Output Distributions of Stack Filters Based on Mirrored Threshold Decomposition

Transcription

1 1454 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001 Output Distributions of Stack Filters Based on Mirrored Threshold Decomposition Ilya Shmulevich, Member, IEEE, José Luis Paredes, Gonzalo R. Arce, Fellow, IEEE Abstract The output distribution formula for stack filters based on mirrored threshold decomposition is derived. This formula allows one to compute the cumulative distribution function of the output of a stack filter for a given independent identically distributed (i.i.d.) input noise distribution. Mirrored threshold decomposition permits us to analyze the input output characteristics of the stack filter in the binary domain. The sliding window operation of the stack filter is modeled by a deterministic finite automaton. The output distribution of the filter is obtained by interpreting the automaton as a Markov chain whose transition probabilities depend on the probabilistic description of the binary input signal. An example illustrating the use of the output distribution formula is provided. Index Terms Deterministic finite automaton, Markov chain, mirrored threshold decomposition, output distribution, stack filter. I. INTRODUCTION LINEAR filters have long been used in many signal image processing applications. They can be easily analyzed implemented. However, many noise contamination situations, such as impulsive, speckle, or signal-dependent noise, dem the use of nonlinear filters [1], [2]. Several important nonlinear filter 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 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 dferentiate them from the more powerful stack filter structures defined in the binary domain of mirrored threshold decomposition. Stack filters based on mirrored threshold decomposition are much more versatile, being empowered not only with lowpass filtering characteristics but with bpass or highpass filtering characteristics as well. Much like the stack smoother framework used in the definition of weighted order Manuscript received October 12, 1999; revised April 3, The associate editor coordinating the review of this paper approving it for publication was Dr. Kenneth Kreutz-Delgado. I. Shmulevich was with the Tampere International Center for Signal ocessing Laboratory, Tampere University of Technology, Tampere, Finl. He is now with the University of Texas MD Anderson Cancer Center, Houston, TX USA. J. L. Paredes G. R. Arce are with the Department of Electrical Computer Engineering, University of Delaware, Newark, DE USA ( paredesj@eecis.udel.edu; arce@eecis.udel.edu). Publisher Item Identier S X(01) statistic (WOS) weighted median (WM) smoothers, the new stack filter framework naturally leads to the definition of WOS WM filters that admit positive negative weights. Consequently, this allows such filters to possess arbitrary frequency-selective characteristics [11], [12]. To date, statistical analysis for stack-type filters has been limited to the case of stack smoothers [6] [9]. In this paper, we develop statistical analysis tools for the more general class of stack filters, in particular, deriving the output distribution formula. This formula enables us to compute the cumulative distribution function of the output of any stack filter with a given input noise distribution, which is assumed to be i.i.d. The recently introduced analytical tool called the mirrored threshold decomposition [10] forms the basis of our derivation in that it permits us to analyze the input output characteristics of the stack filter in the binary domain, despite the input output signals being real valued. Subsequently, the sliding window operation of the stack filter is modeled by a deterministic finite automaton. The output distribution of the filter is obtained by interpreting the automaton as a Markov chain whose transition probabilities depend on the probabilistic description of the binary input signal resulting from the mirrored threshold decomposition operation. As with stack smoothers, the output distribution formula for stack filters can play a key role in their statistical optimization. The output distribution of any stack filter can be obtained in terms of the input distribution, making it possible to optimize stack filters in the mean square sense [2]. In other words, the knowledge of the input distribution will allow one to find a stack filter or a set of stack filters that minimize the output variance, which, in turn, is a measure of the filter s noise attenuation capability. Thus, the statistical analysis tools derived in this paper will lead to new stack filter optimization methods that were previously available for stack smoothers only. Section II introduces mirrored threshold decomposition the new class of stack filters based on it. Section III is devoted to the modeling of the stack filter operation by a deterministic finite automaton the derivation of the output distribution formula, which is stated as Theorem 4. An example illustrating the use of the output distribution formula is given in Section III-C. Finally, Section IV contains some concluding remarks. II. MIRRORED THRESHOLD DECOMPOSITION AND STACK FILTER REPRESENTATION Threshold decomposition provides the foundation needed for the definition of stack smoothers. The class of stack filters has been recently defined in a similar fashion through a more general threshold decomposition architecture referred to as mirrored threshold decomposition [10] X/01$ IEEE

2 SHMULEVICH et al.: OUTPUT DISTRIBUTIONS OF STACK FILTERS 1455 Consider the set of integer-valued samples forming the vector. For purposes of simplicity, the input signals are quantized into a finite set of values with. Unlike threshold decomposition, mirrored threshold decomposition of generates two sets of binary vectors, each consisting of vectors. The first set consists of the vectors associated with the traditional definition of threshold decomposition. The second set of vectors is associated with the decomposition of the mirrored vector of, which is defined as. Since takes on symmetrical values about the origin from, is referred to as the mirror sample of or simply as the signed sample. Threshold decomposition of leads to the second set of binary vectors. The th element of is specied by Much like threshold decomposition can be extended to admit real-valued signals, mirrored threshold decomposition can be extended in a similar fashion. Consider the real-valued vector. Mirrored threshold decomposition maps this real-valued vector to an infinite set of binary vectors,, where sgn sgn sgn sgn sgn sgn can be recon- Since threshold decomposition is invertible, structed from its binary representation as (4) (5) (1) whereas the th element of is defined by The thresholded signal can be written as sgn, where sgn denotes the sign function defined as sgn are both reversible from their corresponding set of decomposed signals, consequently, an integer-valued signal has a unique mirrored threshold signal representation vice versa: where denotes the one-to-one mapping provided by the mirrored threshold decomposition operation. Thus, the original integer-valued signal can be exactly reconstructed from its binary representation through the inverse process as As an example, the representation of the vector in the binary domain of mirrored threshold decomposition is (2) (3) for A. Stack Filters The output of a stack filter is the result of a sum of a stack of binary operations acting on thresholded versions of the input samples their corresponding mirrored samples. The stack filter output is defined by (6) where, are the thresholded samples defined in (1) (2), where is a -variable positive Boolean function (PBF) that contains only uncomplemented input variables in its minimal sum-of-products form. Given an input vector, its mirrored vector, their set of thresholded binary vectors, ;,, it follows from the definition of threshold decomposition that the set of thresholded binary vectors satisfies the partial ordering That is, stack, i.e., for all. Consequently, stack filtering of the thresholded binary vectors by the PBF also satisfies (7) The stacking property in (7) ensures that the decisions on dferent levels are consistent. Thus, the filter at a given time location decides that the signal is less than, then the filter outputs at levels greater must draw the same conclusion. As defined in (6), stack filter input signals are assumed to be quantized to a finite number of signal levels. Following an approach similar to that with stack smoothers, the class of stack filters admitting real-valued input signals is defined next.

