N. C. Gallagher and G. L. Wise, "A theoretical analysis of the properties of median filters", IEEE Trans. Acoust., Speech, Signal Proc., 29: 1136-1141, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... filter imposes a bias if sampling is aperiodic, which can be counteracted by weighting samples [Hmlinen 94] The following highlights the most characteristic properties of median filters; more comprehensive discussions can be found in the literature of digital image and speech processing and in [Gallagher 81] It follows from the definition of the median filter that monotonic signals pass unchanged, yet with a delay of k samples. Moreover, a signal passes unchanged if any two of its successive monotonic subsequences (of opposite direction) are separated by a sequence of at least k constant samples ....

....It follows from the definition of the median filter that monotonic signals pass unchanged, yet with a delay of k samples. Moreover, a signal passes unchanged if any two of its successive monotonic subsequences (of opposite direction) are separated by a sequence of at least k constant samples [Gallagher 81] Informally speaking, a median filter of length 2k 1 favours courses where the turnaround takes at least k samples. Because window width is its only design parameter, this observation may be used as a design guideline. The most characteristic property of median filters is, however, that they ....

NC Gallagher, GL Wise (1981) "A theoretical analysis of the properties of median filters", IEEE Transactions on Acoustics, Speech, and Signal Processing 29(6), 1136--1141

....In addition, this thesis introduces a two dimensional filter that is logically equivalent to the onedimensional pseudomedian filter and develops some of its properties. 4 Chapter 2 Root Signal Analysis This chapter develops some formal properties of the pseudomedian filter. Gallagher and Wise [2] and Tyan [3] first demonstrated the root signal set (that is, the set of signals unchanged by filtering) for the median filter. The root signal analysis that follows demonstrates the close relationship between the median and pseudomedian filters while pointing out important and useful ....

....However, the analysis presented below also holds true for extended signals. These signals are finite in length and have N constant points equal to the first value in the signal appended to the beginning of the signal, and N constant points equal to the last value appended to the end. See also [2]. The signal characteristics defined below create a precise vocabulary for the theorems presented in this chapter. The terms apply to a signal to be operated on by a filter of window width 2N 1. 1. A constant neighborhood is an area of N 1 or more consecutive points that have the same value. ....

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Gallagher, Jr., Neal C. and Gary L. Wise. "A theoretical analysis of the properties of median filters." IEEE Trans Acoust Speech Signal Process, v. ASSP-29 n. 12 (1981), pp. 1136-1141.

....WMMR tends to sharpen such edges by making them more steplike. For a filter window spanning samples, the WMMR is implemented by first selecting the values in the filter window having a minimum range. The output is computed by a weighted sum of the values. These filters, like the median filter [9], 15] have an interesting root signal analysis. Indeed, the root signals (signals that remain unchanged by filtering) of a WMMR filter of width are those signals that are PICO .It has also been shown that repeated passes of a WMMR filter eventually produces a PICO root signal. To achieve a ....

.... interesting properties of the median filter that first led to the introduction of the concept of locally monotonic regression [18] Just as the PICO signals are the fixed points of the WMMR filters, LOMO signals are the fixed points of median filters (with a well established 1 D fixed point theory [9], 15] Similar arguments may be made in favor of LOMO regression and LOMO image estimation as were made for PICO based methods. Since repeated filtering with a median filter leads inevitably to a LOMO signal, then median filtering may be seen as a method for inducing LOMOness on a signal. LOMO ....

N. C. Gallagher and G. L. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 1136--1141, 1981.

....of signal scale that allows a causal scale space: one in which new features are not created with increasing scale. It is worthwhile to note that for , a LOMO signal is also LOMO . The concept of local monotonicity first appeared in the analysis of the root signals produced by the median filter [6]. Since that time, algorithms have been designed that directly generate locally monotonic signals via nonlinear regression [10] 12] In the regression approach, the computation of a LOMO signal that resembles the original (and possibly noisy) signal is treated as a combinatorial optimization ....

....diffusion mechanism. The PDE based method has advantages over the median filter roots in computational complexity while maintaining closeness to the original signal. Furthermore, LOMO diffusion is not subject to the pathological behavior of the median filter given oscillating signals as input [6]. Compared with the regression based solutions, LOMO diffusion has advantages in computational cost. A major contribution of the LOMO diffusion technique is the convergence to a nontrivial class of signals. In addition, two other problems that are found with current diffusion algorithms are ....

