Morphological Analysis of Discrete Random Shapes
In this paper I discuss a number of theoretical issues regarding the morphological analysis of discrete random shapes by means of Matheron’s random set theory. I revisit this theory by limiting myself to the discrete case, since most image data are available in a discrete form. Although it may seem that the transition from the continuous to the discrete case is straightforward (since most of Matheron’s theory is general enough to incorporate the discrete case as a special case), this transition is often challenging and full of exciting and, surprisingly, pleasant results. I introduce the concept of the cumulative-distribution functional of a discrete random set and review some fundamental properties of the capacity functional (a fundamental statistical quantity that uniquely defines a random set and relates random set theory to mathematical morphology). In analogy to a recent result and under a natural boundness condition, I show that there exists a one-to-one correspondence between the probability-mass function of a discrete binary random field and the corresponding cumulative-distribution functional. The relationship between the cumulative-distribution functional and the capacity functional of a discrete random set is also established. The cumulative-distribution and capacity functionals are related to the higher-order moments of a discrete binary random field, and, therefore, their computation is equivalent to computing these moments. A brief discussion of how to perform such computations for a certain class of discrete random sets is provided. The capacity functional of a morphologically transformed, continuous random set cannot be associated to the capacity functional of the random set itself, except in the case of dilation. I show that the derivation of such an association is possible in the discrete case and for the cases of dilation and erosion and more complicated morphological transformations, such as opening and closing. These relationships are then used to derive a fundamental result regarding the statistical behavior of opening and closing morphological filters. I also show that the probability-mass function of a discrete binary random field may be expressed in terms of the cumulative-distribution functional or the capacity functional of a morphologically transformed discrete random set by means of a hit-or-miss transformation. I also introduce moments for discrete random sets, which permit generalization of the concepts of autocorrelation and contact distribution. Furthermore, I demonstrate the fact that the class of opening-based size distributions, introduced axiomatically by Matheron, are higher-order moments of a discrete random set, therefore statistically demonstrating that size distributions are good statistical summaries for shape. Finally, convex random sets are viewed in the discrete domain. My final result regarding convexity, is similar to Matheron’s. However, the tools used here for the derivation of such a result are different from the ones used by Matheron, whose approach to this subject is limited to the continuous case.
Key wordsbinary random fields discrete random shapes mathematical morphology random set theory shape analysis
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