3 1456 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001 Definition 1 (Continuous Stack Filters): Given a set of real-valued samples, the output of a stack filter defined by a PBF is given by where the thresholding function is defined in (1). The link between the continuous stack filter the corresponding PBF is given by the following property. operty 2: Let be a real-valued vector its corresponding mirrored vector that are inputted to a stack filter defined by the positive Boolean function. The PBF with the sum of products expression state labels represent the contents of the filter s window at a given point in time at threshold. The second half of the state labels represents the mirrored real-valued samples thresholded at level. That is,. Let us also suppose that the s are i.i.d. with cumulative distribution function. Let be the value immediately to the right of the filter window, that is, when the window slides by one sample to the right, will be in its right-most position. Furthermore, in favor of notational simplicity, let. Thus, on any given threshold level, we can think of the pair as being the input to the automaton, even though there is only one real-valued input. Hence, the input alphabet consists of four symbols: 1) 2) 3) 4) We can now define the state transition function as (8) where are subsets of, has the stack filter representation Finally, to complete the description of the stack filter operation using the dfa, we can define the output function as with not having the th element at once. Thus, given a positive Boolean function that characterizes a stack filter, it is possible to find the equivalent filter in the real domain by replacing the binary AND OR operations acting on the s s with max min operations acting on the real-valued samples. III. OUTPUT DISTRIBUTIONS OF STACK FILTERS A. Modeling Stack Filters By Deterministic Finite Automata The automaton model that we will use to describe the operation of stack filters is commonly referred to as the Mealy machine [13]. We give its definition below. Definition 3: A deterministic finite automaton (dfa) is a system in which is the input alphabet, is the output alphabet, is the set of states, are the transition output functions, respectively. A useful way to visualize a dfa is to draw its state transition diagram. We can draw a graph in which each state is represented by a vertex;, then there is an arrow from to over which we write. Let be the positive Boolean function describing the stack filter. Consider the binary signal obtained by thresholding the real-valued signal at level. With each binary vector, we will associate a state. The first Thus, the output does not depend on the input value, we can call the output of state. We now turn to the derivation of the output distribution formula for stack filters. B. Derivation of the Output Distribution Formula As there are four possible input values, we can proceed to compute the probability of each, by doing so, we construct a Markov chain with the computed transition probabilities. From the definition of mirrored threshold decomposition, it follows that for However, to obtain the transition probabilities, we need to compute the joint probabilities Let us first consider for (9) Since are functionally related, all the probability masses are on the line. Thus, the joint distribution can be expressed in terms of. Two cases need to be distinguished, depending on whether is positive or negative. If is negative, then the probability mass is located on the line within the intersection of two half planes, namely,. This is illustrated in Fig. 1, where

4 SHMULEVICH et al.: OUTPUT DISTRIBUTIONS OF STACK FILTERS 1457 Fig. 1. Computation of fx = 01; s =+1g when t 0. the probability mass is located on the line between points A B is equal to. When is positive, it can easily be seen that no probability mass will be located in the intersection of the two half planes, thus, the joint probability in (9) is equal to zero. Summarizing, we have (10) In a similar fashion, we can obtain the joint probabilities for the other three possible inputs. These probabilities are (11) (12) (13) The joint probabilities in (10) (13) as functions of are shown in Fig. 2, where is used. Let us denote the four functions in (10) (13) by,,,, respectively. When the densities of the input rom variables are symmetrical with respect to the origin, such as Gaussian, Laplacian, Cauchy, Unorm densities with mean equal to zero, it follows that. Using this fact, (10) (13) reduce to (14) (15) (16) As can be seen from (16), the inputs are probabilistically indistinguishable. This can also be seen from Fig. 2(c) (d). The above probabilities completely define the Markov chain that models the sliding window process. The total number of states is, there are four possible transitions from each every state with the probabilities in (10) (13). These probabilities uniquely specy the state transition matrix of the Markov chain. Then, by solving the equations, where, we can obtain the steady-state probabilities for each state or, equivalently, the invariant distribution of the Markov chain. Summing up the probabilities over all states on which the function is equal to, we immediately obtain the output cumulative distribution function of the stack filter in question. Unfortunately, solving for the row eigenvectors of is analytically burdensome when its dimensions are arbitrarily large (i.e., ), even though the matrix possesses a well-defined structure. A more intuitive method is afforded by the state labels themselves, which inherently capture a finite amount of memory of the Markov chain, that is, a state label gives us information about the last transitions that have taken place [see (8)]. Thus, the process, which is in steady state, is stopped at a rom point in time, the

