Using genetic algorithm optimizing stack filters based on MMSE criterion
Introduction
Stack filters are a class of nonlinear, sliding-window filters that satisfy a weak superposition property known as the threshold decomposition and an ordering property called the stacking property. Researchers pay more and more attention to stack filters because they are useful in suppressing noise whilst retaining image detail. Since the theory of stack filters [1] presented by P.D. Wendt, some methods of optimizing stack filters have been presented. However, almost all these methods are based on MMAE criterion [2], [3], [4]. Jisang Yoo presented a fast method based on MMAE criterion [5]. Though the method is faster than other adaptive algorithms and stack filters optimized by it can protect details of images excellently, but they cannot suppress noise effectively. Visual effect would be affected if noise exists in images, so noise should be suppressed as more as possible in image processing. Optimal stack filters based on MMSE criterion solve this problem. C.E. Savin presented a method that optimizes stack filters based on MMSE criterion [6]. First, the algorithm constructs the mathematical model of optimizing problem, then solve optimizing problem by linear program (LP). In the process of solving the optimizing problem, there are many constraints. So the main drawback of this algorithm is the computational complexity of their configuration. To avoid this drawback, genetic algorithm is presented to solve the optimization problem in this paper. Moreover, the performances of simple genetic algorithm and adaptive genetic algorithm are analyzed in this paper. Simulated experiments show that adaptive genetic algorithm [7] can converge faster than simple genetic algorithm [8] and has better performance.
Section snippets
Basic theory of stack filters
Stack filters are a class of nonlinear filters including many kinds of nonlinear filters such as median filters, weighted median filters, order statistics filters and some morphological filters. Stack filters have threshold decomposition and stacking property. Stack filters have parallel structure by threshold decomposition, the analysis of real signal is changed into the analysis of binary signal by this way, so it is very useful to application of stack filters in real time signal processing.
Optimization model based on MMSE criterion
MSE is mean square error between output signal and desired signal. Mr C.E. Savin et al. constructed optimization model based on MMSE criterion [6], it is referred here for readers convenience.
For stack filters, at n time, if the desired signal is denoted as S(n), and the signal in the filter window is , then the MSE would be denoted as (4) according to the definition of MSE.
Apparently, we can find a stack filter minimize the MSE between the desired signal and
Solving the knapsack problem by using genetic algorithm
The challenge of stack filter design is in determining which positive Boolean function (PBF) to use. In this paper, we code PBF into a binary vector by setting f(bj) 0 or 1. The length of the vector is when the window size of the stack filter is N×N. In genetic algorithm, first, we generate initial population stochastically. However, the individuals of the population may not satisfy stacking property. So they should be checked and constrained in order to guarantee that they satisfy stacking
Experimental results
In this paper, simple genetic algorithm and adaptive genetic algorithm are simulated with using MATLAB program language on a Pentium IV 2.4G computer. Stack filters whose size are 3×3 are trained to restore images corrupted by additive impulse noise from noise free images. We use a 256×256, 8 b/pixel image and a 512×512, 8 b/pixel image as noise free images, showed as, respectively, Fig. 2(a) and (b). We generated noise-corrupted images by using salt and peppers noise with different occurrence
Conclusion
The mathematical model of the optimization of stack filters under MMSE criterion is derived in this paper. The model shows that the optimization of stack filters is equivalent to a one-zero knapsack problem. Simulated experiments show that the model is right. Experimental results show that adaptive genetic algorithm has better performances than simple genetic algorithm. How to select parameters of AGA is discussed in this paper. Under MMSE and MMAE criterion, the optimal stack filters are
Acknowledgements
This study has been supported by the College Doctoral Subject Science Fund of China (No. 20020217006), the Excellent Youth Science Fund of Hei Longjiang Province (No. JC-02-07), the Fund of the author of the national excellent dissertation (No.200037) and TRAPOYT.
Zhao Chunhui was born in Heilongjiang, China, in 1965. He received the BS and MS degree from Harbin Engineering University, in 1986 and 1989, respectively. He received his PhD degree in Department of Automatic Measure and Control at Harbin Institute of Technology in 1998. He was a postdoctoral research fellow in the College of Underwater Acoustical Engineering of Harbin Engineering University. At present, he is working in the College of Information and Communication Engineering of Harbin
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Cited by (0)
Zhao Chunhui was born in Heilongjiang, China, in 1965. He received the BS and MS degree from Harbin Engineering University, in 1986 and 1989, respectively. He received his PhD degree in Department of Automatic Measure and Control at Harbin Institute of Technology in 1998. He was a postdoctoral research fellow in the College of Underwater Acoustical Engineering of Harbin Engineering University. At present, he is working in the College of Information and Communication Engineering of Harbin Engineering University as a professor and doctoral supervisor. He is a senior member of Chinese Electronics Academy. He has published four works and more than 100 papers. His research interests include digital signal and image processing, mathematical morphology, nonlinear filters.
Zhang Wencheng was born in Jiangxi, China, in 1980. He received the BS and MS degree from Harbin Engineering University, in 2000 and 2002, respectively. Now he is a doctoral candidate in the College of Information and Communication Engineering of Harbin Engineering University. He has published 10 papers. His research interests are image processing and nonlinear filters.