The median filter was introduced in 1974 by Tukey [1], who used the moving median as a smoothing technique in time series analysis. Since then, there have been many studies on deterministic and statistical properties of median filtering operator, such as properties of roots of median filters ([2-6]), structures of recurrent sequences of median filters ([7-9]), convergence properties of a signal ([2,10-12]), and the output distribution of mediantype filters under known distributions [13]. L-filters, a generalization of median filters, were introduced [14], their output is given by a linear combination of the order statistics of the input sequence. Assuming a constant signal in white noise, the coefficients in the linear combination are chosen to minimize the output MSE for several noise distributions, i.e., the optimal design of Lfilters, and a new methodology for the design of L-filters is presented in which the L-filter coefficients are obtained by approximating the covariance matrix of the ordered samples through the use of Taylor expansion [15]. Most practical applications deal with unknown signals, which means that the noise distributions of signals are difficulty to characterize. There are cases such as in detection, and an example was given [11], which is an original two-dimensional image obtained by Dr. J. Johnson of Redstone Arsenal with an experimental laser imaging system at the far infrared wavelength of 1.2 mm. It is very difficult to characterize the noise source in the image. In these situations, the statistical methods above do not work. Therefore, a significant filtering problem is:Given a signal with some unknown noise and a filter, how do we numerically measure its filtering behavior for the signal? This problem is general and complicated. So, in this paper, we will study this problem only for stack filters.

The organization of this paper is as follows. In Section 2 we review stack filters and their threshold decomposition property, and define the generalized correlativity between of a discrete binary input signal and its output signal of stack filters. In Section 3 we prove the optimality of median filters based on the generalized correlativity in terms of stack filters. In Section 4 we give some properties of the median filtering operator to show when and how to use the filtering operator better.

In this section, first we briefly review the definition of stack filters. Stack filters are a class of sliding window, nonlinear digital filters that possess the stacking property and the threshold decomposition architecture [11]. The threshold decomposition of an M-valued signal is the set of binary signals, called threshold signals, , which are defined by

Note that the sum of the threshold signals is, i.e.,

Also, the’s are ordered, i.e.,

This property is called the stacking property of sequences. A very important property of median filters operating within a sliding window was observed by Fitch et al. [16]: Applying a median filter to an M-valued signal is equivalent to decomposing the signal to M – 1 binary threshold signals, filtering each binary signal separately with the median filter, and then adding the binary output signals together. This is called the threshold decomposition architecture of median filters.

In the above architecture, the median operation on binary input signals reduces to a Boolean function that possesses the stacking property.

Definition 1. A 2k + 1-input Boolean function is said to possess the stacking property if whenever two input vectors x and y stack, i.e., for each, then also their outputs stack.

It has been shown that a Boolean function has the stacking property if and only if it can be expressed as a Boolean expression that contains no complements of input variables [11]. Such functions are called positive Boolean functions (PBF).

Definition 2. The stack filter based on the PBF is defined as follows:

where

and

.

Stack filters include median filters. Since stack filters possess the threshold decomposition property and the stacking property, a complete characterization of the effect of the stack filter on binary signals is sufficient to characterize the behavior of the M-valued filter. Since any with can be transformed into with by the operation: In what follows, attention will therefore be focused mainly on stack filters with binary input taking values 1 and, and k is a fixed positive integer, Z is the set of integers.

Thus, as Definition 1, we can definite the PBF, and for each, let, for each

We call a stack filtering operator with width. Let denote the set of stack filtering operators with width.

From the definition, it is clear that for each, depends on . Therefore, we will investigate the relation between and. Usually, if two sequences and belong to, that is,

then we use

to study the correlativity of and, but all and do not belong to. Therefore, we first introduce such a window sequence with width: satisfying

•

Further, we define a real sequence satisfying

• for each

•

For each, we have

Since, Theredore,

. This is why we introduce such a real sequence.

Again we give the following definition.

Definition 3. Suppose is a window filtering operator with width. For each, let

We call it the generalized correlativity of and.

In particular, let satisfy that for each,

Then is the median filtering operator and.

It is clear that the generalized correlativity is a measure of the relation between and. From a view of signal processing, it is shown that if we apply a stack filtering operator to a signal with noise, then the generalized deviation and may be a measure of the’s capability of removing noise in the signal.

In the optimal design of filters, it is usual to select the optimal filter using MSE criterion or MAE criterion, for example, the optimal design of L-filters is based on MSE criterion. Both MSE criterion and MAE criterion mean that output signal is closest to input signal, which implies their high correlativity. Therefore, we have the following definition.

Definition 4. Suppose. If, for each ,

then we say that is better than in terms of filtering behavior, denoted by “”.

Based on the definition, we have the following result.

Theorem 1. For any, we have

Proof: Suppose that and. Since, we have

Now we prove that for each,

In fact, since for i = –k, –k + 1,

. Therefore,

Case 1.

In this case, we have

Thus

where

By the definition of sequence, for any with, we have

and

for

and

for

for

So we have

Thus

Therefore, (1) holds.

Case 2.

In this case, we have

Thus

By (2), we have

Thus

Therefore, (1) holds.

For any, if, then

. By the definition of,

. Again, so

. Therefore,

If, then. By the definition of,. Again, so

. Therefore,

So

Therefore, for any

that is,

This completes the proof of Theorem 1.

This result shows that, in the sense of the generalized correlativity, the median filtering operator is optimal of the stack filtering operators. Also it is globally optimal, compared with the optimal design of L-filters.

In this paper we defined the generalized correlativity of input signal and output signal of a stack filtering operator, which can be used for numerously measuring these filtering operators’s behavior in removing noise in signals. In the sense of the criterion of the generalized correlativity, the median filtering operator is optimal of stack filtering operators, which implies that this filtering operator possesses better filtering behavior than the others.