In this book, the coauthors have set out to provide a shape description scheme that is a notational system for expressing the shapes of objects. This is also a way of writing the shape information symbolically to avoid both ambiguity and obscurity, just as we use notation to express music or electronic circuitry. We were interested in the algebraic structure hidden in the shapes, and we wanted to answer the following question: “Even if we identify that a given set of objects possesses an algebraic structure, how much is gained in practice from this discovery?” Of course, we have come to know that the set of objects is closed under some algebraic composition law, and that if it becomes possible to identify its set of generators, we may construct the whole set from that subset. However, can we conceive of some “stronger” kind of structure than this?
In this book, we take a morphological and set-theoretic approach to answering the above question. Then, we show the capability of this approach for image processing and computer graphics by presenting a simple shape model using two basic morphological and set-theoretic shape operators, which are called Minkowski addition and decomposition. The mathematical characteristics of these operators and their significance are explored in some detail, with the aim of eventually arriving at a formal theory of shape description.