A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoustics, Speech, Signal Proc., vol. 31(6), pp. 1342--1349, Dec. 1983.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a hybrid system that incorporates both linear are non linear filtering components. By including both linear and morphological perceptron functionality, a shared weight generalized perceptron network can implement this type of hybrid architecture. Other examples of this approach are L filters [19] where multiple order statistic filters are combined with a linear combination. In another approach [20] the outputs from a bank of linear phase FIR filters are combined with a median or order statistic filter. A comprehensive account of nonlinear filters, particularly with respect to hybrid ....

Bovik, A.C., T. Huang, and D. Munson, A generalization of median filtering using linear combinations or order statistics. IEEE Trans. Acoust., Speech, Signal Processing, 1983. 31: p. 1342-1350.

....median (CWM) filter [7] which gives more weight only to the central value of the window. It is also reasonable to give emphasis to the central sample, because it is one that is the most correlated with the desired estimate. The median filter, as well as its modifications and generalizations [8] are typically implemented invariantly across an image. They tend to alter pixels undisturbed by noise. Additionally, they are prone to edge jitter in cases where the noise ratio is high. As a result, their effectiveness in noise suppression is often at the expense of blurred and distorted image ....

A. C. Bovik, T. Huang and D.C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 1342-1350, Jun. 1983

....value of MEM is attained at . If we stretch the tail of the distribution, thereby increasing , the efficiency will increase and tend to as .A limiting case is the Cauchy distribution ( distribution with ) in which so that ARE . A generalization of the median filters is the class of L filters [16]. Its output is given by where are the filter coefficients. It is easy to show that the MEM filter is a special case of L filters with corresponding weights if otherwise Since the relaxed median filter provides improvement over the standard median filter in preserving details and removes noise ....

A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 1342--1350, Dec. 1983.

....nonlinearity of the median filter permits it to smooth an image without the degree of blurring that a linear filter with similar smoothing characteristics can introduce. A variety of hybrid techniques that employ both median and linear filters combine the characteristics of both approaches [2] [3], 16] 20] 33] A median filter can, however, introduce spurious artifacts into the transformed image. The filter removes or attenuates image features that are smaller than the filter window in at least one dimension. This is the strength of the filter, or a weakness, depending on the ....

Bovik, A. C., T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust. Speech Signal Process., vol. 31, pp. 13421350, December 1983.

....If all a i =1, the resulting filter is sample mean; and if the middle weight is one and all the rest are zeros, then the resulting filter is median. Thus, the L filter provides a compromise between median and mean. The weights a i for a 3 3 neighborhood are listed in Table 1. They were taken from [7] optimized for a Laplacian distribution, but this is not critical as many arbitrary choices did work well. The idea is to put more weight on the center of the ordered values (as in median) and also include the rest to the sum (as in mean) Tab l e 1 3 3 L filter Coefficients Used in Eq. 2) a0 ....

A.C. Bovik, T.S. Huang, and D.C. Munson, Jr., "A generalization of median filtering using linear combinations of order statistics," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 31, pp. 1342--1350, 1983.

.... The midrange filter is a wellknown estimator in the order statistics literature; see, for example, 25 27] The midrange filter is optimal in the mean square sense among all filters that are linear combinations of order statistics for removing uniformly distributed noise from a constant signal [26]. The midrange filter is also the maximum likelihood estimator for uniformly distributed noise [26] The notation for the midrange filter is given in equation (3.1) below. midr(f; N) 1 2 (f EK f ED ) 3.1) 42 Pseudomedian Filter The pseudomedian filter was originally defined in 1985 by ....

.... [25 27] The midrange filter is optimal in the mean square sense among all filters that are linear combinations of order statistics for removing uniformly distributed noise from a constant signal [26] The midrange filter is also the maximum likelihood estimator for uniformly distributed noise [26]. The notation for the midrange filter is given in equation (3.1) below. midr(f; N) 1 2 (f EK f ED ) 3.1) 42 Pseudomedian Filter The pseudomedian filter was originally defined in 1985 by Pratt, Cooper, and Kabir [28] They defined the filter in one dimension to be the average of the ....

A. C. Bovik, T. S. Huang, and D. C. Munson, Jr., "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Process., vol. 31, no. 6, pp. 1342-1350, 1983.

