We first explore recent developments in multiresolution analysis. Essential terminology is introduced in the scope of our general overview, which includes the coverage of sparsity and sampling, best dictionaries, overcomplete representation and redundancy, compressed sensing and sparse representation, and morphological diversity.

Then we describe a range of applications of visualization, filtering, feature detection, and image grading. Applications range over Earth observation and astronomy, medicine, civil engineering and materials science, and image databases generally.

SPARSE REPRESENTATION

Introduction

In the last decade, sparsity has emerged as one of the leading concepts in a wide range of signal-processing applications (restoration, feature extraction, source separation, and compression, to name only a few applications). Sparsity has long been an attractive theoretical and practical signal property in many areas of applied mathematics (such as computational harmonic analysis, statistical estimation, and theoretical signal processing).

Recently, researchers spanning a wide range of viewpoints have advocated the use of overcomplete signal representations. Such representations differ from the more traditional representations because they offer a wider range of generating elements (called atoms). Indeed, the attractiveness of redundant signal representations relies on their ability to economically (or compactly) represent a large class of signals. Potentially, this wider range allows more flexibility in signal representation and adaptivity to its morphological content and entails more effectiveness in many signal-processing tasks (restoration, separation, compression, and estimation). Neuroscience also underlined the role of overcompleteness.