GRASP Special Seminar – Yanxi Liu, Penn State University, “Computational Symmetry”

Abstract: Symmetry is an essential mathematical concept, as well as a ubiquitous, observable phenomenon in nature, science and art. Either by evolution or by design, symmetry implies an efficiency coding that makes it universally appealing, especially so to computational science. Recognition and categorization of symmetry and regularity is the first step towards capturing the essential skeleton of a real world problem, while at the same time minimizing computational redundancy. However, symmetry group detection from real world (digital) data turns out to be a challenging problem that has been puzzling computer vision, computer graphics and psychology researchers for decades.

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’.