A Two-dimensional Inverse Frame Operator Approximation Technique
This work provides a theoretical and practical extension of a numerical frame approximation technique to two-dimensional problems, which is incremental for the field of applied mathematics.
The authors extend the admissible frame method to two dimensions, proving convergence and demonstrating its effectiveness for non-uniform Fourier sampling with numerical experiments.
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.