Weighted Order Statistic Filters Research Articles

The Generalization of WOSF and Corresponding Outputs

Weighted order statistic filters (WOSF) generate linearly separable Boolean functions. In some cases, 2 different WOSF may generate the same Boolean function. We thus can collect all of the WOSF together, which generate the same Boolean function. In this paper, we construct equivalent classes of WOSF, the BF equivalent class and the global equivalent class. Besides, we use minimum weight vectors as base vectors to build all global classes of same order and characterize WOSF to generalize three properties of global classes. Finally, we propose 3 translated formulas to fast generate corresponding outputs of BF equivalent classes, not perform extra machine training or mathematical computation.

Some issues in improving quality of noisy periodic signals and estimating their parameters and characteristics numerically by using a cluster approach: Problem statement

In this paper, a technique is proposed to process and analyze noisy periodic signals recorded at discrete moments of time. The technique includes two stages: (a) signal quality improvement (processing) with the use of weighted order statistics filters, and (b) cluster analysis of processing results. Basic definitions and specific features of this type of filtration are given, as well as formulations and definitions of cluster analysis, which enable problems to be stated for periodic signal analysis. The efficiency of cluster analysis with weighted order statistics filters is proved based on results of numerical modeling of a noisy frequency-modulated signal.

Dichotomous Learning Algorithm for the Optimal Design of Weighted Order Statistics Filters

Weighted order statistics (WOS) filters are highly effective, in processing digital signals, due to their simple window structure. This paper proposes a fast and efficient learning algorithm that both improves learning speed and reduces the complexity of designing WOS filters. The algorithm uses a dichotomous approach to reduce the Boolean functions from 255 levels to two levels which are separated by an optimal hyperplane. The design concept of this algorithm is similar to that of support vector machines (SVMs), which use two separate sets of data to determine the optimal hyperplane. A conjugate gradient algorithm is adopted, to solve the orthant-constrained optimum problem, in order to improve memory storage for the large amounts of data required in the design process. Prior literature includes three different schemes for learning: one approach updates the parameters via pattern-mode learning, while the others involve batch-mode learning and semi-batch mode learning. Our proposed method approximates the optimal weighted order statistics filters far more rapidly than either Yoo's algorithm or adaptive neural filters.

The Optimal Design of Weighted Order Statistics Filters by Using Support Vector Machines

Support vector machines (SVMs), a classification algorithm for the machine learning community, have been shown to provide higher performance than traditional learning machines. In this paper, the technique of SVMs is introduced into the design of weighted order statistics (WOS) filters. WOS filters are highly effective, in processing digital signals, because they have a simple window structure. However, due to threshold decomposition and stacking property, the development of WOS filters cannot significantly improve both the design complexity and estimation error. This paper proposes a new designing technique which can improve the learning speed and reduce the complexity of designing WOS filters. This technique uses a dichotomous approach to reduce the Boolean functions from 255 levels to two levels, which are separated by an optimal hyperplane, Furthermore, the optimal hyperplane is gotten by using the technique of SVMs. our proposed method approximates the optimal weighted order statistics filters more rapidly than the adaptive neural filters.

A Practical Parallel Architecture for Stacks Filters

Stack filters belong to the class of non-linear filters and include the well-known median filter, weighted median filters, order statistic filters and weighted order statistic filters. Any stack filter can be implemented by using the parallel threshold decomposition architecture which allows implementing their non-linear processing by means of a collection of identical binary filters (Boolean logic circuits). Although it is conceptually simple and useful to study the filter properties, this architecture is not practical for direct hardware implementation because as many as (M ? 1) binary filters are required for a M-valued input signal and M is large in many applications. In this paper we introduce a new parallel architecture for stack filter implementations. The complexity is now proportional to the window width L of the filter, instead of to M. In most applications L is much smaller than M which translates into efficient hardware implementations. The attractive characteristic of ease of design exhibited by the threshold decomposition architecture is kept. In fact, for a given stack filter both in the conventional implementation and in the proposed one, the same binary filter is required. The key concept supporting the new architecture is a modified decomposition scheme which generates L binary signals for a multi-valued input. As an application example, a complex WOS filter is designed and prototyped in an FPGA.

