P. Wendt et al., "Stack filters", IEEE Trans. Acoust., Speech, Signal Processing, pp. 898-911, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....property is quite simple, forcing functions have been implicated in a number of phenomena related to discrete dynamical systems as well as nonlinear filters. For example, they have been used to study the convergence behavior of an important class of nonlinear digital filters called stack filters [1], 2] 3] Stack filters are a class of sliding window nonlinear digital filters, defined by monotone Boolean functions I . Due to the so called threshold decomposition property, they are able to operate in the real valued domain. This filter class includes standard and weighted median filters, ....

....no loops. In addition, information about all forcing variables along with their forcing values as well as the forced function values, is contained in the vector c0, Let us give an example. Example: Let f (x, x2, x3) X2X3 be a Boolean function. The truth table of f is equal to f = [1, 1, 1, 1, 0, 0, 0, 1] T. Thus, taking the forcing transform, we have [0 0 0 0 1 1 1 1] a(0,1) f= 0 0 I I 0 0 I I x [11 I I 0 0 0 11 = 1 a The other vectors are equal to c = 4 2 2 ]w, cO,O) 3 1 1 ]w, andc ( 0 2 2 ] It can be seen that c ) 4. This means that f is a forcing function and the forcing ....

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P. Wendt, E. Coyle, N. Gallagher, "Stack Filters," IEEE Trans. Acoust., Speech, Signal Processing, Vol. 34, pp. 898-911, 1986.

....and for . In other words, this is the joint density between two corresponding shrunk signal values on consecutive scale levels and . Note that the Jacobian of the transformation from to is equal to 1. We begin by expressing and as outputs of different stack filters and with the same inputs [23]. Specifically, are three identity filters for and is the median filter med In order to compute , we use the well known formula for the joint distribution of the outputs of two stack filters having i.i.d. inputs with distribution [24] 10) where and Hamming weight of ; complement of ; ....

P. D. Wendt, E. J. Coyle, and N. C. J. Gallagher, "Stack filter," IEEE Trans. Acoustics, Speech, and Signal Processing, vol. ASSP-34, pp. 898--911, 1986.

.... classes that have gained much popularity are classes based on positive (monotone) Boolean functions (PBFs) including the class of stack smoothers based on threshold decomposition and the recently introduced class of stack filters based on mirrored threshold decomposition [10] Stack smoothers [3] [5] which have been defined in the binary domain of threshold decomposition, have traditionally been referred to in the literature as stack filters, although, as detailed in [10] 12] they are limited to lowpass operations. In this paper, we denote these structures as smoothers to ....

P. Wendt, E. J. Coyle, and N. C. Gallagher Jr, "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, Aug. 1986.

.... to impulsive noise, the standard median filter remains the most popular for image processing applications [1] The median filter, however, often tends to remove fine details in the image, such as thin lines and corners [1] In recent years, a variety of median type filters such as stack filters [2], multistage median [3] weighted median [4] rank conditioned rank selection [5] and relaxed median [6] have been developed to overcome this drawback. The output of the relaxed median filter with parameters and is determined by comparing lower and upper order statistics to the center sample in ....

P. D. Wendt, E. J. Coyle, and N. C. Gallagher, "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 898--919, Aug. 1986.

....that impose structural or statistical constraints. Filter parameters such as window size, window shape, or rank can be adapted as a function of the input [10] 19] 26] 35] Variations or generalizations of the median, such as rank order filters and stack filters have been investigated [31]. Some of these permit the design of specially tailored filters to enhance or suppress certain structures in specific types of images [11] 18] 32] 33] At least one design computes an output that depends on the spatial locations of the samples within the transform windows [10] A. Image ....

Wendt, P. D., E. J. Coyle, and N. C. Gallagher, "Stack filters," IEEE Trans. Acoust. Speech Signal Process., vol. ASSP-34, no. 4, August 1986.

....decomposition of a function into cross sections. Adapted from [5] A set processing filter f(X) is said to be increasing [5] if for any two sets A and B where A B, the filtered sets maintain the same set relationship; that is, f(A) f(B) This property is also called the stacking property [17]. A discrete, binary set processing filter possesses the stacking property if and only if its output can be expressed as a Boolean function that does not contain the complement of any of the input variables [17] Such expressions are called positive Boolean functions. 12 Set processing filters ....

