LEAST-SQUARES ORDER STATISTIC FILTERS FOR SIGNAL RESTORATION

This paper develops the fundamental theory for restoring discrete waveforms immersed in independent noise using order statistic (OS) filters, using the least squares criterion as a fidelity measure. OS filters have previously been demonstrated to afford robust structure-preserving noise-smoothing performance for signals of arbitrary dimensionality. Here, nondynamical least squares OS filter design methods are extended to the case of arbitrary discrete waveforms immersed in independent noise. A method for incorporating local structural constraints into the optimization process is introduced. A small number of these constraints can ensure that the designed filter is sensitive to local high-information signal structures. This is accomplished within a natural framework by appending the constraints into the objective function to be minimized using a Lagrangian approach. The principal drawback of OS filter design remains the complexity of computing the temporal/spatial correlations of the OS, although some relatively fast algorithms have been developed for computing OS distributions. For large problems, the computation is still excessive. A suboptimal approximation technique is developed that yields good results when applied to the image restoration problem. Finally, some directions for future research are explored in the conclusion.