Tight frames, partial isometries, and signal reconstruction

TL;DR

This paper presents a method to convert non-tight frames into Parseval frames with explicit linear combinations, enabling improved signal reconstruction and error estimation in finite measurement scenarios.

Contribution

It introduces a procedure to transform arbitrary frames into Parseval frames with explicit formulas, applicable to various function spaces and practical measurement settings.

Findings

Conversion procedure for frames to Parseval frames

Explicit formulas for linear combinations in the new frames

Error estimates for finite measurement-based reconstruction

Abstract

This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The procedure can be applied to the construction of Parseval frames for L^2(B(0,R)), the space of square integrable functions whose domain is the ball of radius R. When a finite number of measurements are used to reconstruct a signal in L^2(B(0,R)), error estimates arising from such approximation are discussed.