The Stochastic Matched Filter and Its Applications to Detection and De-Noising

In several domains of signal processing, such as detection or de-noising, it may be interesting to provide a second-moment characterization of a noise-corrupted signal in terms of uncorrelated random variables. Doing so, the noisy data could be described by its expansion into a weighted sum of known vectors by uncorrelated random variables. Depending on the choice of the basis vectors, some random variables are carrying more signal of interest informations than noise ones. This is the case, for example, when a signal disturbed by a white noise is expanded using the Karhunen-Loeve expansion (Karhunen, 1946; Loeve, 1955). In these conditions, it is possible either to approximate the signal of interest considering, for the reconstruction, only its associated random variables, or to detect a signal in a noisy environment with an analysis of the random variable power. The purpose of this chapter is to present such an expansion, available for both the additive and multiplicative noise cases, and its application to detection and de-noising. This noisy random signal expansion is known as the stochastic matched filter (Cavassilas, 1991), where the basis vectors are chosen so as to maximize the signal to noise ratio after processing. At first, we recall some general considerations on a random 1-D discrete-time signal expansion in section 2. In particular, we study the approximation error and the second order statistics of the signal approximation. Then, in section 3, we describe the stochastic matched filter theory for 1-D discrete-time signals and its extension to 2-D discrete-space signals. We finish this section with a study on two different noise cases: the white noise case and the speckle noise case. In the next section, we present the stochastic matched filter in a de-noising context and we briefly discuss the estimator bias. Then, the de-noising being performed by a limitation to order Q of the noisy data expansion, we propose to determine this truncature order using a mean square error criterion. Experimental results on synthetic and real data are given and discussed to evaluate the performances of such an approach. In section 5, we describe the stochastic matched filter in a detection context and we confront the proposed method with signals resulting from underwater acoustics. Finally, some concluding remarks are given in section 6. 15