Blind de-mixing with unknown sources
Harold H Szu, C. Hsu · Proceedings of International Conference on Neural Networks (ICNN'97) · 2002
Multispectral and multisensor data processing requires the identification of the mixing of cluttered sources using a new multichannel de-mixing technique. As an application, we consider the contraband detection in which both the source mixing matrix W/sub ij/ (for the i-th object and the j-th channel) and V/sub j/, the sources in question, are unknown due to uncooperative subjects. For weak signals, we can assume a linear mixing model U/sub j/=/spl Sigma//sub j=i//sup N/W/sub (/ij)V/sub i/ of which the total probability of the set of unknowns V/sub j/ must be added up to one /spl Sigma//sub i+1//sup N/V/sub j/=1. We postulate that for i-1 each case of real positive unknowns (W/sub ij/ and V/sub j/) there exists a maximum entropy E(V/sub j/) constrained with the measurements in terms of Lagrangian multiplies /spl lambda//sub i/. The entropy function becomes a Hopfield-like energy function when we implement E(V/sub j/) in terms of neurons. Then both the Lagrangian variational calculus and the Hopfield neural network are used to estimate both unknowns as follows: (i) Given measurement {U/sub i/}, we make an initial guess of a set of /spl lambda//sub i/ and W/sub ij/ which allow us to compute by definition /spl lambda//sub 0/ and sources V/sub j/ and then we use the Lagrangian variational calculus (derived at the extremum /spl part/E//spl part/V/sub j/=0) to improve the set of /spl lambda//sub i/ through their changes /spl Delta//spl lambda//sub i/ that minimize the departures from the measurements (ii) alternatively, the gradient descent toward the extremum via Hopfield energy landscape (/spl part/U1//spl part/t=-/spl part/E//spl part/V/sub i/) determines the mixing weight matrix W/sub ij/. We refer to the double recursions methodology (i)-(ii) as the Lagrangian-Hopfield neural network for solving the double unknowns. Three identical simulations are let to discover 3, 2, and 4 unknown sources given three different initial values. Both unknowns V/sub j/ and W/sub ij/ are plotted in the iteration time steps to show the approach to the convergence in terms of the ratio between U/sub i/ and /spl Sigma//sub j=1//sup N/W/sub ij/V/sub j/ that should approach the unity rapidly.