Quantization and finite frames

Abstract

Frames are a tool for providing stable and robust signal representations in a wide variety of pure and applied settings. Frame theory uses a set of frame vectors to discretely represent a signal in terms of its associated collection of frame coefficients. Dual frames and frame expansions allow one to reconstruct a signal from its frame coefficients—the use of redundant or overcomplete frames ensures that this process is robust against noise and other forms of data loss. Although frame expansions provide discrete signal decompositions, the frame coefficients generally take on a continuous range of values and must also undergo a lossy step to discretize their amplitudes so that they may be amenable to digital processing and storage. This analog-to-digital conversion step is known as quantization. We shall give a survey of quantization for the important practical case of finite frames and shall give particular emphasis to the class of Sigma-Delta algorithms and the role of noncanonical dual frame reconstruction.