications, rather than claiming universal superiority over all existing methods in every metric. The work suggests potential for these new supercompact bases in solving other mathematical and engineering problems in the future.

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