Parametric analysis of weighted order statistics filters - Semantic Scholar
~
95
IEEE SIGNAL PROCESSING LElTERS, VOL. 1, NO. 6. JUNE 1994
Parametric Analysis of Weighted Order Statistics Filters Ruikang Yang, Moncef Gabbouj, and Pao-Ta Yu
Abstract-In this letter, we shall 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. It will be shown 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 will be derived.
Unfortunately, many WOS filters are excluded as they do not satisfy this condition. We shall show, in this letter, that this restriction is not necessary and can be eliminated. In other words, any symmetric threshold logic function will make an input converge to a root or oscillate in a cycle of 2. 11. CLASSIFICATION OF WOS FILTERS
The weighted order statistics filter can be defined in the following way. Definition 1: In the real domain, the output of a WOS fitler with window size N = R L 1 and weight vector W
+ +
I. INTRODUCTION
EIGHTED order statistics (WOS) filters, including median and weighted median filters as their special cases, belong to the class of stack filters. They have been studied extensively and successfully applied in many applications such as image processing, due mainly to their good performance for edge preservation and excellent suppression of impulsive noise [ 1]-[4]. Since weighted order statistics filters are nonlinear operations, their behavior and performance are assessed based on their statistical and deterministic properties. The former deals with, e.g., output distributions, and noise attenuation capability and has been the subject of several papers [3], [SI. The latter is related to the concept of root signals and deterministic convergence and is the focus of this letter. One of the contributions in this letter is to introduce a set of parameters to characterize the convergence properties of WOS filters. Based on the parameters, WOS filters can be classified into five categories, all of them except one possess convergence property. Study is then focused on the remaining category. It is shown that the WOS filters in this category are type-3 stack filters. A condition under which WOS filters will make any input signal converge to a root in a finite number of filtering is derived. Another contribution of the letter is the elimination of the restrictions imposed by Wendt [ 2 ] .In his paper, Wendt showed that in order to make a symmetric threshold logic function converge or oscillate in a cycle of 2, the threshold logic function must preserve the roots of the standard median filters. Manuscript received April 4, 1994; revised April 25, 1994. The associate editor coordinating the review of this letter and approving it for publication was Prof. V. Mathews. R. Yang is with the Audio-visual Processing Laboratory, Nokia Research Center, Tampere, Finland. M. Gabbouj is with the Signal Processing Laboratory, Tampere University of Technology, Finland. P.-T. Yu is with the Institute of Computer Science and Information Engineering, National Chung Cheng University, Taiwan. IEEE Log Number 9402922.
w =(W-L'..,"J,''.,WR) is given by
Y , = T : thlargest[W-L 0 X,-L, ... WO O x t , . " , W R O X z + R ]
(1)
-
where T is called the threshold of the WOS filter and the 0 denotes duplication n times
n O X =X:..,X.
The weights W, can be real numbers. Usually, we select
R=L=K. The filtering procedure can be stated as follows: Sort the samples inside the filter, and add up the corresponding weights from the upper end of the sorted list until the sum just exceeds the threshold T ; the output of the WOS filter is the sample corresponding to the last weight added. We shall concentrate on the convergence behavior of WOS filters in the binary domain, due to the threshold decomposition property According to previous research on convergence properties of stack filters, the center weight WOof a WOS filter plays an important role to govem the convergence behavior of the WOS filter. This motivates us to introduce two parameters ( a and p) to represent the center weight and the sum of the rest of the weights, respectively, i.e.
[a
K
a = ~ oand
w,.
p=
(2)
z=-K
We define the threshold T as a linear function of a parameter t
1070-9908/94$04.00 0 1994 IEEE
T = g ( t ) = (1 - t ) a + t/?: for -m < t < m.
