Almost-Periodic Interpolation and Approximation

Abstract

In this chapter, following [41], we present a method to interpolate or approximate a given function $f:{\mathbb G}\rightarrow {\mathbb C}$ (or $F:{\mathbb H}\rightarrow {\mathbb C}$) by an AP functions in ${{\mathrm{AP}}}_F({\mathbb G})$, i.e., AP functions whose Fourier transform is supported in a given discrete and finite set $F\subset \widehat{\mathbb G}$. (See Sect. 2.3.1.) In order to do this, we generalize the well-known decomposition of the 2D Fourier transform on the plane in polar coordinates, via the Fourier-Bessel operator. In the first part of the chapter, exploiting the deep connection between (4.1) and the group of rototranslations SE(2), we generalize the former to ${{\mathrm{AP}}}_F({\mathbb G})$ functions. In particular, we show how a discrete operator that we call the generalized Fourier-Bessel operator plays a crucial role in this generalization. We then consider the problem of interpolating functions $\psi :{\mathbb G}\rightarrow {\mathbb C}$ on ${\mathbb K}$-invariant finite sets $\tilde{E}\subset {\mathbb G}$ via ${{\mathrm{AP}}}_F({\mathbb G})$ functions. The last part of the chapter is devoted to particularize (and slightly generalize) the above results to the relevant case for image processing, i.e., ${\mathbb G}=SE(2,N)$. Indeed we present numerical algorithms for the (exact) evaluation, interpolation, and approximation of ${{\mathrm{AP}}}_F(SE(2,N))$ functions on finite sets of spatial samples $\tilde{E}\subset {\mathbb R}^2$, invariant under the action of ${\mathbb Z}_N$. This is an instance of a very general problem, and can be seen as a generalization of the discrete Fourier Transform and its inverse, that act on regular square grids, i.e., invariant under the the action of SE(2, 4).