Estimation and structural based approach for the design of optimal stack filters

Moncef Gabbouj, Purdue University

Abstract

Two approaches have been used in the past to design or choose a rank-order based filter to estimate a signal from a noise corrupted observation of that signal: the structural and the estimation approaches. In the structural approach, the goal is to find a filter which preserves those shapes that are part of the signal while removing those that are part of the noise. In the estimation approach, the goal is to find a filter which best estimates the desired signal, given the noise corrupted version of the signal as the filter's input. The first part of this thesis develops a theory for the structural behavior of stack filters. This theory provides: a test which can determine if a given stack filter has any root signals; a method for classifying the root signal behavior of any stack filter found to have roots; and, perhaps most importantly, a method for designing stack filters with specific root signals or other structural behavior. This theory of root signals for stack filters is then combined with the theory of minimum mean absolute error stack filtering. This new, unified theory allows the designer to pick a filter which minimizes noise subject to constraints on its structural behavior. The second part of the thesis deals with the convergence behavior of stack filters. First, stack filters (or positive Boolean functions (PBFs), in the binary case) are classified into four different types, called type-0 through type-3. It is shown that PBFs of type-0 through type-2 possess the convergence property, while type-3 PBFs do not all share this property. The rates of convergence for a subset of convergent PBFs, namely stack filters of of type-0 through type-2, is also determined. The convergence behavior and rates of convergence is then generalized to include stack filters of type-0 through type-2 with index i. In the final part of this thesis, a new optimization theory for stack filters is presented. This new theory is based on the minimax error criterion rather than the mean absolute error criterion used in (Coy) and (CoL). In the binary case, a methodology is designed to find the optimal stack filter that minimizes the maximum absolute error between the input and the output signals. The most interesting feature of this optimization procedure is the fact that it can be solved using a linear program, just like in the MMAE case (CoL). When generalizing to multiple inputs, complexity problems will arise and two alternative approaches will be suggested.

Degree

Ph.D.

Advisors

Coyle, Purdue University.

Subject Area

Electrical engineering|Mathematics|Statistics

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