Approximating the Inverse Frame Operator from Localized Frames

TL;DR

This paper introduces a new method for approximating the inverse frame operator using well-localized frames, improving accuracy and robustness, with applications in MRI and a technique for frame data projection.

Contribution

It develops a practical numerical approximation method for the inverse frame operator and analyzes its convergence, including a frame projection technique for non-well-localized data.

Findings

Sampling with well-localized frames enhances accuracy and robustness.

The proposed method converges asymptotically based on function smoothness.

Numerical examples demonstrate the effectiveness of the approach.

Abstract

This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that sampling with {\em well-localized frames} improves both the accuracy of the numerical frame approximation as well as the robustness and efficiency of the (finite) frame operator inversion. Moreover, in applications such as magnetic resonance imaging, where the given data often may not constitute a well-localized frame, a technique is devised to project the corresponding frame data onto a more suitable frame. As a result, the target function may be approximated as a finite expansion with its asymptotic convergence solely dependent on its smoothness. Numerical examples are provided.