EXPLICIT CALCULATION OF SINGULAR INTEGRALS OF TENSORIAL POLYADIC KERNELS

The Riesz transform of u : S(Rn) & RARR; S?(Rn) 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 S(Rn) & RARR; S?(Rnxnx...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.

Automated summary

This study extends the Riesz transform to higher order Riesz transforms and proposes an integral transform method for function transformation. By computing the Fourier multiplier, a recursive algorithm for the coefficients of the transformed kernel is obtained, and experimental results are presented.