.... relationship; that is, f(A) f(B) This property is also called the stacking property [17] A discrete, binary set processing filter possesses the stacking property if and only if its output can be expressed as a Boolean function that does not contain the complement of any of the input variables [17]. Such expressions are called positive Boolean functions. 12 Set processing filters that are increasing (possess the stacking property) may be converted to function processing filters by performing the set processing operations on the individual cross sections of a signal; the filtered signal is ....

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P. D. Wendt, E. J. Coyle, and N. C. Gallagher, "Stack filters," IEEE Trans. Acoust., Speech, Signal Process., vol. 34, no. 4, pp. 898-911, 1986.

....where an integer value m is segmented and represented as the vector [0 0 1 . 1] T . The number of ones in the vector is precisely m, so that the sum of the components equals m. Since the vector is binary, boolean operations can be applied to the set of decomposed elements. Stack filters [22] and boolean microstatistic filters 3 [1] have been developed from this concept. The segmentation procedure was extended to real values in the work of Chen and Arce [4] The concept is simple: a real value x can be represented as a vector of segments defined by a set of thresholds. As an example ....

P. D. Wendt, E. J. Coyle, and N. C. Gallagher, "Stack filters", IEEE Trans. Acoust., Speech, Signal Processing, Vol. 34, No. 4, pp. 898-911, August 1986.

....with hard nonlinearities while simultaneously, the estimation of coefficients of the localized polynomial expansion remains a linear problem. 2. THRESHOLD DECOMPOSITION Threshold decomposition is an operator that was originally developed in the context of integer numbers for stack filtering [7]. In a previous paper we extended the threshold decomposition operator to the field of real numbers and used it to develop piecewise linear filters [3] and approximation networks with adaptive piecewise polynomial kernels [4] In this paper, we show that tensor operations on thresholddecomposed ....

P. D. Wendt, E. J. Coyle, and N. C. Gallagher, "Stack filters", IEEE Trans. Acoust., Speech, Signal Processing, Vol. 34, No. 4, pp. 898-911, August 1986.

....to improvement of detail preservation. Index Terms Chow parameter, detail preservation, monotone Boolean function, optimization, sample selection probability, stack filter. I. INTRODUCTION S TACK filters constitute an important class of nonlinear filters based on monotone Boolean functions [1]. A design method for stack filters based on minimization of the mean absolute error was demonstrated in [2] Statistical properties of stack filters have been studied in terms of output distributions and moments for independent and identically distributed (i.i.d. input signals [3] 6] ....

....as well as present some notation and definitions. Section III discusses the proposed SSP based design algorithm. Finally, an example of stack filter design using the proposed method is illustrated in Section IV. II. PROBLEM FORMULATION A stack filter is defined by a monotone Boolean function [1]. For such a representation, we will 1070 9908 00 10.00 2000 IEEE 190 IEEE SIGNAL PROCESSING LETTERS, VOL. 7, NO. 7, JULY 2000 need to use the notion of an dimensional hypercube, which we denote by . The th layer ( of the cube contains only those binary vectors that have exactly components ....

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P. Wendt, E. Coyle, and N. Gallagher, "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 898--911, Aug. 1986.

....well suited for image processing, where their non linear effects are useful. The standard median filter, for example, removes impulsive noise and preserves sharp edges. Many median type filters, e.g. weighted median and order statistic filters, can be thought of as special cases of stack filters [9] and thus expressed as combinations of MIN and MAX operations. The systems that we mainly concentrate on in this paper belong to this class. Statistical properties of median and stack filters have been studied in [6, 7, 9, 10, 11, 12, 15] Focus has been on properties like output distribution ....

....and order statistic filters, can be thought of as special cases of stack filters [9] and thus expressed as combinations of MIN and MAX operations. The systems that we mainly concentrate on in this paper belong to this class. Statistical properties of median and stack filters have been studied in [6, 7, 9, 10, 11, 12, 15]. Focus has been on properties like output distribution functions, joint distributions, and other statistical descriptions. In the present paper we will emphasise scale dependency by examining run length distributions of these systems. Bovik [6] analyses streaking in median filtered images. ....

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P. D. Wendt, E. J. Coyle and N. C. Gallagher Jr., "Stack filters", IEEE Transactions on Acoustics, Speech and Signal Processing, vol. ASSP-34, no. 4, pp. 898-911, Aug. 1986.

.... The success of these filters is due to their ability to preserve edges as well as efficiently attenuate noise in impulsive noise environments (see [4] and [5] for overviews) Many median type filters, such as weighted median filters and order statistic filters, are special cases of stack filters [6]. If the filter length is , the median filtering operation can be denoted as med where and are the input and output sequences, respectively. A simple extension of the median filter is the recursive median filter (RMF) 7] which is obtained by replacing some input values by previous output ....