5 1458 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001 Fig. 2. Joint probabilities as functions of t. (a) fx = +1; s =+1g. (b) fx = 01; s = 01g. (c) fx = 01; s =+1g. (d) fx =+1;s = 01g. probability of finding it in any given state can be obtained by direct inspection of the state labels. For example, the state is, we can conclude that the last three inputs must have been,,, which occur with probabilities,,, respectively. Consequently, the steady-state probability of that particular state must be equal to. Before stating the above results in a more formal manner, let us clary some notation. Since are binary variables taking values in, the operations of conjunction complement are defined in the usual manner, having replaced 0 by 1 by. That is,. Moreover, for any two vectors,,. Let be the Hamming weight of, that is, the number of elements equal to. Finally, let be a binary vector of length. Then, we have the following Theorem 4: The output cumulative distribution function of the stack filter defined by the positive Boolean function is equal to TABLE I EXAMPLE OF A COMPUTATION OF THE OUTPUT DISTRIBUTION FUNCTION At this point, let us consider an example illustrating the application of the above Theorem. C. Example Suppose we wish to compute the output distribution of the stack filter defined by when the input variables are i.i.d. In the domain of real numbers, this stack filter can be expressed as (17) with i.i.d. input distribution,,, denoting the joint probabilities in (10) (13). Upon direct inspection of the above expression, it can be seen that the output of the stack filter is restricted to be non-negative, even though the inputs can take positive or negative values. To see this, suppose all four terms inside the operator

6 SHMULEVICH et al.: OUTPUT DISTRIBUTIONS OF STACK FILTERS 1459 Fig. 3. (a) Input (N(0; 1) output probability density functions. (b) Input output cumulative distribution funtions. are negative, a contradiction will result. To compute the output distribution, we must first list the vectors on which. These vectors are given in Table I. The last column of the table shows the terms associated with each vector, which have been obtained using (17). Finally, the output cumulative distribution function of the stack filter can be obtained by summing all these terms, which results in Fig. 3(a) shows the input output probability density functions, whereas Fig. 3(b) shows the input output cumulative distribution functions for this example. Note that as expected, all output values are non-negative. It is interesting to note that some states, such as in this example, have zero steady-state probability, as indicated by the sixth row in the table. The reason for this is that, then. On the other h,, then [see (10) (11)]. Thus, given any threshold, both events cannot occur. This can be stated more generally in the following oposition 5: Let. If, then, the steady-state probability for state is equal to zero. IV. CONCLUDING REMARKS The output distribution formula has played an important role in the statistical analysis of stack smoother properties. For instance, the so-called rank selection probabilities, which are probabilities that the output of the filter is equal to an input sample with a given rank, can be easily calculated by using this formula. In addition, the output distribution formula forms the basis of statistical optimization of stack filters. The rank selection probabilities can, in turn, be used to specy robustness constraints during the optimization stage. The newly introduced class of stack filters is more general powerful than the class of stack smoothers. In fact, a stack smoother is simply a stack filter whose PBF does not depend on the variables (i.e., they are fictitious). Consequently, the well-known output distribution formula for stack smoothers is a special case of the formula given in Theorem 4. It should be mentioned that the usual approach taken for the derivation of the output distribution for stack smoothers, by partitioning the input space into disjoint events [2], is not applicable to stack filters, which are based on mirrored threshold decomposition. This is because the input samples used by a stack filter are no longer independent. The dynamic approach, based on Markov chains, taken in this paper is more general, as it can serve as an alternate straightforward method for deriving the classical output distribution formula for stack smoothers as well as for the recursive counterparts of these filters [9], where the independence assumption is once again violated. The newly developed output distribution formula paves the way toward statistical optimization of stack filters based on mirrored threshold decomposition, allowing a richer variety of problems to be tackled. This should be the subject of future research. REFERENCES [1] I. Pitas A. N. Venetsanopoulos, Nonlinear Digital Filters: inciples Applications. Boston, MA: Kluwer, [2] J. Astola P. Kuosmanen, Fundamentals of Nonlinear Digital Filtering. New York: CRC, [3] P. Wendt, E. J. Coyle, N. C. Gallagher Jr, Stack filters, IEEE Trans. Acoust., Speech, Signal ocessing, vol. ASSP-34, Aug [4] O. Yli-Harja, J. Astola, Y. Neuvo, Analysis of the properties of median weighted median filters using threshold logic stack filter representation, IEEE Trans. Acoust., Speech, Signal ocessing, vol. 39, pp , Feb [5] M. Gabbouj E. J. Coyle, Minimum mean absolute error stack filtering with structural constraint goals, IEEE Trans. Acoust., Speech, Signal ocessing, vol. 38, pp , June [6] O. Yli-Harja, Formula for the joint distribution of stack filters, IEEE Signal ocessing Lett., vol. 1, pp , June [7] P. Kuosmanen, Statistical Analysis Optimization of Stack Filters. Helsinki, Finl: Acta Polytechnica Scinavica, 1994.

