Stack filter design using selection probabilities

A new fuzzy filter, called fuzzy stack filter (FSF), is proposed to extend the filtering capability of conventional stack filter (SF), which is based on the positive Boolean function (PBF) as its window operator. We fuzzify the onset and off-set of a given PBF to obtain two types of fuzzy PBFs. Then, we adopt the architecture of threshold decomposition to develop this new fuzzy filter with a fuzzy PBF as its window operator. Each fuzzy PBF is associated with a set of control parameters. Therefore, the original PBF can be estimated from above and below by two fuzzy PBFs with appropriate control parameters. Furthermore, we can apply the fuzzy modifiers to modify the fuzzy PBFs such that the PBFs can be completely estimated by the fuzzy PBFs. Hence, the stack filter is a special case of fuzzy stack filter. Since some control parameters are added in this new filter, the neural learning algorithms can be easily developed under the flexibility of the given control parameters. We first propose the fuzzy (m,n) rank-order filter to test our proposed learning algorithm. In this simple learning algorithm, we can remove the noise-corrupted images very well in contrast to the filtering behavior of rank-order filters. We believe that the results presented will lead to more fruitful research on more advanced and powerful learning algorithms dedicated to the appropriate applications.

The signed binary number representation (SBNR) is used to reduce the number of PBF calculation stages in stack filters. The approach is based on the possibility of minimization of signed power-of-two terms in the SBNR representation of input data. A coder and a decoder for the mutual transformation of binary-weighted code and the minimal SBNR are proposed. An efficient algorithm and a processor structure for stack filtering based on the minimal SBNR are presented. The time complexity of the proposed filter is of order kT/sub PBF/, where k is the number of bits in the input signals and T/sub PBF/ is the delay in the incorporated circuit for the positive Boolean function (PBF). A stack filtering algorithm for the generalized nonadjacent form (GNAF) of signed digit number representation (SDNR) is also presented.

Weighted medians — positive Boolean functions conversion algorithms

Stack filters are a class of nonlinear filters for removing noise that is uncorrelated with signal. Their design is formulated as a high nonlinear optimization problem. An improved immune memory clonal selection algorithm (IMCSA) is suitable for designing stack filters. But as many algorithms for designing stack filters have to face that evaluation of each candidate solution is still takes the most computation complexity. By representing the positive Boolean functions (PBF) in objective function, a hybrid computation approach (software and hardware) for the calculation tasks has been proposed. Compared with available methods, the proposed method can improve the efficiencies of design stack filters and the design time is significantly reduced. The procedure of design and implementation for stack filters can be reached at the same time by the method

Some methods and algorithms of solution of systems of linear Diophantine equations over the naturals are briefly reviewed. Criteria and an incremental algorithm of efficient solution of the problem of consistency for a system of linear Diophantine equations and inequalities over the naturals are given.

In this paper, we extend the concept of the M<SUB>i</SUB>'s of weighted median filters to stack filters. A fast algorithm is proposed to compute M<SUB>i</SUB> of stack filters. The problem of synthesis of stack filters by rank selection probabilities through the M<SUB>i</SUB>'s is addressed. The necessary and sufficient condition for M<SUB>i</SUB> to be a stack filter is presented. A procedure is proposed to synthesize a stack filter by a given set of rank selection probabilities.

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

A polynomial algorithm is proposed to construct the minimal generating set of solutions and the basis of the solution set for systems of linear Diophantine equations over the ring of integer. The algorithm is based on a modified TSS method.

A new algorithm for fording the minimal solutions of systems of linear Diophantine equations has recently been published. In its description the emphasis was put on the mathematical aspects of the algorithm. In complement to that, in this paper another presentation of the algorithm is given which may be of use for anyone wanting to implement it.

Another constructive proof is presented for the fact that a system of linear equations with integer coefficients in bounded integer variables is equivalent to a single equation, which is a linear combination of the original ones. The equation is obtained in a number of steps; in each step two equations are replaced by a single one. This replacement is performed subject to the condition that the remaining equations hold and a final equation with relatively small coefficients is obtained. It may be inefficient however to calculate small coefficients, as the original coefficients can be used to represent the final ones in a suitably chosen number system.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Two new methods to solve linear systems of Diophantine equations are proposed - modular (CRT) and p-adic (Hensel). Each of them allows to obtain solutions of a system with the size n x m with the complexity O(nsm). For quasi-square systems, the p-adic method allows to obtain solution with the complexity O(n 3), and the modular method with complexity O(n s+1). Both estimates have the accuracy up to the logarithmic multipliers, s being the power in the estimation of matrix multiplication time.

In this paper, a classifier based on the positive Boolean function (PBF) is proposed for the supervised pattern classification. A PBF is exactly represented as one sum-of-product form without any negative components. The PBF possesses the well-known threshold decomposition and stacking properties. The classification errors can be calculated from the summation of the absolute errors incurred at each level. The optimal PBF are found and designed to be a classifier by minimizing the classification error rate along the training samples. The experimental results were given to show the validity of our proposed approaches.

This paper presents a new supervised classification technique for hyperspectral imagery, which consists of two algorithms, referred to as greedy modular eigenspace (GME) and positive Boolean function (PBF). The GME method is designed to extract features by a simple and efficient GME feature module. The GME makes use of the data correlation matrix to reorder spectral bands from which a group of feature eigenspaces can be generated to reduce dimensionality. It can be implemented as a feature extractor to generate a particular feature eigenspace for each of the material classes present in hyperspectral data. The residual reconstruction errors (RRE) are then calculated by projecting the samples into different individual GME-generated modular eigenspaces. The PBF is further developed for classification. It is a stack filter built by using the binary RRE as classifier parameters for supervised training. It implements the minimum classification error (MCE) as a criterion so as to improve classification performance. It utilizes the positive and negative sample learning ability of the MCE criteria to improve classification accuracy. The performance of the proposed method is evaluated by MODIS/ASTER airborne simulator (MASTER) images for land cover classification during the Pacrim II campaign. Experimental results demonstrate that the GME feature extractor suits the nonlinear PBF-based multi-class classifier well for classification preprocessing. The proposed approach is not only an effective method for land cover classification in earth remote sensing but also dramatically improves the eigen-decomposition computational complexity compared to the conventional principal components analysis (PCA).

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