....are the input and output sequences, respectively. A simple extension of the median filter is the recursive median filter (RMF) 7] which is obtained by replacing some input values by previous output values as med (1) Statistical properties of median and stack filters have been studied in [2] [6], and [8] 11] With the assumption of a constant input signal corrupted by additive white noise, as is often done in the analysis of linear filters, the output distribution of any stack filter can be obtained in terms of the input distribution. Hence, the output variance, which is a measure of ....

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P. Wendt, E. Coyle, and N. Gallagher, "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 898--911, 1986.

....stack filters. However, if the automaton is not state minimal, the solution may be intractable, which is the motivation for this work. Formulae for the number of states in state minimal automata are presented. 1 INTRODUCTION Statistical properties of median and stack filters have been studied in [1, 9, 10, 11]. It has been shown that by using finite automata and Markov Chain theory we can compute statistical characteristics, such as output distributions, of stack filters [6] The calculation of runlength distributions [13] and breakdown probabilities [3, 4] is also straightforward. The computation of ....

P. Wendt, E. Coyle, N. Gallagher, "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. 34, pp. 898--911, 1986.

....finding a mapping between reduced input space clusters and the filter space, will result in rules for selecting the best suited stack filter for a given region. The supervised clustering procedures are shown to surpass significantly the global filtering approach. 1 INTRODUCTION Stack filters ([5]) are a wide class of nonlinear digital filters which includes a large number of filter classes with various types of behavior. Consequently, the successful design inside the stack filter class is a challenging task. Theoretical approaches to optimal stack filter design under the mean absolute ....

P.D.Wendt, E.J.Coyle, N.C.Gallagher, "Stack Filters" IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-34,No.4, pp. 898-911, Aug.1986. 3 Supervised filtering No clustering Clustering Local mean and var. Edge parameters Binary windows prob.

....techniques that impose structural or statistical constraints. Filter parameters such as window size, window shape, or rank can be adapted as a function of the input [10, 19, 26, 35] Variations or generalizations of the median, such as rank order filters and stack filters have been investigated [31]. Some of these permit the design of specially tailored filters to enhance or suppress certain structures in specific types of images [11, 18, 32, 33] At least one design computes an output that depends on the spatial locations of the samples within the transform windows [10] 1.1 Image features ....

Wendt, P. D., E. J. Coyle, and N. C. Gallagher, "Stack filters," IEEE Trans. Acoust. Speech Signal Process., vol. ASSP-34, no. 4, August 1986.

....to show how these fast algorithms can be used to accelerate the Hough transform. 1 Introduction Stack filters for adaptive applications are ideally suited to hardware implementation on reconfigurable FPGAs. These filters have proven their worth in noise reduction and morphological operations [2,3,4]. The two primitive morphological operators, erosion and dilation, can be easily performed by a stack filter. This allows the more sophisticated morphological filters to be built up using these base blocks. The median filter is the best known of the stack filters. It has excellent noise ....

....stack filter is guaranteed to be equal to one of the samples in the current input window. 2 Stack Filter Architectures Three main architectures have been proposed for stack filtering: i. Threshold decomposition ii. Input range compression iii. Binary refinement The threshold decomposition [4] architecture operates by decomposing each digital input sample of i bits into 2 i 1 binary threshold signals. Each of these threshold signals is then processed by a Positive Boolean Function (PBF) before being summed or stacked to obtain the output value as shown in Figure 1. The PBFs must be ....

P. D. Wendt, E. J. Coyle, and N. C. Gallagher, Jr., "Stack Filters" IEEE Trans. Acoustics, Speech, and Signal Processing, Vol. ASSP-34, no. 6, pp. 898-911, August 1986.

....can be classified as array, sorting network, or stack filter based architectures. In array architectures, each element in the window is operated on by maintaining its rank value. In sorting network architectures, the elements in the window are completely sorted for each output. A stack filter [7] translates the filtering operation to the binary domain through the use of threshold decomposition. The existing rank order filter architectures can be extended to WOS filters architectures [5] in a straightforward manner by repeating each sample according to its weight and then operating on the ....

Wendt P., Coyle E., and Gallagher N. "Stack Filters". IEEE Trans. On Acoustics, Speech, and Signal Processing. vol. ASSP-34, no. 4, August 1986.

