Closure—Hierarchies of Gestalten

Gestalt perception comes in hierarchies. Human observers are quick and reliable in reconstructing from pictorial data constructions like a row made of reflection symmetric parts, where each wing of the parts is again a lattice of reflection symmetric sub-parts, etc. Machines hardly compete with these skills up to now, in particular when projection distorted the patterns, or occlusion deleted some parts of it. This chapter is meant as an attempt to code machines so that they become better. To this end, the combinatorial nature of such “Gestalt sentences” must first be understood. Instead of the traditional way of defining grammars, here an algebraic view is taken, understanding what used to be a “pattern language” as the closure of the Gestalt Algebra. The operations of this algebra are the Gestalt operations given in the previous chapters. Of course such closure is infinite, but we can prove that almost all Gestalten in it have assessments close to zero. The set of hierarchical Gestalten assessed better than an \(\epsilon >0\) and resulting from a finite set of primitives is finite. Still, this set can be very huge. In particular if we use the operations simply, i.e., only taking into account the features of immediately preceding Gestalten. However, in this chapter we also present ways to propagate the Gestalt laws through the hierarchy, if necessary, down to the preceding primitives. This greatly reduces numbers and efforts. The literature on symmetry recognition sees a problem in the non-local nature of, e.g., reflection symmetry. Search for correspondence cannot be bounded by less than quadratic complexity. Here hierarchy can actually help. Correspondence between very distant small objects is established by use of hierarchy. This can be implemented in sub-quadratic complexity. Thus, what looks at first glance like a combinatorial nightmare, turns out to be a proposal for the solution of an old and hard combinatorial correspondence problem.