Weighted order statistics filter for real-time signal processing applications based on pass transistor logic

The authors describe the rationale of the design of complex weighted order statistics filters (WOS filters) suitable for real-time signal processing applications. It is based on a simple stack-like architecture that decomposes the M-valued signals into a reduced number of binary signals which are filtered by identical boolean logic circuits (binary filters). The main distinguishing feature of the proposed procedure is the efficient implementation of both the decomposition stage and the binary filters. The former is achieved on the basis of introducing a new decomposition scheme which can be economically translated to hardware. The latter comes from realising that the binary filters required for WOS filtering are threshold functions and from the development of a systematic methodology for building pass-transistor networks that implement them. Results for a complex example filter show a sample frequency of over 175 MHz in a 0.35 μm MOS technology at 3.3 V.

New direct design method for weighted order statistic filters

The paper describes a new method for the design of optimum weighted order statistic (WOS) filters. WOS filters form a general class of increasing filters which generate an output based on the weighted rank ordering of the samples within the filter window. They include the median, weighted median, stack filters and morphological filters with flat structuring elements. This new design method is applicable to all of these operators. It has the advantage over existing techniques in that the filter weights are calculated directly from the training set observations and require no iteration. The method makes an assumption about the training set observations known as the weight monotonic property. It tests if the training set corruption is suitable for correction with an increasing filter. Where the assumption does not hold then a WOS filter should not be used for that training set. The paper includes two examples to demonstrate the design method and a justification for a training set approach to image restoration problems. Other benefits arising from the new design method are outlined in the paper. These include a method to determine the minimum MAE possible for a given training set and filter window.

0.8 μm cmos implementation of weighted-order statistic image filter based on cellular neural network architecture

In this paper, a very large scale integration chip of an analog image weighted-order statistic (WOS) filter based on cellular neural network (CNN) architecture for real-time applications is described. The chip has been implemented in CMOS AMS 0.8 /spl mu/m technology. CNN-based filter consists of feedforward nonlinear template B operating within the window of 3 /spl times/ 3 pixels around the central pixel being filtered. The feedforward nonlinear CNN coefficients have been realized using programmable nonlinear coupler circuits. The WOS filter chip allows for processing of images with 300 pixels horizontal resolution. The resolution can be increased by cascading of the chips. Experimental results of basic circuit building blocks measurements are presented. Functional tests of the chip have been performed using a special test setup for PAL composite video signal processing. Using the setup real images have been filtered by WOS filter chip under test.

VLSI architectures for weighted order statistic (WOS) filters

The class of median filters has been extended to include weighted order statistics (WOS) filters, to improve the flexibility of the filtering operation. The WOS filter weights each input within a sample window, and thus retains the original temporal order information. In this paper, we present efficient VLSI architectures for WOS filters which maintain a weighted rank for each sample in the sample window and update the weighted ranks for each window shift. We present novel (i) array architectures, (ii) stack filter architectures and (iii) sorting network architectures for non-recursive and recursive WOS filters which implement the above procedure. Our analysis shows that the bit-serial stack filter implementation is the one with the smallest area while the bit-parallel stack filter is the one with the smallest input–output latency. The sorting network architecture (based on updating a sorted list) has the best area–time performance. Physical implementations verify our analysis.

L/sub p/ norm design of stack filters

This paper addresses the problem of designing optimal stack filters by employing an Lp norm of the error between the desired signal and the estimated one. It is shown that the Lp norm can be expressed as a linear function of the decision errors at the binary levels of the filter. Thus, an Lp-optimal stack filter can be determined as the solution of a linear program. The conventional design of using the mean absolute error (MAE), therefore, becomes a special ease of the general Lp norm-based design developed here. Other special cases of the proposed approach, of particular interest in signal processing, are the problems of optimal mean square error (p=2) and minimax (p-->infinity) stack filtering. Since an Linfinity optimization is a combinatorial problem, with its complexity increasing faster than exponentially with the filter size, the proposed Lp norm approach to stack filter design offers an additional benefit of a sound mathematical framework to obtain a practical engineering approximation to the solution of the minimax optimization problem. The conventional MAE design of an important subclass of stack filters, the weighted order statistic filters, is also extended to the Lp norm-based design. By considering a typical application of restoring images corrupted with impulsive noise, several design examples are presented, to illustrate the performance of the Lp-optimal stack filters with different values of p. Simulation results show that the Lp-optimal stack filters with p=or>2 provide a better performance in terms of their capability in removing impulsive noise, compared to that achieved by using the conventional minimum MAE stack filters.