....for implementation of TBFs in a general purpose fully parallel architecture, and at the same time obtain a systematic procedure to generate analogic programs for the CNN UM. 1 Introduction Threshold Boolean Filters form a wide class of nonlinear filters that includes Stack Filters (SF) [3], Order Statistic (OS) filters [4,5] and morphological filters [6] as special cases. Such filters find application in removal of non Gaussian noise and feature extraction [3,7,8,9] and are therefore used for many purposes in image processing. Architectures for realization of TBFs have been ....

....CNN UM. 1 Introduction Threshold Boolean Filters form a wide class of nonlinear filters that includes Stack Filters (SF) 3] Order Statistic (OS) filters [4,5] and morphological filters [6] as special cases. Such filters find application in removal of non Gaussian noise and feature extraction [3,7,8,9], and are therefore used for many purposes in image processing. Architectures for realization of TBFs have been proposed in the literature [1,3,10] generally as sequential processors exploiting the threshold decomposition property, to be defined below. In this paper, we shall show that TBFs can ....

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Wendt, P.D., Coyle, E.J., Gallagher, N.C., "Stack Filters", IEEE Trans. on Acoust., Speech, and Signal Processing, vol. 34, 1986, pp. 898-911.

....TBF approaches are provided in Section IV. II. STACK FILTERS A. Properties of Stack Filters Stack filters are a class of nonlinear filters that satisfy two properties: the weak superposition property known as the threshold decomposition and the ordering property called the stacking property in [5]. To define these properties and establish the notation used throughout this paper, we must introduce the threshold decomposition of an image. Images will be assumed to take values on a lattice , with each point in the lattice denoted by . A gray scale image with pixel values ranging between 0 ....

P. D. Wendt, E. J. Coyle, and N. C. Gallagher, Jr., "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 898--911, Aug. 1986.

....the weighting of errors in the mean absolute error criterion so it more closely matches a perceptual error criterion. This new approach requires less CPU time to design a stack filter, yet provides a dramatic improvement in the visual appearance of the filtered image. 1 Introduction Stack filters[1] [4] are a class of discrete time, nonlinear filters which are defined in terms of positive Boolean functions and a weak superposition property called the threshold decomposition. Algorithms for determining a stack filter which minimizes the mean absolute error criterion have been developed and ....

P.D. Wendt, E. J. Coyle, N. C. Gallagher, Jr., "Stack filters", IEEE Trans. Acoustics, Speech, and Signal Proc., Vol. 34, pp. 898-911, Aug. 1986

....work was supported in part by NSF Grants CDA 9015696, CDA 9422250, and CDA 9617388. 1 Introduction Stack filters are a class of nonlinear, sliding window filters that satisfy a weak superposition property known as the threshold decomposition and an ordering property called the stacking property [1, 2, 3]. They are especially useful in such image processing applications as image enhancement [4, 5, 6] and detection of intensity edges in noisy images [7] One of the main strengths of stack filters is the existence of an analytical technique for determining a stack filter that is optimal for ....

....for this new algorithm. Performance comparisons between current algorithms and the new algorithm are provided in Section 5. 2 Review of Optimal Stack Filtering Algorithms 2. 1 Stack Filters Every stack filter possesses both the threshold decomposition property and the stacking property [1, 2, 3]. To define these properties, we must first define the threshold decomposition of an image. Images will be assumed to take values on a lattice S. Each point in the lattice will be denoted by s 2 S. A gray scale image X with pixel values ranging between 0 and M may be represented as the sum of a ....

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P. D. Wendt, E. J. Coyle, and N. C. Gallagher, Jr., "Stack filters," IEEE Trans. Acoust. Speech, Signal Processing, vol. 34, pp.898-911, August 1986.

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P. Wendt et al., "Stack filters", IEEE Trans. Acoust., Speech, Signal Processing, pp. 898-911, 1986.

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P. Wendt, E. Coyle, and N. C. Gallagher, "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 898--911, 1986.

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P. D. Wendt, E. J. Coyle, and N. C. Gallagher, Jr., "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 898--911, Aug. 1986.

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P. Wendt, E. Coyle, and N. C. Gallagher, "Stack filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-34, pp. 898--911, 1986.

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P. D. Wendt, E. J. Coyle, and N. C. Gallagher, Jr., "Stack Filters" IEEE Trans. Acoustics, Speech, and Signal Processing, Vol. ASSP-34, no. 6, pp. 898-911, August 1986.

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