[hal-00813969, v1] Morphological Diversity and Sparsity for Multichannel Data Restoration

Over the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichannel data are of growing interest, it seems essential to devise sparsity-based tools accounting for such specific multichannel data. Sparsity has proved its efficiency in a wide range of inverse problems. Hereafter, we address some multichannel inverse problems issues such as multichannel morphological component separation and inpainting from the perspective of sparse representation. In this paper, we introduce a new sparsity-based multichannel analysis tool coined multichannel Morphological Component Analysis (mMCA). This new framework focuses on multichannel morphological diversity to better represent multichannel data. This paper presents conditions under which the mMCA converges and recovers the sparse multichannel representation. Several experiments are presented to demonstrate the applicability of our approach on a set of multichannel inverse problems such as morphological component decomposition and inpainting. Introduction This paper addresses several multichannel data recovery problems such as multichannel morphological component decomposition and inpainting. We first need to define with care what multichannel data are. Such data are often physically composed of m observations (a colour layer in colour images, an observation at a fixed frequency for multispectral data and so on). One classical example of such multichannel data are the hyperspectral data provided by satellite observations; a fixed geographic area is observed at m different frequencies. More formally, we assume that each observation is made of t samples. We will write each observation as a 1 × t row vector {xi}i=1,··· ,m. For convenience, those m vectors are stacked in a m × t matrix X = [ x1 · · ·x T m ]T . In a wide range of applications, the data X are often degraded by the acquisition system (convolution, missing ha l-0 08 13 96 9, v er si on 1 16 A pr 2 01 3 Author manuscript, published in "Journal of Mathematical Imaging and Vision 33, 2 (2009) 149-168" DOI : 10.1007/s10851-008-0065-6