We explore a formal and computational characterization of real world regularity using a hierarchical model of symmetry groups as a theoretical basis, embedded in a well-defined Bayesian framework. Such a formalization simultaneously facilitates (1) a robust and comprehensive algorithmic treatment of the whole regularity spectrum, from regular (perfect symmetry), near-regular (approximate symmetry), to various types of irregularities; (2) an effective detection scheme for real world symmetries and symmetry groups; and (3) a set of computational bases for measuring and discriminating quantified regularities on diverse data sets.
Besides some theoretical background on crystallographic groups in particular, I shall illustrate various recent results of applications of computational symmetry in texture analysis/synthesis, tracking, and manipulation; perceptual grouping and 3D modeling from a single view, human gait and activity recognition; symmetry- based dance analysis/synthesis; grid-cell clustering; automatic geo-tagging; and image ‘de- fencing’.