Symmetry as a Continuous Feature

| Symmetry is treated as a continuous feature and a Continuous Measure of Distance from Symmetry in shapes is deened. The Symmetry Distance (SD) of a shape is deened to be the minimum mean squared distance required to move points of the original shape in order to obtain a symmetrical shape. This general deenition of a symmetry measure enables a comparison of the \amount" of symmetry of diierent shapes and the \amount" of diierent symmetries of a single shape. This measure is applicable to any type of symmetry in any dimension. The Symmetry Distance gives rise to a method of reconstructing symmetry of occluded shapes. We extend the method to deal with symmetries of noisy and fuzzy data. Finally, we consider grayscale images as 3D shapes, and use the Symmetry Distance to nd the orientation of symmetric objects from their images, and to nd locally symmetric regions in images.