Mathematics for Multimedia Signal Processing II: Discrete Finite Frames and Signal Reconstruction

Certain signal reconstruction problems can be understood in terms of frames and redundant representations. The redundancy is useful because it leads to robust signal representations, that is, representations in which partial loss of data can be tolerated without misbehavior or adverse effects. This chapter begins by presenting a few engineering problems in which robust data representations play a central role. It turns out that these problems, which occur in signal processing, spectrum analysis, information theory, and fault-tolerant computing, are closely related or even equivalent. However, perhaps surprisingly, the connections between them have gone nearly unnoticed so far. Frames, and in particular discrete finite frames, provide one of the ways of understanding certain of these problems, including the important missing data problem. Some of the methods that can be used to recover from missing data errors are examined, emphasizing finite-dimensional theory because of its simplicity and practical usefulness, and interpreting the results in terms of discrete finite frames. The connection between the frame algorithm and a few other iterative reconstruction methods, such as POCS and the Papoulis-Gerchberg iteration, is detailed.