(3)
IEEE SIGNAL PROCESSING LETTERS, VOL. I , NO. 6, JUNE 1994
96
It is easy to check that any WOS filter can be obtained by varying the parameter t properly. For instance- Parmeter t corresponding to weighted median filters is the following:
111. CONVERGENCE BEHAVIOR OF SYMMETRIC WOS FILTERS Symmetric WOS filters are widely used in practice. A WOS filter is said to be symmetric if
w,= w-,
Wendt [2] proposed a very useful approach to analyze the convergence behavior using some results in threshold automaton. He showed that any symmetric threshold function F ( .) will make an input sequence converge to a root or oscillate in a cycle of period 2, provided that the threshold function preserves all roots of some median filter. However, many symmetric threshold functions do not satisfy this restriction. For instance, any symmetric threshold function with threshold
For standard median, t becomes
t=-
.
for 2 = 1,.. . K .
K 2K - 1'
In the following, the definition about type-0 through type-3 stack filters are summarized [7]. Definition 2 : A stack filter with its positive Boolean function f ( . ) of 2K 1 variables belongs to one of the following T four classes of stack filters, B P T < -2 or T > n ! + 21) Type-0 (trivial) stack filters if and only if f ( Z 1 , z2. . . , Z Z K + l ) = 0,f ( Z 1 . x2,.. . , 2 2 K + 1 ) = 1, does not satisfy this restriction although it is still a type-3 or f ( . r 1 , ~ 2 , . . . , ~ 2 =~Z+K1 + )~. stack filters. Actually, this restriction is not necessary; in the 2) Type-1 (decreasing) stack filters if and only if the on set following, we shall show that this restriction can be eliminated on(f) is a proper subset of xKlsK,which is the set of using some strategy, that is, the convergence property of the all 2K + 1 dimensional binary vectors where the center symmetric threshold automaton is valid for all symmetric binary variable is a 1. o n ( f ) is defined as follows: threshold functions. In order to make the letter self-contained, here, we briefly o n ( f ) = {U E {1,0}2K+1: f ( v ) = l}. review some concepts and results in threshold automaton. Definition 3: The threshold automaton A: {0,1}" + Note that o n ( f ) must not be empty; otherwise, it be(0, l}"consists of a set of n threshold functions Q z defined by comes a type-0 filter. the real r b x 71 matrix A = ( a t J )and the n-element threshold 3) A type-2 (increasing) stack filters if and only if the off vector 8 = ( & > .. . as follows: set o f f ( f ) is proper subset of xKOxK, which is the set of all 2 K f l dimensional binary vectors where the center binary variable is a 0. o f f ( f ) is defined as follows:
+
'
,e,)
o f f ( f ) = {U E
{l,o}'K+1:f(U)
Note also that o f f ( f ) must not be empty; otherwise, it becomes a type-0 stack filter. 4) A type-3 stack filter if and only if it is not a type-0, type-1, or type-2 stack filter. Now we can classify WOS filters based on parameter t , a, and p. According to the definitions of type-I through type3 stack filters, we have the following theorem, which states that WOS filters can be classified into five categories; two of them belong to type-1 stack filters and two of them belong to type-2 stack filters, whereas the last one belongs to type-3 stack filters. Theorem f : WOS filters must fall into one of the following five categories: I) If N 2 /3 and t 0, type-1 decreasing stack filters. 11) If N 2 B and t > 0, type-2 increasing stack filters. 111) If N < p and t 5 0, type-2 increasing stack filters. IV) If (1 < ,O and t > 1, type-1 decreasing stack filters. V) If N < @ and 0 < t < 1, type-.? mixed stack filters. Since the convergence properties of type-1 and type-2 are known [ 7 ] , we conclude that all WOS filters falling into category i) through iv) will converge in a finite number of filtering. The convergence behavior of WOS filters in category v ) is the subject of the next section.
.Then, if
{ %;:
for .r7>E (0, l},, n,(yll(t))
t = 1,2,.
(5)
' '
there exists a positive integer t' such that
.yl,(t
+ 2) = y r 2 ( t )
for all t
2 f'.
