Bayesian Morphology : Fast Unsupervised

We consider the problems of image segmentation and classi cation, and image restoration when the true image is made up of a small number of (unordered) colors. Our emphasis is on both performance and speed; speed has become increasingly important for analyzing large images and multispectral images with many bands, processing large image databases, real-time or near real-time image analysis, and the online analysis of video. Bayesian image analysis (Geman and Geman 1984) provides an elegant solution to these problems, but it is computationally expensive, and the solutions it provides may be sensitive to unrealistic global properties of the models on which it is based. The ICM algorithm (Besag 1986) is faster and based on the local properties of the models underlying Bayesian image analysis; parameter estimation is performed iteratively via pseudo-likelihood. Mathematical morphology (Matheron 1975) is faster again and is widely considered to perform well, but lacks a statistical basis; method selection (analogous to parameter estimation) is done in a rather ad hoc manner. We propose Bayesian morphology, a synthesis of these methods that attempts to combine the speed of mathematical morphology with the principled statistical basis of ICM. The key observation is that when the original image is discrete (or if an initial segmentation has been carried out), then, assuming a Potts model for the true scene and channel transmission noise, (1) the ICM algorithm is equivalent to a form of mathematical morphology; and (2) the segmentation is insensitive to the precise values of the model parameters. Unlike in standard Bayesian images analysis and ICM, it is feasible to do maximum likelihood estimation of the parameters in this setting. For grey-level or multispectral images, we propose an initial segmentation based on the EM algorithm for a mixture model of the marginal distribution of the pixels. The resulting algorithm is much faster than ICM, with gains that increase for more bands and larger images, and has good performance in experiments and for real examples.

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