Mathematical Morphology
Historically, the study of shape took place over a long period of time and resulted in a highly variegated set of algorithms and methods. Over the past 20 years the formalism of mathematical morphology was set up and provided a background theory into which many of the individual advances could be slotted. This chapter takes a journey through this interesting subject but aims to steer an intuitive path between the many mathematical theorems, concentrating particularly on finding practically useful results.
• how the concepts of expanding and shrinking are transformed into the more general concepts of dilation and erosion.
• how dilation and erosion operations may be combined to form more complex operations whose properties may be predicted mathematically.
• how the concepts of closing and opening are defined, and how they are used to find defects in binary object shapes, via residue (or “top-hat”) operations.
• the hit-and-miss transform and its applications.
• how thinning is brought into the formalism.
• how mathematical morphology is generalized to cover gray-scale processing.
This chapter on morphology follows last in Part 1 of this volume because, being rather mathematical, it could demotivate inexperienced readers if placed earlier. In any case, the theory is largely valuable by the way in which it integrates the other topics. Nevertheless, once the methods have been learned, morphology should be of distinct value in taking the earlier ideas forward and optimizing any algorithms that use them.