Simple gestalt algebra
The laws of Gestalt perception rule how parts are assembled into a perceived aggregate. This contribution defines them in an algebraic setting. Operations are defined for mirror symmetry, repetition in rows, and arrangement in rotational symmetry patterns respectively. While the mirror operation is a classical binary operation, the other two operations are of arity n > 1. Thus the Gestalt domain with its three operations forms a general algebra. Deviations from the perfect mutual positioning are handled using positive and differentiable assessment functions achieving maximal value for the case of perfect symmetry and approaching zero if the parts mutually violate the Gestalt laws. Theorems of closure are proven, stating that any of the operations on any Gestalten will produce again a well-defined new Gestalt. It is also proven that no neutral elements and no inverse Gestalten exist for the three operations. Practically, these definitions and calculations can be used in two ways: 1. Images with Gestalts can be rendered by using random decisions with the assessment functions as densities; 2. given an image (in which Gestalts are supposed) Gestalt-terms are constructed successively, and the ones with high assessment values are accepted as plausible, and thus recognized.