Modern data are often composed of two or more morphologically distinct constituents, and one typical goal is the extraction of those components. Recently, sparsity methodologies have been successfully utilized to solve this problem, both theoretically as well as empirically. The key idea is to choose a deliberately overcomplete representation made of several frames each one providing a sparse expansion of one of the components to be extracted. The morphological difference between the components is then encoded as incoherence conditions of those frames. The decomposition principle is to minimize the ℓ1 norm of the frame coefficients. This chapter shall serve as an introduction to and a survey of this exciting area of research as well as a reference for the state of the art of this research field.

Introduction

Over the last few years, scientists have faced an ever growing deluge of data, which needs to be transmitted, analyzed, and stored. A close analysis reveals that most of these data might be classified as multimodal data, i.e., being composed of distinct subcomponents. Prominent examples are audio data, which might consist of a superposition of the sounds of different instruments, or imaging data from neurobiology, which is typically a composition of the soma of a neuron, its dendrites, and its spines. In both these exemplary situations, the data has to be separated into appropriate single components for further analysis. In the first case, separating the audio signal into the signals of the different instruments is a first step to enable the audio technician to obtain a musical score from a recording.