Adaptive Nonlinear Filters

Abstract

The nonlinear filters described in the previous chapters are usually optimized for a specific type of noise and sometimes for a specific type of signal. However, this is not usually the case in many applications of nonlinear filtering, especially in image processing. Images can be modeled as two-dimensional stochastic processes, whose statistics vary in the various image regions. Images are nonstationary processes. Furthermore the noise statistics, e.g., the noise standard deviation and even the noise probability density function, vary from application to application, as was described in chapter 3. Sometimes, the noise characteristics vary in the same application from one image to the next. Such cases are the channel noise in image transmission and the atmospheric noise (e.g., the cloud noise) in satellite images. In these environments non-adaptive filters cannot perform well because their characteristics depend on the noise and signal characteristics, which are unknown. Therefore, adaptive filters are the natural choice in such cases. Their performance depends on the accuracy of the estimation of certain signal and noise statistics, namely the signal mean and standard deviation and the noise standard deviation. The estimation is usually local, i.e., relatively small windows are used to obtain the signal and noise characteristics. An important property of these estimators is their robustness to impulse noise, which is present in many image processing applications. Another reason for using adaptive filters is the fact that edge information is very important for the human eye and must be preserved. Certain filters, e.g., the moving average, perform well in homogeneous image regions but fail close to edges. The opposite is true for other filters, e.g., for the median. A combined filter which performs differently in the image edges than in the image plateaus can be used in such a case. These filters are also called decision directed filters because they employ an edge detector to decide if an edge is present or not. Decision directed filtering can also be used in the cases of mixed additive white noise and impulsive noise. Impulses can be detected and removed before the additive noise filtering is performed. Another approach related to decision directed filtering is the two-component model filtering. An image is assumed to consist of two components, the low-pass and the high-pass component. The first one is mainly related to homogeneous image regions, whereas the second one is related to edge information. These two components can be processed in different ways. The output of the two corresponding filters can be recombined to give the final filtered image. The two-component image processing model has been used both for noise removal and image enhancement applications.