7 1460 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 49, NO. 7, JULY 2001 [8] S. Agaian, J. Astola, K. Egiazarian, Binary Polynomial Transforms Digital Filters. New York: Marcel Dekker, [9] I. Shmulevich, O. Yli-Harja, K. Egiazarian, J. Astola, Output distributions of recursive stack filters, IEEE Signal ocessing Lett., vol. 6, pp , July [10] J. L. Paredes G. R. Arce, Stack filters, stack smoothers, mirrored threshold decomposition, IEEE Trans. Signal ocessing, vol. 47, pp , Oct [11] G. R. Arce, A general weighted median filter structure admitting negative weights, IEEE Trans. Signal ocessing, vol. 46, pp , Dec [12] G. R. Arce J. L. Paredes, Recursive weighted median filters admitting negative weights their optimization, IEEE Trans. Signal ocessing, vol. 48, pp , Mar [13] O. Kuznetsov G. Adelson-Velskii, Discrete Mathematics for the Engineer. Moscow, Russia: Energoatomizdat, José L. Paredes was born in Mérida, Venezuela. He received the Diploma in electrical engineering from the Universidad de Los Andes, Mérida, in 1995, where he was the first student to graduate Summa cum Laude from his department. In 1999, he received the M.S. degree in electrical engineering from the University of Delaware, Newark, where he is currently pursuing the Ph.D degree. From 1995 to 1996, he was an Assistant ofessor with the Electrical Engineering Department, Universidad de Los Andes. Since September 1996, he has being a Research Assistant with the Department of Electrical Computer Engineering, University of Delaware. He has consulted with industry in the areas of digital communication signal processing. His research interests include robust signal image processing, nonlinear filter theory, adaptive signal processing, digital communications. Mr. Paredes received the Best Paper award from the U.S. Army s ATIRP Federated Laboratory Consortium in In 1995, he received several medals awards for being the first electrical engineering student to graduate from the Universidad de Los Andes with the highest honors. He also was awarded scholarships from the Government of Venezuela in 1988, 1992, Ilya Shmulevich (M 97) received the Ph.D. degree in electrical computer engineering from Purdue University, West Lafayette, IN, in From 1997 to 1998, he was a postdoctoral researcher with the Nijmegen Institute for Cognition Information, University of Nijmegen, National Research Institute for Mathematics Computer Science, University of Amsterdam, The Netherls, where he studied computational models of music perception recognition. From 1998 to 2000, he was a senior researcher at the Tampere International Center for Signal ocessing, Signal ocessing Laboratory, Tampere University of Technology, Tampere, Finl. esently, he is with the Cancer Genomics Laboratory, University of Texas MD Anderson Cancer Center, Houston. His research interests include nonlinear signal image processing, computational learning theory, computational biology, music recognition perception. Gonzalo R. Arce (F 00) was born in La Paz, Bolivia. He received the B.S.E.E. degree with the highest honors from the University of Arkansas, Fayetteville, in 1979 the M.S. Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in , respectively. Since 1982, he has been with the Department of Electrical Computer Engineering, the University of Delaware, Newark, where he is currently ofessor Chair a Fellow with the Center for Advanced Studies. He has consulted for several industrial organizations in the general areas of signal image processing digital communications. His research interests include robust signal processing its applications, communication theory, image processing, electronic imaging. He holds two U.S. patents. He was Guest Editor for the Optical Society of America s Optics Express, he is a Senior Editor of EURASIP s Applied Signal ocessing Journal. Dr. Arce has served as Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, as Guest Editor of the IEEE TRANSACTIONS ON IMAGE PROCESSING. He is a member of the Digital Signal ocessing Technical Committee of the IEEE Circuits Systems Society is a member of the board of nonlinear signal image processing.