Explicit calculation of singular integrals of tensorial polyadic kernels
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The Riesz transform of u : 𝒮(ℝ^n) →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 𝒮(ℝ^n) →𝒮'(ℝ^ n × n ×… 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.
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