Analysis of data separation and recovery problems using clustered sparsity

Data often have two or more fundamental components, like cartoon-like and textured elements in images; point, filament, and sheet clusters in astronomical data; and tonal and transient layers in audio signals. For many applications, separating these components is of interest. Another issue in data analysis is that of incomplete data, for example a photograph with scratches or seismic data collected with fewer than necessary sensors. There exists a unified approach to solving these problems which is minimizing the ℓ₁ norm of the analysis coefficients with respect to particular frame(s). This approach using the concept of clustered sparsity leads to similar theoretical bounds and results, which are presented here. Furthermore, necessary conditions for the frames to lead to sufficiently good solutions are also shown. © 2011 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).