....Refinement level 4 is achievable on a higher end workstation. Refinement level 5 is currently in the supercomputer league. 2.4. Relation to Order Statistic Filters Finite element filters with piecewise linear interpolation have an interesting relationship with order statistic filters (OSF) [5]. If one makes the (stronger) symmetry assumption that the ordering of pixels within a 3x3 window doesn t matter, then the result is the permutation group S9 the symmetric permutation group on nine elements. If a single hypercube element is used, and the basis functions are constructed to be ....

Alan C. Bovik, Thomas S. Huang, and David C. Munson. A generalization of median filtering using linear combinations of order statistics. IEEE Transactions on Acoustics, Speech, and Signal Processing, 31(6):1342-- 1349, December 1983.

....0.7 0.8 Figure 1.2: Probability density functions of the Gaussian, Laplacian and Uniform distributions p n (n) 1 p 2oe e Gamma p 2jnj oe (1. 6) Nonlinear estimators can provide a much more accurate estimate of the mean of a stationary Laplacian random variable than the linear average [6]. ffl Uniform noise is not often encountered in real world imaging systems, but provides a useful comparison with Gaussian noise. The linear average is a comparatively poor estimator for the mean of a uniform distribution. This implies that nonlinear filters should be better at removing uniform ....

....X which uses a linear combination of order statistics: F (X 1 ; X 2 ; XN ) ff 1 X (1) ff 2 X (2) ff N X (N) 1. 9) Order Statistic Filters have long been known to statisticians as L estimators, but were re christened and applied to image processing problems by Bovik et al. [6]. Some common filters which fit the order statistic filter framework are: ffl The linear average, which has coefficients ff i = 1=N (1.10) ffl The median filter, which has coefficients ff i = 1 i = N 1) 2 0 otherwise (1.11) For image processing applications, N is almost always odd, so ....

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A. C. Bovik, T. S. Huang, and D. C. Munson. A generalization of median filtering using linear combinations of order statistics. IEEE Transactions on Acoustics, Speech, and Signal Processing, 31(6):1342--1349, December 1983.

....the original pixel value s0 using a nonlinear function F (x0 ; xn Gamma1 ) of pixels from the degraded image. For example, a 3x3 filter would have as inputs the pixel to be restored (x0 ) and its eight neighbours: x8 x4 x5 x3 x0 x1 x7 x2 x6 Median filters, order statistic filters [2], and Lee s local statistics filter [5] are well known examples of such filters. For a signal and degradation process satisfying certain weak conditions, a function F which is optimal in the minimum mean squared error (MMSE) sense exists. This MMSE optimal filter tends to have no computationally ....

....to handle detail regions, and F2 is a point estimator used to restore flat regions. The vector x contains pixel values from the local neighbourhood. For additive white Gaussian noise, F2 is a local average; for other noise distributions, order statistic filters provide better noise reduction [2]. The function fi(x) ranges between 0 for flat regions and 1 for highly detailed regions. The equation for fi is borrowed from the Lee filter structure [5] Given a variance estimate oe 2 x for a local window, fi is given by: fi = max oe 2 x Gamma oe 2 n oe 2 x ; 0 (2) where ....

[Article contains additional citation context not shown here]

A. C. Bovik, T. S. Huang, and D. C. Munson. A generalization of median filtering using linear combinations of order statistics. IEEE Transactions on Acoustics, Speech, and Signal Processing, 31(6):1342--1349, December 1983.

....nonlinearity of the median filter permits it to smooth an image without the degree of blurring that a linear filter with similar smoothing characteristics can introduce. A variety of hybrid techniques that employ both median and linear filters combine the characteristics of both approaches [2, 3, 16, 20, 33]. A median filter can, however, introduce spurious artifacts into the transformed image. The filter removes or attenuates image features that are smaller than the filter window in at least one 1 dimension. This is the strength of the filter, or a weakness, depending on the features and the noise ....

Bovik, A. C., T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust. Speech Signal Process., vol. 31, pp. 1342-1350, December 1983.

