Blind de-mixing with unknown sources

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_ij (for the i-th object and the j-th channel) and V_j, the sources in question, are unknown due to uncooperative subjects. For weak signals, we can assume a linear mixing model U_j=Σ_j=i^NW_(/ij)Vsub i/ of which the total probability of the set of unknowns V_j must be added up to one Σ_i+1^NV_j=1. We postulate that for i-1 each case of real positive unknowns (W_ij and V_j) there exists a maximum entropy E(V_j) constrained with the measurements in terms of Lagrangian multiplies λ_i. The entropy function becomes a Hopfield-like energy function when we implement E(V_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_i}, we make an initial guess of a set of λ_i and W_ij which allow us to compute by definition λ₀ and sources V_j and then we use the Lagrangian variational calculus (derived at the extremum ∂E/∂V_j=0) to improve the set of λ_i through their changes Δλ_i that minimize the departures from the measurements (ii) alternatively, the gradient descent toward the extremum via Hopfield energy landscape (∂U1/∂t=-∂E/∂V_i) determines the mixing weight matrix W_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_j and W_ij are plotted in the iteration time steps to show the approach to the convergence in terms of the ratio between U_i and Σ_j=1^N W_ijV_j that should approach the unity rapidly