  The nonnegative matrix factorization (NMF) is widely used in signal and image
processing, including bio-informatics, blind source separation and
hyperspectral image analysis in remote sensing. A great challenge arises when
dealing with a nonlinear formulation of the NMF. Within the framework of kernel
machines, the models suggested in the literature do not allow the
representation of the factorization matrices, which is a fallout of the curse
of the pre-image. In this paper, we propose a novel kernel-based model for the
NMF that does not suffer from the pre-image problem, by investigating the
estimation of the factorization matrices directly in the input space. For
different kernel functions, we describe two schemes for iterative algorithms:
an additive update rule based on a gradient descent scheme and a multiplicative
update rule in the same spirit as in the Lee and Seung algorithm. Within the
proposed framework, we develop several extensions to incorporate constraints,
including sparseness, smoothness, and spatial regularization with a
total-variation-like penalty. The effectiveness of the proposed method is
demonstrated with the problem of unmixing hyperspectral images, using
well-known real images and results with state-of-the-art techniques.
