OTB
6.7.0
Orfeo Toolbox

#include <otbMDMDNMFImageFilter.h>
This filter implements unmixing based non negative matrix factorization (NMF) which finds simultaneously the end members and abundances matrix which product is the closer to the observed data, based on the following works: K. S. F.J. Theis and T. Tanaka, First results on uniqueness of sparse nonnegative matrix factorisation. M. G. A. Huck and J. BlancTalon, IEEE TGRS, vol. 48, no. 6, pp. 25902602, 2010. A. Huck and M. Guillaume, in WHISPERS, 2010, Grenoble.
Let be the matrix of the hyperspectral data, whose columns are the spectral pixels and the rows are the vectorial spectral band images. The linear mixing model can be written as follow : The columns of contain the spectral pixels and the columns of hold their respective sets of abundance fractions. The rows of are the abundance maps corresponding to the respective endmembers. The columns of are the end members spectra, and is the signal matrix. Both and are unknown.
The basic NMF formulation is to find two matrices and such as: NMF based algorithms consider the properties of the dual spaces and , in which . The positiveness is then a fundamental assumption and is exploited to restrict the admissible solutions set.
A common used solution is to minimize the reconstruction quadratic error : . In order to satisfy the sumtoone constraint for hyperspectral data, a regularization term can be added to the objective function.
A generic expression for the optimized function is in which is the connectiontothedata term, and are regularization parameters for end members and abundances constraints . Huck propose an other regularization term, , which ensures low spectral dispersion on endmembers. The physical motivation is based on the assuption that in most situations, the whole set of pure materials do not appear in each pixel, but selectively in multiple piecewise convex sets. As a consequence, the mean value of the abundance, , is the least likely one. The maximum abundance dispersion condition is given by . MDMDNMF algorithm minimizes the following function , the sumtoone constraint.
In the literature, NMF based optimization algorithms are mainly based on multiplicative rules, or else alternate gradient descent iterations, or else on alternate least square methods. In MDMDNMF, the update rules at each iteration become : where and are the step sizes. Huck propose a multiscale method to determine the coefficients and . The projection operator at each step ensures the positivity constraint for and , and and include the sumtoone constraint: .