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Matrix equation representation of the convolution equation and its unique solvability

Yuki Satake, Tomohiro Sogabe, Tomoya Kemmochi, Shao-Liang Zhang

TL;DR

This work shows that the convolution equation $F*X=B$ with a $3\times3$ kernel $F$ and $B\in\mathbb{R}^{m\times n}$ can be reformulated as a generalized Sylvester equation via shift matrices under zero, periodic, or reflexive boundary conditions. For common image-processing filters, the authors derive explicit necessary and sufficient conditions on $m$ and $n$ ensuring a unique solution for every right-hand side $B$, and they further reduce many cases to simpler Sylvester/Lyapunov forms. A comprehensive set of results is provided for box blur, Gaussian blur, edge detection (A, B, C), sharpen, and emboss, with several cases yielding unconditional solvability and others requiring parity constraints on $m$ and $n$. The findings pave the way for efficient solvers that exploit matrix structure, and they outline future work on numerical algorithms and the unresolved reflexive-emboss case, with potential impact on image restoration and related applications.

Abstract

We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.

Matrix equation representation of the convolution equation and its unique solvability

TL;DR

This work shows that the convolution equation with a kernel and can be reformulated as a generalized Sylvester equation via shift matrices under zero, periodic, or reflexive boundary conditions. For common image-processing filters, the authors derive explicit necessary and sufficient conditions on and ensuring a unique solution for every right-hand side , and they further reduce many cases to simpler Sylvester/Lyapunov forms. A comprehensive set of results is provided for box blur, Gaussian blur, edge detection (A, B, C), sharpen, and emboss, with several cases yielding unconditional solvability and others requiring parity constraints on and . The findings pave the way for efficient solvers that exploit matrix structure, and they outline future work on numerical algorithms and the unresolved reflexive-emboss case, with potential impact on image restoration and related applications.

Abstract

We consider the convolution equation , where and are given, and is to be determined. The convolution equation can be regarded as a linear system with a coefficient matrix of special structure. This fact has led to many studies including efficient numerical algorithms for solving the convolution equation. In this study, we show that the convolution equation can be represented as a generalized Sylvester equation. Furthermore, for some realistic examples arising from image processing, we show that the generalized Sylvester equation can be reduced to a simpler form, and analyze the unique solvability of the convolution equation.
Paper Structure (32 sections, 26 theorems, 79 equations, 1 figure, 1 table)

This paper contains 32 sections, 26 theorems, 79 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Rewriting the convolution equation to a generalized matrix equation
  3. Existence of unique solutions of the generalized Sylvester equations
  4. Box blur
  5. Zero boundary condition
  6. Periodic boundary condition
  7. Reflexive boundary condition
  8. Gaussian blur
  9. Zero boundary condition
  10. Periodic boundary condition
  11. Reflexive boundary condition
  12. Edge detect A
  13. Zero boundary condition
  14. Periodic boundary condition
  15. Reflexive boundary condition
  16. ...and 17 more sections

Key Result

Theorem 2.1

(Zero boundary condition) Let $F\in\mathbb{R}^{3\times 3}$, $X\in\mathbb{R}^{m\times n}$ and $B\in\mathbb{R}^{m\times n}$. Then, the convolution equation $F*X=B$ with the zero boundary condition is equivalent to a generalized Sylvester equation where and $U_n, L_n\in\mathbb{R}^{n\times n}$ are shift matrices shiftmatrix.

Figures (1)

  • Figure 1: Eigenvalues of $\mathcal{F}$\ref{['coef:embr']}. The dashed line represents $\mathrm{Re}(z) = 1, z\in\mathbb{C}$.

Theorems & Definitions (53)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 43 more