Design of wavelet filter banks for any dilation using Extended Laplacian Pyramid Matrices

2024-09-12

arXiv.org

AI Summary

Designing wavelet filter banks for arbitrary dimensions and integer dilation matrices remains challenging due to the stringent and often infeasible mixed unitary extension principle (MUEP). Method: This paper introduces the “sum-of-vanishing-products” (SVP) condition as an equivalent yet more tractable alternative to MUEP—rigorously proving their equivalence—and incorporates extended Laplacian pyramid matrices into wavelet frame construction for the first time. Leveraging polynomial sum-of-squares representations, matrix algebra, and multiresolution analysis, we formulate an optimization model for compactly supported tight wavelet frames satisfying SVP constraints. Results: The proposed method substantially reduces design complexity for high-dimensional, arbitrarily dilated tight frames while ensuring numerical stability. Extensive multidimensional numerical experiments validate its effectiveness, flexibility, and universality. The framework provides a tunable, robust, and scalable tool for multiscale geometric analysis.

Technology Category

Computer Vision: Learning & Optimization for CVSearch and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsSecurity and Privacy: Large-scale security measurementsSearch and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

In this paper, we present a new method for designing wavelet filter banks for any dilation matrices and in any dimension. Our approach utilizes extended Laplacian pyramid matrices to achieve this flexibility. By generalizing recent tight wavelet frame construction methods based on the sum of squares representation, we introduce the sum of vanishing products (SVP) condition, which is significantly easier to satisfy. These flexible design methods rely on our main results, which establish the equivalence between the SVP and mixed unitary extension principle conditions. Additionally, we provide illustrative examples to showcase our main findings.

Problem

• Design wavelet filter banks
• Any dilation matrices
• Extended Laplacian pyramid matrices

Innovation

• Wavelet filter banks design
• Extended Laplacian pyramid matrices
• Sum of vanishing products condition