....For various signal processing applications it is sometimes useful to mix in the same system both nonlinear and linear filtering strategies. Thus, hybrid systems, composed by linear and nonlinear sub systems, have been frequently proposed in the research literature. For example, Bovik et al. [3] investigated the L filters, whose output is defined as a linear combination of the order statistics of the input signal. Bednar and Watt [4] analyzed the alpha trimmed means, representing a special case of the L filters. Heinonen and Neuvo [5] proposed the FIR median hybrid filters, where the ....

A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. 31, pp. 1342--1350, Dec. 1983.

....Since this technique is specific to coding applications, it will not be developed here. 2.2. 1 Particular FSM L predictors By constraining the parameters of the FSM L predictor and selecting a suitable FSM modeller, we can distinguish between following classes: linear predictors, L predictors[1], L predictors[5] adaptive soft Boolean predictors[6] adaptive Boolean predictors[4] 7] 3 Application to Lossless Compression of Gray Scale Images The contexts are supposed to split the highly nonstationary natural images into several stationary subsources. Suitable conditional probabilities ....

A.C.Bovik, T.S. Huang, and D.C.Munson Jr. A generalization of median filtering using linear combinations of order statistics. IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP-31:1342-- 1350, Dec. 1983.

.... estimator for removing signal dependent camera noise from image sequences [2,3] Various forms of order statistics estimators have been introduced in the literature, ranging from simple median, minimum and maximum filters to adaptive estimators using linear combinations of ordered observations [4,5,6]. Due to the ordering of the observations prior to the actual filtering operation, OS estimators often outperform linear estimators. In this correspondence we propose a new technique to estimate the weights of an adaptive OS estimator. The applicability of this technique is not limited to the ....

....Since, however, 0 # 1, the expectation and variance of the noise n i j k # ( is dependent of the signal f(i,j,k) 3. MSE OPTIMAL ORDER STATISTIC ESTIMATOR Weighted order statistic (OS) estimators have proved to be very useful in various image (sequence) processing problems [4 6,10]. Any weighted OS estimator forms an estimation of a signal s value using a linear combination of ordered observations taken from a local observation window or windows. The various OS estimators presented in the literature differentiate in how they determine the observation window(s) and how they ....

[Article contains additional citation context not shown here]

A.C. Bovik, T.S. Huang, and D.C. Munson, "A generalization of median filtering using linear combinations of order statistics", IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 31, pp.1326-1337, 1983.

....but even in a more general (gray level) situation, non linear reconstruction filters can be designed to explore the inherently non band limited nature of sharpedges present in most images. The non linear reconstruction filter we analyze in this paper is a modified rank order filter (an L filter)[11]. It consists of averaging the result of the samples at rank .50 and .51 when using the weights in the mask of Figure 2a. A non linear filter cannot generally be decomposed into polyphase sub filters. Nevertheless, since we only apply this filter to the up sampled signal, a polyphase decomposition ....

A. C. Bovik, T. S. Huang, and J. D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. 31, pp. 1342--1350, Dec. 1983.

....signal with tone interference and additive contaminated Gaussian noise. 26 ii 1 Introduction Considerable attention has been given in the signal processing literature to the definition and testing of rank order filters and their generalizations [2, 7, 13, 16, 22, 25, 28, 29]. These filters have their roots in order statistics, which provides a mechanism that attains advantages over traditional linear filters; they can be designed (a) to be robust in environments where the assumed statistics deviate from Gaussian models and are possibly contaminated with outliers, and ....

....x r = x (1) x (2) x (n) The median filter is perhaps the most widely used rank order filter whose output, at each time interval, is the sample median of the observation vector. To generalize median and rank order filter, order statistic filters, or L filters, were introduced [7]. L filters take as their output a linear combination of the samples in the sorted vector x r , rather than simply a single order statistic. Nevertheless, rank order information alone is not sufficient in many important applications. Order statistic based estimators suffer from a fundamental ....

A. C. Bovik, T. S. Huang, and D. C. Munson, Jr., "A generalization of median filtering using linear combinations of order statistics," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 31, no. 12, December, 1983.

....produce piece wise constant surfaces and thus the images appear blocky; 3) It is not clear how scale is defined. Median filtering was first proposed by Tukey [7] for smoothing data. It has been used for speech processing and image restoration. Its generalized versions, order statistic (OS) filters [8] and variations have also been widely studied and applied. However, it is known that median or OS filters do not provide sufficient smoothing for nonimpulsive noise. Another recent approach was the line processes introduced in [2] A line process is a binary variable which signals the presence ....