A new representation of weighted order statistic filters

The Weighted Order Statistic Filters (WOSF) are useful tools in digital signal processing. Their standard representation is not canonical, which is a major drawback in comparison or design issues. A new representation is proposed, which is based on the statistical study of the WOSF. Efficient conversion algorithms are presented. This representation is of interest for analyzing, comparing and designing WOSF.

Extended permutation filters and their application to edge enhancement

Extended permutation (EP) filters are defined and analyzed. In particular, we focus on extended permutation rank selection (EPRS) filters. These filters are constrained to output an order statistic from an extended observation vector. This extended vector includes N observation samples and K statistics that are functions of the observation samples. The rank permutations from selected samples in this extended observation vector are used as the basis for selecting an order statistic output. We show that by including the sample mean in the extended observation vector, the filters exhibit excellent edge enhancement properties. We also show that several previously defined classes of rank-order-based edge enhancers (CS, LUM, and WMMR sharpeners) can be formulated as subclasses of EPRS filters. These sharpening subclasses are in addition to the smoothing subclasses, which include rank conditioned rank selection, permutation stack, and weighted order statistic filters. Thus, this novel class of filters provides a broad framework within which many rank-order-based smoothers and edge enhancers can be unified. Edge enhancement properties are developed and an L(n) norm EPRS filter optimization procedure is presented. Finally, extensive computer simulation results are presented, comparing the performance of EPRS and other sharpening filters in edge enhancement applications.

Linear combination of weighted order statistic filters: canonical structure and optimal design

A new class of filters, called linear combination of weighted order statistic (LWOS) filters, is introduced. This filter is a combination of L-filters and weighted order statistic (WOS) filters. Based on the observation that this filter possesses the threshold decomposition property, a representation of LWOS filters, named the canonical representation, is developed. It is shown that most nonrecursive filters having the threshold decomposition property can be thought of as special cases of the canonical LWOS filter. This result indicates that this class of LWOS filters encompasses a variety of filters which include median-type nonlinear filters and linear FIR filters. A procedure for designing an optimal canonical LWOS filter under the mean square error (MSE) criterion has been developed. The optimization of LWOS filters yields an FIR Wiener filter when the input is zero-mean Gaussian and a median-type nonlinear filter for non-Gaussian inputs. Experimental results in image restoration are presented to compare the relative performances of the LWOS and existing filters.

Permutation weighted order statistic filter lattices

We introduce and analyze a new class of nonlinear filters called permutation weighted order statistic (PWOS) filters. These filters extend the concept of weighted order statistic (WOS) filters, in which filter weights associated with the input samples are used to replicate the corresponding samples, and an order statistic is chosen as the filter output. PWOS filters replicate each input sample according to weights determined by the temporal-order and rank-order of samples within a window. Hence, PWOS filters are in essence time-varying WOS filters. By varying the amount of temporal-rank order information used in selecting the output for a given observation window size, we obtain a wide range of filters that are shown to comprise a complete lattice structure. At the simplest level in the lattice, PWOS filters reduce to the well-known WOS filter, but for higher levels in the lattice, the obtained selection filters can model complex nonlinear systems and signal distortions. It is shown that PWOS filters are realizable by a N! piecewise linear threshold logic gate where the coefficients within each partition can be easily optimized using stack filter theory. Simulations are included to show the advantages of PWOS filters for the processing of image and video signals.

Input compression and efficient VLSI architectures for rank order and stack filters

Rank-order-based filters include rank-order filters, stack filters, and weighted order statistic filters. The output of a rank-order-based filter is always one of the sample points in its input window; which one is chosen depends only upon the ranks and positions of the samples within the window. This paper introduces new architectures for rank-order-based filters. They all achieve fast, efficient operation by exploiting an algorithm called input compression. Under this algorithm, the sample points in the input window are first mapped to their relative ranks — the sample points in a window of size N + 1 would thus be mapped to the integers 0− N. The rank-order-based filter to be implemented is then applied directly to this compressed input, and the rank chosen is then mapped back to the sample of that rank in the original data to obtain the final output. This approach has been used to implement rank-order filters, in which case the same rank is always chosen from the compressed data. In this paper, which rank is chosen also depends on the positions of the ranks in the compressed data. Implementations employing input compression have several advantages. They are computationally efficient like running order sorters, yet can be pipelined to a fine degree like sorting networks. In stack filter implementations, the threshold decomposition circuitry can be eliminated when input compression is combined with unary encoding of the ranks. Weighted order statistic filter implementations based on input compression can support programmable, noninteger weights.

