A Two-dimensional Inverse Frame Operator Approximation Technique

TL;DR

This paper extends the admissible frame method to two-dimensional inverse frame operator approximation, enabling efficient reconstruction from non-uniform Fourier samples with proven convergence and practical numerical demonstrations.

Contribution

It introduces a two-dimensional admissible frame approach with convergence proof and demonstrates its effectiveness through numerical experiments.

Findings

Convergence of the 2D admissible frame method is established.

Numerical experiments show successful reconstruction from non-uniform Fourier samples.

Method is applicable to practical sampling patterns inspired by real applications.

Abstract

The ability to efficiently and accurately construct an inverse frame operator is critical for establishing the utility of numerical frame approximations. Recently, the admissible frame method was developed to approximate inverse frame operators for one-dimensional problems. Using the admissible frame approach, it is possible to project the corresponding frame data onto a more suitable (admissible) frame, even when the sampling frame is only weakly localized. As a result, a target function may be approximated as a finite frame expansion with its asymptotic convergence solely dependent on its smoothness. In this investigation, we seek to expand the admissible frame approach to two dimensions, which requires some additional constraints. We prove that the admissible frame technique converges in two dimensions and then demonstrate its usefulness with some numerical experiments that use sampling patterns inspired by applications that sample data non-uniformly in the Fourier domain.