We first explore recent developments in multiresolution analysis. Essential terminology will be introduced in the scope of our general overview. This includes coverage of: sparsity and sampling; best dictionary; 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

1.1.1 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, compression, to name only a few). Sparsity has long been an attractive theoretical and practical signal property in many areas of applied mathematics (such as computational harmonic analysis, statistical estimation, 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 basis 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, estimation). Neuroscience also underlined the role of overcompleteness. Indeed, the mammalian visual system has been shown to be probably in need of overcomplete representation (Field 1999; Hyvarinen and Hoyer 2001; Olshausen and Field 1996a; Simoncelli and Olshausen 2001). In that setting, overcomplete sparse coding may lead to more effective (sparser) codes.

The interest in sparsity has arisen owing to the new sampling theory, compressed sensing (also called compressive sensing or compressive sampling), which provides an alternative to the well-known Shannon sampling theory (Candès and Tao 2006; Donoho 2006a; Candes et al. 2006b). Compressed sensing uses the prior knowledge that signals are sparse, while Shannon theory was designed for frequency band-limited signals. By establishing a direct link between sampling and sparsity, compressed sensing has had a huge impact in many scientific fields such as coding and information theory, signal and image acquisition and processing, medical imaging, geophysical and astronomical data analysis. Compressed sensing acts today as wavelets did two decades ago, linking together researchers from different fields.