Fast adaptation and performance characteristics of FIR-WOS hybrid filters

Fast adaptive algorithms are developed for training weighted order statistic (WOS) filters and FIR-WOS hybrid (FWH) filters under the mean absolute error (MAE) criterion. These algorithms are based on the threshold decomposition of real-valued signals introduced in this paper. With this method an N-length WOS filter can be implemented by thresholding the input signals at most N times independent of the accuracy used. Beside saving in computations, the proposed algorithms can be applied to process arbitrary real-valued signals directly. Performance characteristics of FWH filters in 1-D and 2-D signal restoration are investigated through computer simulations. We show that both in restoration of signals containing edges and in the case of heavy tailed nonGaussian noise, considerable improvement in performance can be achieved with FWH filters over WOS filters, Ll filters, and adaptive linear filters. Two new FWH filter design strategies are found for removal of impulsive noise and for restoration of a square wave, respectively.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parametric analysis of weighted order statistics filters

The authors study the convergence properties of weighted order statistics filters. Based on a set of parameters, weighted order statistics filters are divided into five categories making their convergence properties easily understood. They show that any symmetric weighted order statistics filters will make the input sequence converge to a root or oscillate in a cycle of period 2. This result is significant since a restriction imposed by an earlier research is eliminated making the result applicable for the whole class of symmetric weighted order statistics filters. A condition to guarantee convergence of symmetric weighted order statistics filters is also derived.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Fast algorithms for training stack filters

Stack filters constitute a class of nonlinear digital filters possessing a weak superposition property. In the paper, fast algorithms for training stack filters are introduced. The first algorithm is based on a technique previously developed. Similar to the previous algorithm, this improved method requires only simple arithmetic operations-increment, decrement, and local comparison. The /spl epsiv/-convergence property of the fast algorithm is established. In addition, two fast training algorithms are developed for a subclass of stack filters called weighted order statistic filters. These fast algorithms are variants of the LMS and RLS algorithms in linear filters. All three fast algorithms developed exploit the repetition property of the training input. The factor of speedup is approximately the ratio between the number of quantization levels of the input and the window size.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Weighted order statistics filters-their classification, some properties, and conversion algorithm

Stack filters that are based on threshold logic with nonnegative weights and threshold value are called weighted order statistics (WOS) filters and are classified into four different types: type-0 (trivial), type-1 (decreasing), type-2 (increasing), and type-3 (mixed). A significant property of the threshold logic is that it only needs (n+1) tuples to represent its function representation if the input of this function has it variables. This associative representation is very useful in neural computing, nonlinear filtering, etc. The authors propose a significant classification of threshold logic such that the behaviors of WOS filters can be understood easily based on this classification. In the paper, type-0, type-1, and type-2 WOS filters are shown to possess the convergence property. However, type-3 WOS filters do not necessarily possess the convergence property. Hence, the authors investigate the convergence behaviors of some type-3 WOS filters including symmetric weighted median filters and the type-3 filters proposed in Wendt [1990]. Finally, an efficient algorithm to determine whether a positive Boolean function corresponds to a weighted median filter is proposed. The authors use the property all threshold logics are regular to make the algorithm tractable. >

Adaptive multistage weighted order statistic filters based on the backpropagation algorithm

As a concise representation of stack filters, multistage weighted-order statistic (MWOS) filters are introduced, which correspond to multistage threshold logic gates or multilayer perceptrons in the binary domain. Two adaptive algorithms are derived for finding optimal MWOS filters under the mean absolute error criterion and the mean square error criterion, respectively. Experimental results from image enhancement are provided to compare the performance of adaptive MWOS filters and adaptive stack filters.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>