Pattern Theory: A Unifying Perspective

The term "Pattern Theory" was introduced by Ulf Grenander in the 70s as a name for a field of applied mathematics which gave a theoretical setting for a large number of related ideas, techniques and results from fields such as computer vision, speech recognition, image and acoustic signal processing, pattern recognition and its statistical side, neural nets and parts of artificial intelligence (see [Grenander 76-81]). When I first began to study computer vision about ten years ago, I read parts of this book but did not really understand his insight. However, as I worked in the field, every time I felt I saw what was going on in a broader perspective or saw sonic theme which seemed to pull together the field as a whole, it turned out that this theme was part of what Grenander called pattern theory. It seems to me now that this is the right framework for these areas, and, as these fields have been growing explosively, the time is ripe for making an attempt to reexamine recent progress and try to make the ideas behind this unification better known. This article presents pattern theory from my point of view, which may be somewhat narrower than Grenander's, updated with recent examples involving interesting new mathematics. I want to define pattern theory as: