Extracting non-negative basis images using pixel dispersion penalty

Non-negativity matrix factorization (NMF) and its variants have been explored in the last decade and are still attractive due to its ability of extracting non-negative basis images. However, most existing NMF based methods are not ready for encoding higher-order data information. One reason is that they do not directly/explicitly model structured data information during learning, and therefore the extracted basis images may not completely describe the ''parts'' in an image [1] very well. In order to solve this problem, the structured sparse NMF has been recently proposed in order to learn structured basis images. It however depends on some special prior knowledge, i.e. one needs to exhaustively define a set of structured patterns in advance. In this paper, we wish to perform structured sparsity learning as automatically as possible. To that end, we propose a pixel dispersion penalty (PDP), which effectively describes the spatial dispersion of pixels in an image without using any manually predefined structured patterns as constraints. In PDP, we consider each part-based feature pattern of an image as a cluster of non-zero pixels; that is the non-zero pixels of a local pattern should be spatially close to each other. Furthermore, by incorporating the proposed PDP, we develop a spatial non-negative matrix factorization (Spatial NMF) and a spatial non-negative component analysis (Spatial NCA). In Spatial NCA, the non-negativity constraint is only imposed on basis images and such constraint on coefficients is released, so both subtractive and additive combinations of non-negative basis images are allowed for reconstructing any images. Extensive experiments are conducted to validate the effectiveness of the proposed pixel dispersion penalty. We also experimentally show that Spatial NCA is more flexible for extracting non-negative basis images and obtains better and more stable performance.