Quantitative Analysis of Continuous Symmetry

Symmetry is typically viewed as a discrete feature: an object is either symmetric or non-symmetric. However visual perception treats symmetry as a continuous feature, relating to statements such asòne object is MORE symmetric than another' oràn object is MORE mirror symmetric than rotational symmetric'. With this notion in mind, we view symmetry as a continuous feature and dene a Continuous Symmetry Measure (CSM) to quantify the "amount" of symmetry of dierent shapes and thèamount' of diierent symmetries of a single shape. Computational methods have been developed to compute the CSM values with respect to any point symmetry for any shape or pattern in any dimension. A preliminary study showed that the Symmetry Measure developed, is commensurate with human perceptual experience. The computational approach embeds both the hierarchical and continuous nature of symmetry of objects. Thus global and local features can be evaluated for their symmetry content. The CSM has been applied to real images and to shapes and forms extracted from measured data. The CSM has been extended to deal with symmetries of noisy and fuzzy data as well as reconstructing symmetries of occluded shapes. Additionally, the CSM is used to evaluate symmetries of 3D objects, given as measured data or evaluated from the 2D projection.