Journal of Mathematical Imaging and Vision 2, 137-154 (1992). © Kluwer Academic Publishers. Manufactured in The Netherlands. NonLinear Filtering Structure for Image Smoothing in Mixed-Noise Environments ROBERT L. STEVENSON AND SUSAN M. SCHWEIZER Laboratory for Image and Signal Analysis, Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 Abstract. This paper introduces a new nonlinear filtering structure for filtering image data that have been corrupted by both impulsive and nonimpulsive additive noise. Like other nonlinear filters, the proposed filtering structure uses order-statistic operations to remove the effects of the impulsive noise. Unlike other filters, however, nonimpulsive noise is smoothed by using a maximum a posterior/ estimation criterion. The prior model for the image is a novel Markov random-field model that models image edges so that they are accurately estimated while additive Gaussian noise is smoothed. The Markov random-field-based prior is chosen such that the filter has desirable analytical and computational properties. The estimate of the signal value is obtained at the unique minimum of the a posterior/log likelihood function. This function is convex so that the output of the filter can be easily computed by using either digital or analog computational methods. The effects of the various parameters of the model will be discussed, and the choice of the predetection order statistic filter will also be examined. Example outputs under various noise conditions will be given. Key words, image processing, nonlinear filtering, stochastic image models 1 Introduction In many data acquisition, transmission, and storage systems noise is introduced into the data. When the data are a two-dimensional image, this noise reduces the picture quality of the original signal. Various filtering techniques have been developed to suppress the noise in the signal in order to improve the overall picture qual- ity. For images, linear-filtering operations do not perform well because images usually con- tain many sharp edges and thin structures that tend to be smeared or lost in the linear-filtering process. For these reasons nonlinear filters have been examined. Nonlinear filters are designed to suppress the additive-noise component in an image while preserving the important structural information, such as edges and lines. As with any nonlinear system, the analysis of these filters is difficult. Over the years many nonlinear-filtering ideas have been put forth. For noise environments that are very impulsive, rank-based estimators have been very successful. The median [1], mor- phological [2]-[4], and stack filters are three im- portant and related rank-based filter types. For nonimpulsive noise the rank-based filters do not generally remove noise as well as does the class of linear filters, but, as was pointed out above, the linear filters suffer from poor syntactical per- formance. For this reason several hybrid filter designs, which incorporate both rank-based and linear-based sections in the filter, have been pro- posed [5]-[9]. Although these filters work better in mixed-noise environments, they suffer from some of the same limitations as the ranked- based filters, namely their designs are based on statistical or on syntactical considerations, and the other properties are difficult to determine and quantify. This paper proposes a new nonlinear filter- ing structure based on a maximum a posteri- or~ estimation criterion which uses a Markov random field model for the prior distribution. The form of the Markov random field is such that estimates obtained with the proposed model permit discontinuities in the signal to be accu- rately estimated while additive Gaussian noise 41