(6)
In other words, the parallel iteration of a symmetric threshold automaton will make any input sequence converge to a fixed
97
YANG et al.: PARAMETRtC ANALYSIS OF STATISTICS FILTERS
point, or a cycle of period 2, in a finite number of iteration. The minimum value of t' is called the transient length of A p . Next, we shall show that for any given WOS filter, there is always a threshold automaton A such that its parallel iteration A p is equivalent to the filtering operation of the WOS filter. Note that we do not impose any restriction on the WOS filter. Suppose that the given WOS filter has window size N = 2K + 1 and weight vector
Using the theorem, we can derive a condition under which symmetric WOS filters falling into category v) will make any input signal converge to a root in a number of filtering passes. Theorem 3: For 0 < t < 1, any symmetric WOS filters will make any input sequence converge to root in a finite number of filtering passes if
W = (W-~ , W - K + l , . . . , W O , W ~ : . . . , W K )(7)
Proof: In the following, we show that the necessary condition for the output of a WOS filter oscillating at the period of 2 is
and the threshold T . If the unappended signal has length L, denote the first sample and the last sample by ~ ( 1and ) s(L), respectively. By appending K samples with the value of z( 1) a < $0. and K samples with the value of s ( L ) to both ends of the signal, the initial signal to the threshold automaton has length Then by Theorem 2, Theorem 3 follows. of L+2K. Now, we construct a ( L + 2 K ) x (L+2K) Toeplitz Given a symmetric WOS filter, which is denoted with matrix A = CL,^) whose elements are specified by , : . , W K ) Suppose . the output weight ( W K *, . . ,W1,W OW1,. of the filter oscillates at the cycle of 2. Hence, there exist X if (i - j l 5 K 1.2'( = otherwise and Y , which belong to {O,l}L, where L is the length of the input signal, such that the output of the filter is X if the and the threshold vector tJspecified by input is Y and vice verse. Note that X # Y . Let the filter window centered at the point where the first change occurs c, for 1 5 i 5 K when the sequence X is filtered by the filter. Without loss of T forK+l
95
IEEE SIGNAL PROCESSING LElTERS, VOL. 1, NO. 6. JUNE 1994
Parametric Analysis of Weighted Order Statistics Filters Ruikang Yang, Moncef Gabbouj, and Pao-Ta Yu
Abstract-In this letter, we shall 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. It will be shown 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 will be derived.
Unfortunately, many WOS filters are excluded as they do not satisfy this condition. We shall show, in this letter, that this restriction is not necessary and can be eliminated. In other words, any symmetric threshold logic function will make an input converge to a root or oscillate in a cycle of 2. 11. CLASSIFICATION OF WOS FILTERS
The weighted order statistics filter can be defined in the following way. Definition 1: In the real domain, the output of a WOS fitler with window size N = R L 1 and weight vector W
+ +
I. INTRODUCTION
EIGHTED order statistics (WOS) filters, including median and weighted median filters as their special cases, belong to the class of stack filters. They have been studied extensively and successfully applied in many applications such as image processing, due mainly to their good performance for edge preservation and excellent suppression of impulsive noise [ 1]-[4]. Since weighted order statistics filters are nonlinear operations, their behavior and performance are assessed based on their statistical and deterministic properties. The former deals with, e.g., output distributions, and noise attenuation capability and has been the subject of several papers [3], [SI. The latter is related to the concept of root signals and deterministic convergence and is the focus of this letter. One of the contributions in this letter is to introduce a set of parameters to characterize the convergence properties of WOS filters. Based on the parameters, WOS filters can be classified into five categories, all of them except one possess convergence property. Study is then focused on the remaining category. It is shown that the WOS filters in this category are type-3 stack filters. A condition under which WOS filters will make any input signal converge to a root in a finite number of filtering is derived. Another contribution of the letter is the elimination of the restrictions imposed by Wendt [ 2 ] .In his paper, Wendt showed that in order to make a symmetric threshold logic function converge or oscillate in a cycle of 2, the threshold logic function must preserve the roots of the standard median filters. Manuscript received April 4, 1994; revised April 25, 1994. The associate editor coordinating the review of this letter and approving it for publication was Prof. V. Mathews. R. Yang is with the Audio-visual Processing Laboratory, Nokia Research Center, Tampere, Finland. M. Gabbouj is with the Signal Processing Laboratory, Tampere University of Technology, Finland. P.-T. Yu is with the Institute of Computer Science and Information Engineering, National Chung Cheng University, Taiwan. IEEE Log Number 9402922.
