Rank Selection Probabilities Research Articles

Calculating the output distribution of stack filters that are erosion-dilation cascades, in particular LU LU -filters

Two procedures to compute the output distribution φS of certain stack filters S (so called erosion-dilation cascades) are given. One rests on the disjunctive normal form of S and also yields the rank selection probabilities. The other is based on inclusion-exclusion and e.g. yields φS for some important LU LU -operators S. Properties of φS can be used to characterize smoothing properties of S. Also, in the same way as our polynomials φS are computed one could compute the reliability polynomial of a connected graph, or more generally the reliability polynomial w.r.t. any positive Boolean function.

Computing the Output Distribution and Selection Probabilities of a Stack Filter from the DNF of Its Positive Boolean Function

Many nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function (PBF). Our algorithm avoids to enumerate the models of the PBF one by one, and thus is considerably more efficient than previous methods. The so called rank selection probabilities can be computed along the way.

Stack filter design using selection probabilities

Selection probabilities, viz. rank and sample selection probabilities, quantify the impulse suppression, robustness, lowpass filtering ability, and detail preserving ability of stack filters. Closed-form expressions exist for computing selection probabilities from positive Boolean function describing the stack filter. We first present an analysis framework that interprets a Boolean vector to denote membership in a partition set. Using this framework a new algorithm to compute selection probabilities from positive Boolean functions is derived. Next a method to synthesize a stack filter from specified selection probabilities is presented. An underdetermined system of linear Diophantine equations is derived in the unknown elements of the C-matrix and the desired selection probabilities. This system of equations is solved to obtain the candidate solutions for the C-matrix. For each candidate C-matrix, an algorithm based on partition sets is used to derive one or more positive Boolean functions of the stack filter or show that none exists.

On the robustness of the class of stack filters

A widely held view in the nonlinear signal processing community is that the class of stack filters is robust. Although this is supported by extensive experimental evidence, no systematic theoretical justification exists, despite the availability of analytical tools for studying robustness of individual stack filters. We focus on rank selection probabilities (RSPs) as measures of robustness as it is well known that other statistical characterizations of stack filters, such as output distributions, breakdown probabilities and output distributional influence functions can be represented in terms of RSPs. We show, in a very general sense, that the class of stack filters is highly robust. It is also shown that almost all stack filters have very similar output distributions for independent and identically distributed (i.i.d.) input signals and, thus, very similar statistical behavior.

An efficient algorithm to calculate sample and rank selection probabilities for weighted median filters

Sample and rank selection probabilities are important in the analysis of weighted median filters as well as in comparing linear and nonlinear filters. We propose a new, efficient algorithm to calculate these probabilities for weighted median filters. The computational savings of this new algorithm are substantial and compare favorably with other known algorithms.

Optimal stack filters under rank selection and structural constraints

A new expression for the moments about the origin of the output of stack filtered data is derived in this paper. This expression is based on the A and M vectors containing the well-known coefficients A i of stack filters and numbers M( Φ, γ, N, i) defined in this paper. The noise attenuation capability of any stack filter can now be calculated using the A and M vector parameters in the new expression. The connection between the coefficients A i and so called rank selection probabilities r i is reviewed and new constraints, called rank selection constraints, for stack filters are defined. The major contribution of the paper is the development of an extension of the optimality theory for stack filters presented by Yang et al. and Yin. This theory is based on the expression for the moments about the origin of the output, and combines the noise attenuation, rank selection constraints, and structural constraints on the filter's behaviour. For self-dual stack filters it is proved that the optimal stack filter which achieves the best noise attenuation subject to rank selection and structural constraints can usually be obtained in closed form. An algorithm for finding this form is given and several design examples in which this algorithm is used are presented in this paper.

Fast algorithms for analyzing and designing weighted median filters

In this paper, two fast algorithms are developed to compute a set of parameters, called M i 's, of weighted median filters for integer weights and real weights, respectively. The M i 's, which characterize the statistical properties of weighted median filters and are the critical parameters in designing optimal weighted median filters, are defined as the cardinality of the positive subsets of weighted median filters. The first algorithm, which is for integer weights, is about four times faster than the existing algorithm. The second algorithm, which applies for real weights, reduces the computational complexity significantly for many applications where the symmetric weight structures are assumed. Applications of these new algorithms include design of optimal weighted filters, computations of the output distributions, the output moments, and the rank selection probabilities, and evaluation of noise attenuation for weighted median filters.

Stack filter lattices

Lattice theory, rank selection probabilities (RSPs), and sample selection probabilities (SSPs) are used to develop methods for comparing and ordering stack filters. New recursive algorithms for calculating RSPs and SSPs are developed. The structure of these algorithms assists in the derivation of several interesting properties of RSPs and SSPs. The identification of some important sublattices of the stack filter lattice leads to an improved characterization of the impulsive noise suppression capabilities of stack filters.

Stack filters and selection probabilities

Based on the fact that the output of a given stack filter can be determined if the ranks of the samples in the input window are known and that this output always equals one of the samples in the input window, rank and sample selection probabilities are defined. The output distribution of the stack filter of size N with independent identically distributed (i.i.d.) inputs can be expressed as a weighted sum of the ith, i=1, 2, ..., N order statistics, where the rank selection probabilities are the weights. The sample selection probabilities equal the impulse response coefficients of a finite impulse response (FIR) filter whose output spectrum is closest, of all linear filters, to that of the stack filter for i.i.d. Gaussian inputs. Results are also derived for correlated inputs. Robustness and detail preserving properties of stack filters are related to the selection probabilities. Other statistical properties are also derived. Finally, methods to compute the selection probabilities of the stack filter from its positive Boolean function and the selection probabilities of the weighted median filter from its weights are given in detail.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On rank selection probabilities

The concept of rank selection probabilities for stack filters has previously been introduced. If the probabilities are known, then the output distribution for i.i.d. input follows easily. There is another expression for the output distribution of a stack filter using certain quantities called A/sub i/. The authors utilize the different forms of output distribution to derive a simple connection between the quantities A/sub i/ and the rank selection probabilities. This also leads to a new fast method for the computation of rank selection probabilities. >