Explicit calculation of singular integrals of tensorial polyadic kernels

TL;DR

This paper develops a method to explicitly compute singular integrals involving tensorial polyadic kernels, extending the classical Riesz transform analysis to higher-order tensor transforms with applications in image analysis.

Contribution

It introduces a general approach to explicitly calculate Fourier multipliers for higher-order tensorial Riesz transforms, including a recursive algorithm for kernel coefficients.

Findings

Explicit formulas for transformed kernels are derived.

A recursive algorithm for coefficient computation is proposed.

Application to image analysis demonstrates practical relevance.

Abstract

The Riesz transform of $u$ : $\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S'}(\mathbb{R}^n)$ is defined as a convolution by a singular kernel, and can be conveniently expressed using the Fourier Transform and a simple multiplier. We extend this analysis to higher order Riesz transforms, i.e. some type of singular integrals that contain tensorial polyadic kernels and define an integral transform for functions $\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S'}(\mathbb{R}^{ n \times n \times \dots n})$. We show that the transformed kernel is also a polyadic tensor, and propose a general method to compute explicitely the Fourier mutliplier. Analytical results are given, as well as a recursive algorithm, to compute the coefficients of the transformed kernel. We compare the result to direct numerical evaluation, and discuss the case $n=2$, with application to image analysis.