A.C. Bovik, T.S. Huang and D.C. Munson, Jr., "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, 31, 1342-1350, 1983.

....vector of the ordered input samples within a sliding window at time step i. The main feature of L filters is that they can be designed to suppress white noise which is nongaussian in nature. The design can be accomplished using two different optimality principles, the mean squared error criterion [2] and the least squares method [3] The filter coefficients can also be computed adaptively using proper modifications of the LMS and RLS algorithm [4] 5] The capability of L filters to deal with nongaussian signals is due to the sorting process which performs an estimation of the cumulative ....

A.C. Bovik, T.S. Huang, and D.C. Munson. A generalization of median filtering using linear combinations of order statistics. IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP31 (6):1342--1349, Dec. 1983.

....by the median filters which preserve the edges and are the optimal estimators for impulsive noise. There is now a multitude of nonlinear filters based on data ordering. Among them are the L filters whose output is defined as a linear combination of the order statistics of the input sequence [4]. It is well known that digital image filtering techniques must take into account the local image content (i.e. the local statistics) because image statistics vary throughout an image. It has been proved both in theory and in practice that adaptive techniques can cope with nonstationary and or ....

A.C. Bovik, T.S. Huang, and D.C. Munson, Jr., "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. on Acoustics, Speech and Signal Processing, vol. ASSP-31, no. 6, pp. 1342--1349, December 1983.

....June 9, 1997 DRAFT 7 The study of local monotonicity (which led to the idea of locally monotonic regression) has a relatively long history in the field of nonlinear filtering. Local monotonicity appeared in the study of the set of root signals of the median filter [19] 20] 21] 22] 23] [24], 25] The median is arguably the most widely known and used nonlinear filter. As it turns out, in 1 D, the set of root signals (fixed points) of a median filter of a certain length (i.e. the set of those signals that are not affected at all as they pass through the given median filter) is the ....

A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics", IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 31, pp. 1342--1349, 1983.

....in nature. This ability is due to the ordering process which performs an explicit estimation of the cumulative distribution function of the input signal. The design of the filter coefficients a k can be done using two different optimality principles, the mean squared error (MSE) criterion [1], J(a) Ef(a T z i Gamma s i ) 2 g min a ; 2) and the least squares (LS) method [2] J i (a i ) i X k=1 i Gammak (a T i z k Gamma s k ) 2 min a i ; 3) where s i is the desired response and the so called forgetting factor. 2 Algorithms Since the structure of the ....

A.C. Bovik, T.S. Huang, and D.C. Munson. A generalization of median filtering using linear combinations of order statistics. IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP-31(6):1342--1349, Dec. 1983.

....misleading false contours would have developed in the solution image, thus distorting possible interpretation of the parenchymal tissues revealed in the mammogram. As a filter comparison to the PILI image estimation method, we applied a specific order statistic (OS) filter to the noisy mammogram [7]. Within a finite window, the filter alge TABLE II PICO AND PILI IMAGE ESTIMATION AND FILTERING RESULTS braically orders the intensities within the window, then linearly weights them using a piecewise linear (triangular) weighting to compute the output. Thus, the filter, called the OS filter ....

....braically orders the intensities within the window, then linearly weights them using a piecewise linear (triangular) weighting to compute the output. Thus, the filter, called the OS filter (triangle OS filter) was selected, since it is near optimal for heavy tailed noise in minimum variance sense [7]; it is highly robust, and it supplies a linear weighting to naturally ordered samples near intensity transitions. This makes it a fair comparison for a piecewise linear fit. It was implemented by applying a 1 D OS filter along both the rows and columns of Fig. 4. Illustrative examples of LOMO 3 ....

A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 1342--1350, 1983.

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A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoustics, Speech, Signal Proc., vol. 31(6), pp. 1342--1349, Dec. 1983.

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A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. 31, pp. 1342--1349, 1983.

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A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 1342--1349, 1983.

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A. C. Bovik, T. S. Huang, and D. C. Munson, "A generalization of median filtering using linear combinations of order statistics," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-31, pp. 1342--1349, 1983.

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