Two new robust nonlinear signal restoration algorithms

Abstract

Signal restoration is a well-known problem is many scientific fields such as image processing and geophysics. Here, we process a signal to enhance its quality in order to uncover features about the original or system input signal z that are not apparent in the noisy and distorted output of the system, u. Once the model is specified, the signal restoration problem is then to find a signal z” that is as close as possible to the original one subject to a suitable optimality criterion. The assumed givens are the output u, system model h, and any a priori knowledge about the solution in the form of constraints. Various methods that relied on different criteria showed attractive results in well-defined situations. Minimum meansquare error, maximum entropy, maximum likelihood, and maximum a posteriori probability are a few criteria that have proven useful. Here, regularization theory is used to tackle the problem. Especially, the stabilizing functional approach is considered [ 1,2 ] . The problem is formulated as the constrained minimization of a certain stabilizing functional a( z, P) [1,2]. The choice of the stabilizing functional or criterion based on the problem at hand is still an open question. In a recent paper [l] , Karayiannis and Venetsanopoulos investigated the choice of a stabilizing functional in the case of 2-D ill-posed inverse problems. These authors emphasized the use of a certain class of quadratic functionals as stabilizing functionals. The use of global nonquadratic functionals was discouraged due to several problems. One main problem is the justification of the choice of a particular nonquadratic functional [l] . Recently, a step toward reducing the set of nonquadratic functionals to an optimal class of functionals useful in speech and image compression, cluster analysis, and pattern classification was presented by Jones [ 31. The considered functionals satisfy an orthogonality condition similar to that enjoyed by linear projections in Hilbert space. The concept of directed orthogonality is known as the correlation matching property in speech analysis and as expected value matching within the space of probability densities [ 41. Jones demonstrates that the Itakura-Saito distortion measure (ISD) of communication theory and the Kullback-Leilber distance measure ( KLM ) of statistics are the only functionals in the smooth Ali-Silvey-Csiszar class and the class of regular ratio distortion measures, respectively, to obey the directed orthogonality principle. Therefore, our focus in this paper is to investigate further these two functionals, i.e., Qisn and RkLM. In Section II, a regularized solution for the ill-posed inverse problem of signal restoration is developed through the use of the stabilizing function approach that uses the entropic stabilizing functionals &, and !l kLM, respectively. These functionals are minimized subject to a mean-squared-error constraint equation. This produces two iterative schemes, called the ISD and KLM algorithms, through which the signal is obtained. In Section III, the derived iterative algorithms are shown to be theoretically sound, leading to a unique, robust solution. In Section IV, examples that demonstrate the restoration achieved by the new methods in several signal-to-noise-ratio (SNR) environments are presented. Finally, we note that this work is an extension of that presented in [ 61.