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N. C. Gallagher and G. L. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 1136--1141, Dec. 1981.

....1( is the indicator function. The maximum number of iterations needed for convergence to a LOMO root signal is r = max IT(x) 0 x N 1] 20) 1294 As mentioned, the median filter produces LOMO root signals. The results from the literature can be summarized by Theorem 2: Theorem 2 [6][2]: The output of a length w = 2m l median filter median(I) equals I if and only if I is LOMO (m 2) Suppose that the 1 D signal I contains at least one monotonic segment of length m l. Then the w = 2m 1 median filter will reduce a length N signal to a root signal that is LOMO (m 2) in at most (N ....

N.C. Gallagher and G.L. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-29, pp. 1136-1141, Dec. 1981.

....sequence of length that is lomo is lomo as well; thus, the lomotonicity of a sequence is defined as the highest degree of local monotonicity that it possesses [1] A sequence is said to exhibit an increasing (resp. decreasing) transition at coordinate if (resp. The following (cf. 1] 16] [17]) is a key property of locally monotonic signals: If is locally monotonic of degree , then has a constant segment (run of identical elements) of length at least in between an increasing and a decreasing transition. The reverse is also true. The study of local monotonicity (which led to the idea of ....

N. C. Gallagher, Jr. and G. W. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 1136--1141, Dec. 1981.

....degree (or lomo or simply lomo in case is understood) if each and every one of its segments of length is monotonic. Throughout the following, we assume that .A sequence is said to exhibit an increasing (resp. decreasing) transition at coordinate if (resp. The following property (cf. 1] [3]) is key in the subsequent development of our fast algorithm. If is locally monotonic of degree , then has a constant segment (run of identical symbols) of length at least in between an increasing and a decreasing transition. The reverse is also true. If , then a sequence of length that is lomo ....

N. C. Gallagher Jr. and G. W. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 1136--1141, Dec. 1981.

....[19] is another example. This regression is the optimal counterpart of iterated median filtering. It involves the concept of local monotonicity, which we need to define. Local monotonicity is a property of sequences that appears in the study of the set of root signals of the median filter [11] [12], 14] 17] it constraints the roughness of a signal by limiting the rate at which the signal undergoes changes of trend (increasing to decreasing or vice versa) Let x be a real valued sequence (string) of length N , and let fl be any integer less than or equal to N.Asegment of x of length ....

N. C. Gallagher, Jr. and G. W. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. 29, pp. 1136--1141, Dec. 1981.

....and other related filters was poorly understood for many years. In the early 1980 s, important results on the statistical behavior of the median filter were presented [7] and a 1 2 new technique was developed that defined the class of signals invariant to median filtering, the root signals [8, 9]. Morphological filters are derived from a more rigorous mathematical background [10 12] which provides an excellent basis for design but few tools for analysis. Statistical and deterministic analyses for the basic morphological filters were not published until 1987 [5, 6, 13] The understanding ....

....in a given situation. An important deterministic property of a nonlinear filter is its root signal set, the set of signals that are unchanged by the operation of the filter. Root signals are also called fixed points of a filter. The root signal set was specified for the median filter in 1981 [8, 9] and for the morphological filters in 1987 [6] The relationship between the root signal sets of the median and morphological filters was also examined by Maragos in [6] 2.2.1. Median Filter Root Signals The median filter is an order statistic (stack) filter that replaces the center value in ....

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N. C. Gallagher, Jr. and G. L. Wise, "A theoretical analysis of the properties of the median filter," IEEE Trans. Acoust., Speech, Signal Process., vol. 29, no. 6, pp. 1136-1141, 1981.

....the mean of a neighborhood. This does remove noise, but also blurs details. Another important and well studied algorithm, is median filtering. In median filtering the current pixel value is replaced with the median value of a local neighborhood, see [ Tukey 1976, Narendra 1981, Huang et al. 1979, Gallagher and Wise 1981 ] This is noticeably better with respect to preserving detail, when the detail is larger than the median window. It still, however, blurs fine detail, e.g. corners, thin lines, rapidly varying texture. To make these statistical algorithms more robust and still preserve details, use of only ....

N.C Gallagher and G.L. Wise. A theoretical analysis of the properties of median filters. IEEE Trans. on Acoustics, Speech and Signal Processing, ASSP-29(6):1136--1141, December 1981.