w =(W-L'..,"J,''.,WR) is given by
Y , = T : thlargest[W-L 0 X,-L, ... WO O x t , . " , W R O X z + R ]
(1)
-
where T is called the threshold of the WOS filter and the 0 denotes duplication n times
n O X =X:..,X.
The weights W, can be real numbers. Usually, we select
R=L=K. The filtering procedure can be stated as follows: Sort the samples inside the filter, and add up the corresponding weights from the upper end of the sorted list until the sum just exceeds the threshold T ; the output of the WOS filter is the sample corresponding to the last weight added. We shall concentrate on the convergence behavior of WOS filters in the binary domain, due to the threshold decomposition property According to previous research on convergence properties of stack filters, the center weight WOof a WOS filter plays an important role to govem the convergence behavior of the WOS filter. This motivates us to introduce two parameters ( a and p) to represent the center weight and the sum of the rest of the weights, respectively, i.e.
[a
K
a = ~ oand
w,.
p=
(2)
z=-K
We define the threshold T as a linear function of a parameter t
1070-9908/94$04.00 0 1994 IEEE
T = g ( t ) = (1 - t ) a + t/?: for -m < t < m.
(3)
IEEE SIGNAL PROCESSING LETTERS, VOL. I , NO. 6, JUNE 1994
96
It is easy to check that any WOS filter can be obtained by varying the parameter t properly. For instance- Parmeter t corresponding to weighted median filters is the following:
111. CONVERGENCE BEHAVIOR OF SYMMETRIC WOS FILTERS Symmetric WOS filters are widely used in practice. A WOS filter is said to be symmetric if
w,= w-,
Wendt [2] proposed a very useful approach to analyze the convergence behavior using some results in threshold automaton. He showed that any symmetric threshold function F ( .) will make an input sequence converge to a root or oscillate in a cycle of period 2, provided that the threshold function preserves all roots of some median filter. However, many symmetric threshold functions do not satisfy this restriction. For instance, any symmetric threshold function with threshold
For standard median, t becomes
t=-
.
for 2 = 1,.. . K .
K 2K - 1'
In the following, the definition about type-0 through type-3 stack filters are summarized [7]. Definition 2 : A stack filter with its positive Boolean function f ( . ) of 2K 1 variables belongs to one of the following T four classes of stack filters, B P T < -2 or T > n ! + 21) Type-0 (trivial) stack filters if and only if f ( Z 1 , z2. . . , Z Z K + l ) = 0,f ( Z 1 . x2,.. . , 2 2 K + 1 ) = 1, does not satisfy this restriction although it is still a type-3 or f ( . r 1 , ~ 2 , . . . , ~ 2 =~Z+K1 + )~. stack filters. Actually, this restriction is not necessary; in the 2) Type-1 (decreasing) stack filters if and only if the on set following, we shall show that this restriction can be eliminated on(f) is a proper subset of xKlsK,which is the set of using some strategy, that is, the convergence property of the all 2K + 1 dimensional binary vectors where the center symmetric threshold automaton is valid for all symmetric binary variable is a 1. o n ( f ) is defined as follows: threshold functions. In order to make the letter self-contained, here, we briefly o n ( f ) = {U E {1,0}2K+1: f ( v ) = l}. review some concepts and results in threshold automaton. Definition 3: The threshold automaton A: {0,1}" + Note that o n ( f ) must not be empty; otherwise, it be(0, l}"consists of a set of n threshold functions Q z defined by comes a type-0 filter. the real r b x 71 matrix A = ( a t J )and the n-element threshold 3) A type-2 (increasing) stack filters if and only if the off vector 8 = ( & > .. . as follows: set o f f ( f ) is proper subset of xKOxK, which is the set of all 2 K f l dimensional binary vectors where the center binary variable is a 0. o f f ( f ) is defined as follows:
