Application of the ICI principle to window size adaptive median filtering
Introduction
Suppose that we are given noisy observations of a signal y(x) with a sampling period Δ:Here εs are i.i.d. random errors with , where E(·) stays for the mathematical expectation.
It is assumed that the underlying signal y(x) belongs to a non-parametric class of piecewise continuous and twice differentiable functions with a small number of discontinuities. The goal is to reconstruct , from the observations (zs)s=1n in such a way that certain desirable features such as jumps or instantaneous slope changes will be preserved and the point-wise mean squared error (MSE) will be as small as possible. As a basic estimator we study the median and its generalization, the weighted median. It is well known that the sliding median estimate with respect to the additive Gaussian noise is nearly as good as the sliding linear average, while the median estimate demonstrates a good resistance to the random impulse noise modeled by heavy-tailed probability density functions. Another important feature of the median estimates is their ability to preserve edges and discontinuities in curves, which are usually smoothed by linear filtering. These properties as well as some advantages of implementation explain the great interest to the median filters widely used in image, speech and signal processing (e.g. [1]). In order to meet the aforementioned goals we introduce a median-based filter which is new in the following two important aspects:
- (1)
We consider in parallel the medians with symmetric and non-symmetric left/right window functions and determine the filter output as a combination of the outputs of these median filters.
- (2)
The window size of these median filters is varying and adaptive to an unknown estimated signal. The intersection of confidence interval (ICI) rule is developed for a data-driven window size selection.
In the following we give an efficient way to obtain the weighted median. Let z(r) be ordered observations zs from (1):Order the normalized vs=w(s)/∑s′=1nw(s′) according to (4). Consider the sum Sq=∑r=1qv(r) of the ordered vs and find the minimal q from the condition Sq⩾1/2, i.e. . Then, the weighted median is given as .
It is well known that the window size h is crucial in the efficiency of these estimators for noisy observations. When the window size N is relatively small, the estimator gives a good approximation of a smooth y(x) in a neighborhood of x=xk and the estimation error has a small bias, but then fewer data are used and is more variable and sensitive to the noise. The best choice of h involves (in statistical terms) a trade-off between the bias and variance, which depends on the sampling period, the standard deviation of the noise and the smoothness (derivatives) of the signal y(x).
There have been several attempts to design median filters with adaptive window length (e.g. [6], [9], [10]) based on the detection of a non-stationary behavior of the signal analyzing its temporal (spatial) correlation, changes in distances between the mean and the median estimates, edge detection, local entropy, etc. To the best of our knowledge, this paper is a first attempt to solve the window size selection problem based on the accuracy optimization for the median estimation. Recently, a new window size selection algorithm was developed for linear estimates known as the ICI rule [4], [7]. This paper presents a modification and development of these results to non-linear median estimates. Some of the presented results have been described briefly in [8].
Section snippets
Performance analysis
It is convenient to present the estimation accuracy analysis for estimates more general than the considered weighted median. Let the estimate of the signal be defined as a solution of the following optimization problem:where the loss function F(x) is convex, bounded and symmetric, F(x)=F(−x). In statistics, the term M-estimate is usually used for estimates obtained by minimizing a sum of non-quadratic loss functions of residuals [5]. Thus, given
The ICI rule for window size selection
According to Proposition 1, the inequality , holds with the probability p=1−α, where χ1−α/2 is (1−α/2)th quantile of the standard Gaussian distribution. Then, using (13)The following derivations are based on the results presented in [4], [7]. Let us introduce a finite set of increasing window sizes ⌊H/Δ⌋={N1<N2<⋯<NJ} and, according to (14), determine a sequence of confidence intervals D(j) of the biased estimates as follows:
Simulation
The performance of the proposed adaptive window size median algorithm is demonstrated in a number of experiments. All signals are of the length n=1024 and given on the segment [0,1], Δ=1/n. The noise is Gaussian and its standard deviation σ is estimated as [2]: . The threshold parameter Γ plays an important role in the performance of the algorithm. Too large or too small Γ results in oversmoothing or undersmoothing data, respectively [7]. The parameter Γ
Conclusion
A novel approach to solve a problem of window size selection for median filtering noisy signals is presented. The algorithm is simple to implement and requires calculation of the estimates and their standard deviations for a set of the window size values. The adaptive estimator is built as J parallel filters, which are different only in their window size , and the selector, which determines the best N+ and the corresponding estimate for every xk. The selector uses the rule
Acknowledgements
We would like to thank two anonymous referees for their helpful comments.
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