.... of locality features of a program written directly by the student or chosen within a set of predefined, very simple programs; in the example shown in detail in this Section, the program is the implementation of the median filter algorithm applied to a 34#34 pixel image with a 3#3 pixel window [Gallagher81]. As shown in the previous Section, during the Program Development phase (Figure 1) a trace can be produced to allow a detailed program locality analysis (number of unique blocks, locality surface, spatial locality) If we define T #i# as the i th reference of a trace T , for each couple fT ....

N. Gallagher and G. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoustics Speech and Signal Proc., vol. 29, pp. 1136--1141, 1981.

....works. The work of Huang [36] and Martinez [62] represents two attempts to achieve temporal median filtering along motion trajectories using a 3 tap median filter. The median filter itself was introduced by Tukey [98] in 1977. The technique is one from the generic set of Order Statistic filters [107, 77, 27, 24]. These filters operate on some window of data. The procedure is first to rank the samples in that window and then the output of the filter is chosen to be one of the order statistic values for that ranked distribution. Thus the output can be the Upper Quartile value for instance. The median ....

....adversely affect the output image quality. Dirt and Sparkle is typically larger than this and so this thesis also spends some effort in introducing new MMF s that can solve this problem. It is possible to quantify the performance of median filters if suitable simplifying assumptions are made [5, 4, 24, 27, 74]. However these analyses are of limited use in practice. Nevertheless there is growing interest in optimal adaptive order statistic filtering which quantifies the effects of this general class of filters [56, 15] This thesis does not investigate the prospect of these optimal schemes, preferring ....

Neal C. Gallagher and Gary L. Wise. A theoretical analysis of the properties of median filters. IEEE Trans. Acoustics and Signal Processing, 29:1136--1141, December 1981. 154

.... of locality features of a program written directly by the student or chosen within a set of predefined, very simple programs; in the example shown in detail in this Section, the program is the implementation of the median filter algorithm applied to a 34#34 pixel image with a 3#3 pixel window [4]. As shown in the previous Section, during the Program Development phase (Figure 1) a trace can be produced to allow a detailed program locality analysis (number of unique blocks, locality surface, spatial locality) If we define T #i# as the i th reference of a trace T , for each couple fT ....

N. Gallagher and G. Wise. A theoretical analysis of the properties of median filters. IEEE Trans. Acoustics Speech and Signal Proc., 29:1136--1141, 1981.

....output respectively and (1.1) y(n) h 0 N k 1 H k [x(n) The Volterra filter is general enough to model many of the classical nonlinear filters, 1. 2) H k [x(n) N 1 i 1 0 N 1 i 2 0 N 1 i k 0 h k (i 1 ,i 2 , i k )x(n i 1 )x(n i 2 ) x(n i k ) including order statistic [2] [4] and morphological filters [5] By adding higher order terms to the Volterra series, its modelling accuracy can be improved. Filters based on the Volterra series and another representation called Wiener series [1] will be referred as polynomial filters. Unfortunately, the design of polynomial ....

....Approach UB 7 Input, 2nd Degree 4 7 7 28 5 Input, 3rd Degree 7 16 8 35 3 Input, 4th Degree 5 11 15 15 network, using block approach. According to the multidimensional Lagrange interpolation formula [18] a 2 dimensional function can be expressed as a polynomial with integer degrees for x 1 e [1,2] and x 2 e [1,2] The approximating function is then realized in the compact (5.7) f(x) 1 x 1 x 2 4.694 3.25x 1 3.25x 2 Const. and 1st degee block 0.722x 2 1 2.25x 1 x 2 0.722x 2 2 2nd degree block 0.5x 2 1 x 2 0.5x 1 x 2 2 3rd degree block 0.111x 2 1 x 2 2 4th degree ....

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N. Gallagher and G. Wise, "A Theoretical Analysis of the Properties of Median Filter," IEEE Trans. on Acoust, Speech, Signal Proc., Vol. ASSP-29, pp. 11361141, Dec. 1981.

....for the resources we currently monitor with the NWS. 4.2 Median based Methods The median value can also serve as a useful predictor, particularly if the measurement sequence contains randomly occurring, asymmetric outliers. Our presentation of these techniques follows the exposition in [19] and [12]. The median over a sliding window of fixed length whose leading edge is the most recent measurement is used as the forecast for the next measurement. That is, we define Sort K = the sorted sequence of the K most recent measurement values, Sort K (j) the jth value in the sorted sequence, and ....

