Explicit calculation of singular integrals of tensorial polyadic kernels : an extension of Riesz transform for image analysis

The Riesz transform of a function is defined as a convolution by a singular kernel, and can be conveniently expressed using the Fourier Transform and a simple multiplier. It is widely used in image treatment / analysis.

We extend these works to higher order Riesz transforms, i.e. some type of singular integrals that contain tensorial polyadic kernels. We compute their Fourier Transform analytically.

This opens application in image processing (dimension 2), but also in convolution neural networks, so as to compute ANALYTICALLY the convolution integrals with some singular kernels of higher dimensions.

This Archive proposes a set of Matlab programs, using recursive programing, that actually makes this analytical computation and permits to compares the result to a direct computation (slow and unaccurate).

An exemple in the field image processing is provided.


Companion article :
M. Perrin, F. Gruy, "Explicit Calculation of Singular integrals of tensorial polyadic kernels", Quarterly of App. Math. (in press).

See also https://arxiv.org/abs/2209.01111