+
'
,e,)
o f f ( f ) = {U E
{l,o}'K+1:f(U)
Note also that o f f ( f ) must not be empty; otherwise, it becomes a type-0 stack filter. 4) A type-3 stack filter if and only if it is not a type-0, type-1, or type-2 stack filter. Now we can classify WOS filters based on parameter t , a, and p. According to the definitions of type-I through type3 stack filters, we have the following theorem, which states that WOS filters can be classified into five categories; two of them belong to type-1 stack filters and two of them belong to type-2 stack filters, whereas the last one belongs to type-3 stack filters. Theorem f : WOS filters must fall into one of the following five categories: I) If N 2 /3 and t 0, type-1 decreasing stack filters. 11) If N 2 B and t > 0, type-2 increasing stack filters. 111) If N < p and t 5 0, type-2 increasing stack filters. IV) If (1 < ,O and t > 1, type-1 decreasing stack filters. V) If N < @ and 0 < t < 1, type-.? mixed stack filters. Since the convergence properties of type-1 and type-2 are known [ 7 ] , we conclude that all WOS filters falling into category i) through iv) will converge in a finite number of filtering. The convergence behavior of WOS filters in category v ) is the subject of the next section.
.Then, if
{ %;:
for .r7>E (0, l},, n,(yll(t))
t = 1,2,.
(5)
' '
there exists a positive integer t' such that
.yl,(t
+ 2) = y r 2 ( t )
for all t
2 f'.
(6)
In other words, the parallel iteration of a symmetric threshold automaton will make any input sequence converge to a fixed
97
YANG et al.: PARAMETRtC ANALYSIS OF STATISTICS FILTERS
point, or a cycle of period 2, in a finite number of iteration. The minimum value of t' is called the transient length of A p . Next, we shall show that for any given WOS filter, there is always a threshold automaton A such that its parallel iteration A p is equivalent to the filtering operation of the WOS filter. Note that we do not impose any restriction on the WOS filter. Suppose that the given WOS filter has window size N = 2K + 1 and weight vector
Using the theorem, we can derive a condition under which symmetric WOS filters falling into category v) will make any input signal converge to a root in a number of filtering passes. Theorem 3: For 0 < t < 1, any symmetric WOS filters will make any input sequence converge to root in a finite number of filtering passes if
W = (W-~ , W - K + l , . . . , W O , W ~ : . . . , W K )(7)
Proof: In the following, we show that the necessary condition for the output of a WOS filter oscillating at the period of 2 is
and the threshold T . If the unappended signal has length L, denote the first sample and the last sample by ~ ( 1and ) s(L), respectively. By appending K samples with the value of z( 1) a < $0. and K samples with the value of s ( L ) to both ends of the signal, the initial signal to the threshold automaton has length Then by Theorem 2, Theorem 3 follows. of L+2K. Now, we construct a ( L + 2 K ) x (L+2K) Toeplitz Given a symmetric WOS filter, which is denoted with matrix A = CL,^) whose elements are specified by , : . , W K ) Suppose . the output weight ( W K *, . . ,W1,W OW1,. of the filter oscillates at the cycle of 2. Hence, there exist X if (i - j l 5 K 1.2'( = otherwise and Y , which belong to {O,l}L, where L is the length of the input signal, such that the output of the filter is X if the and the threshold vector tJspecified by input is Y and vice verse. Note that X # Y . Let the filter window centered at the point where the first change occurs c, for 1 5 i 5 K when the sequence X is filtered by the filter. Without loss of T forK+l