Gallagher, N., and Wise, G. A theoretical analysis of the properties of median filters. IEEE Transactions ASSP (December 1981).

.... alternating sequential filters, that do not impose a rigid geometry on objects, and connected sets have been described[34, 37, 36, 35, 27] that fall within our definition of sieves [38, 2] A separate stream of development has been that of rank filters, including median [39] root median [40], recursive median [41] and, more generally, stack filters [42] Such filters have been developed primarily to remove random noise from signals and images, although there are suggestions for using them for tasks such as shape recognition [43] There is a close relationship between rank filters and ....

Gallagher N C. and Wise G N. A theoretical analysis of the properties of median filters. IEEE Trans. Acoust. Speech, Signal Processing, 21:pp 1136--1141, 1981.

....lomo ff as well; thus, the lomotonicity of a sequence is defined as the highest degree of local monotonicity that it possesses [9] A sequence x is said to exhibit an increasing (resp. decreasing) transition at coordinate i if x(i) x(i 1) resp. x(i) x(i 1) The following (cf. 19] 9] [20]) is a key property of locally monotonic signals: If x is locally monotonic of degree ff, then x has a constant segment (run of identical symbols) of length at least ff Gamma 1 in between an increasing and a decreasing transition. The reverse is also true. June 9, 1997 DRAFT 7 The study of ....

....The reverse is also true. 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 ....

N.C. Gallagher Jr. and G.W. Wise, "A theoretical analysis of the properties of median filters", IEEE Trans. ASSP, vol. ASSP-29, pp. 1136--1141, Dec. 1981.

....is that it is desirable for the MAP estimate to be stable, and not depend on an absolute parameter of scale such as T . On first inspection, it may seem that these goals are incompatible with the requirement of preserving image edges. However, many nonlinear operations such as median filters[29] and stack filters[30] have been developed which preserve edges with out explicit prior knowledge of edge size. Since both of these nonlinear operations are homogeneous, any scaling of the input data results in a proportional scaling of the output image. Therefore, these edge preserving operations ....

....10 15 20 25 30 35 40 45 50 Noisy Signals signal #1 signal #2 Unstable Reconstructions signal #1 signal #2 2 1 0 1 2 3 4 5 6 0 5 10 15 20 25 30 35 40 45 50 (a) b) Figure 3: Unstable reconstruction of two noisy pulses. a) Noisy square pulses with magnitudes 4.2(#1) and 4. 3 (#2) in the interval [20,29], and additive white Gaussian noise of unit variance. b) Resulting MAP estimates using Blake and Zisserman function with T = 1.75, # = 5, and b sr = 1 for adjacent points. Optimization was performed using 10 5 iterations of simulated annealing. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 1 ....

[Article contains additional citation context not shown here]

N. Gallagher and G. Wise, "A Theoretical Analysis of the Properties of Median Filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, no. 6, pp. 1136-1141, Dec. 1981.

No context found.

N. C. Gallagher and G. L. Wise, "A theoretical analysis of the properties of median filters", IEEE Trans. Acoust., Speech, Signal Proc., 29: 1136-1141, 1981.

No context found.

N. C. Gallagher and G. L. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, no. 6, 1981.

No context found.

N. C. Gallagher and G. L. Wise, "A theoretical analysis of the properties of median filters," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, no. 6, 1981.

No context found.

N.C. Gallagher Jr. and G.L. Wise, "A theoretical analysis of the properties of the median filter," IEEE Trans. Acoust., Speech, Signal Process. 29(6), 1136-1141 (1981).

No context found.

N. C. Gallagher, Jr. and G. L. Wise. "A theoretical analysis of the properties of the median filter." IEEE Trans Acoust Speech Signal Process, v. 29 n. 12 (1981) pp. 1136-1141.

No context found.

N.C Gallagher and G.L. Wise. A theoretical analysis of the properties of median filters. IEEE Trans. on Acoustics, Speech and Signal Processing, ASSP-29(6):1136-1141, December 1981.

No context found.

Gallagher, N. C. JR. and Wise, G., "Theoretical Analysis of the Properties of the Median Filters", IEEE transactions on Acoustics, Speech, and Signal Processing, ASSP-29, 6, (1981).

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