Statistical morphology and Bayesian reconstruction

The aim of this paper is to show that basic morphological operations can be incorporated within a statistical physics formulation as a limit when the temperature of the system tends to zero. These operations can then be expressed in terms of finding minimum-variance estimators of probability distributions. It enables us to relate these operations to alternative Bayesian or Markovian approaches to image analysis. We first show how to derive elementary dilations (winner-take-all) and erosions (loser-take-all). These operations, referred to as statistical dilations and erosion, depend on a temperature parameter β=1/T. They become purely morphological as β goes to infinity and become purely linear averages as β goes to zero. Experimental results are given for a range of intermediate values of β. Concatenations of elementary operations can be naturally expressed by stringing together conditional probability distributions, each corresponding to the original operations, thus yielding statistical openings and closings. Techniques are given for computing the minimum-variance estimators. Finally, we describe simulations comparing statistical morphology and Bayesian methods for image smoothing, edge detection, and noise reduction.