Explicit calculation of singular integrals of tensorial polyadic kernels

Mathias Perrin (1, 2) , Frédéric Gruy (3, 4)

1 LOMA - Laboratoire Ondes et Matière d'Aquitaine
2 UB - Université de Bordeaux
3 LGF-ENSMSE - Laboratoire Georges Friedel
4 SPIN-ENSMSE - Centre Sciences des Processus Industriels et Naturels

The Riesz transform of u $u$ : $\mathcal{S}(\R^n) \rightarrow \mathcal{S'}(\R^n)$ 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 $\mathcal{S}(\R^n) \rightarrow \mathcal{S'}(\R^{ n \times n \times \dots n})$. 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.

Domaines: Analyse numérique [math.NA]; Intelligence artificielle [cs.AI]; Vision par ordinateur et reconnaissance de formes [cs.CV]

Mathias Perrin, Frédéric Gruy. Explicit calculation of singular integrals of tensorial polyadic kernels. Quarterly of Applied Mathematics, 2022, 81 (1), pp.65 - 86. ⟨10.1090/qam/1629⟩. ⟨hal-03768198v1⟩