Non-Negative Sparse Mathematical Morphology

International audienceSparse modeling involves constructing a succinct representation of initial data as a linear combination of a few typical atoms of a dictionary. This paper deals with the use of sparse representations to introduce new nonlinear image filters which efficiently approximate morphological operators. Reasons why non-negative matrix factorization (NMF) is a dimensional reduction (i.e., dictionary learning) paradigm particularly adapted to the nature of morphological processing are given. In particular, Sparse-NMF representations are studied and used to introduce first approximations to binary dilations/erosions and then to openings/closings. The idea behind consists of processing exclusively the image dictionary and then, the result of processing each image is approximated by multiplying the processed dictionary by the coefficient weights of the current image. These operators are then extended to gray-scale images and their interest for feature detection is illustrated. The practical relevance of our approach is considered for two applications on multivariate image processing. The first case deals with multispectral texture modeling using Boolean random set theory; the second case with multi-scale decomposition of hy-perspectral images and its interest in spectral-spatial pixel classification