RANKED-ORDER TYPE OPERATORS IN ADAPTIVE QUANTIZATION AND DIGITAL SIGNAL PROCESSING (MEDIAN FILTERS, STACK)

PETER DAVID WENDT, Purdue University

Abstract

Ranked-order and median filters and nonlinear digital filters that use a moving window and a ranking operation; the output of a k('th)-order filter is always the k('th) largest sample in the window, whereas the output of a median filter is the median sample value in the window. This thesis introduces a new application of ranked-order operators in an adaptive quantizer, and extends previous analyses of median filters. Also, a new and very large class of nonlinear filters is introduced which includes all ranked-order filters. First, a robust, wide-range adaptive quantizer that uses ranked-order operators is proposed. It adapts to a new sample by estimating the local input probability density from sample quantiles of the previous n samples, which it obtains with ranked-order operators; it uses the estimate to approximate the optimum distribution of output levels for the new sample. As the adaptation is quite general, the quantizer can adapt to input variance, density shape, and mean. The quantizer approaches the theoretical limits on quantization distortion for both Gaussian and Laplacian inputs. Also, the convergence properties of median filters are considered. A median filter will filter any signal of finite length of an invariant signal or root in a finite number of passes. In many applications, these roots are the desired noiseless signals, so any limit on the number of passes needed for convergence is valuable. For a median filter of window width 3, a recursive formula is derived to count the number of binary signals that will converge to roots in exactly m passes of the filter. Then, a general limit on the number of passes to a root is proven; it is proportional to the signal length but inversely proportional to the window width. Finally, a new class of nonlinear filters called stack filters is introduced; these are generalizations of ranked-order filters. They are based on a parallel filter implementation called the threshold decomposition architecture, which has already proved successful for ranked-order filters. Because of the similarity of stack filters to ranked-order filters, their deterministic and statistical behavior can be analyzed using the tools developed for ranked-order filters. Even for small window widths, the numbers of these stack filters are extremely large, and we have already found several new and useful filters of window width 3.

Degree

Ph.D.

Subject Area

Electrical engineering

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