  Introducing spatial prior information in hyperspectral imaging (HSI) analysis
has led to an overall improvement of the performance of many HSI methods
applied for denoising, classification, and unmixing. Extending such
methodologies to nonlinear settings is not always straightforward, specially
for unmixing problems where the consideration of spatial relationships between
neighboring pixels might comprise intricate interactions between their
fractional abundances and nonlinear contributions. In this paper, we consider a
multiscale regularization strategy for nonlinear spectral unmixing with
kernels. The proposed methodology splits the unmixing problem into two
sub-problems at two different spatial scales: a coarse scale containing
low-dimensional structures, and the original fine scale. The coarse spatial
domain is defined using superpixels that result from a multiscale
transformation. Spectral unmixing is then formulated as the solution of
quadratically constrained optimization problems, which are solved efficiently
by exploring their strong duality and a reformulation of their dual cost
functions in the form of root-finding problems. Furthermore, we employ a
theory-based statistical framework to devise a consistent strategy to estimate
all required parameters, including both the regularization parameters of the
algorithm and the number of superpixels of the transformation, resulting in a
truly blind (from the parameters setting perspective) unmixing method.
Experimental results attest the superior performance of the proposed method
when comparing with other, state-of-the-art, related strategies.
