Estimating true symmetry in scale space

The scale dependency of symmetry is often neglected in transforms designed to identify symmetry. In a single scale transform coarse scale symmetries are inappropriately perturbed by detailed variations in the boundary description. Multi-scale methods for computing symmetry as a function of scale overcome these problems. The result of applying such transforms to a 2-D grey-level image is a 3-D "image" in which the ridges are axes of symmetry. In existing algorithms for computing the multi-scale medial axis, such as HMAT and the credit attribution methods, a single boundary can produce evidence of symmetry leading to the generation of "spurious" symmetries. The voting scheme of the HMAT and the credit attribution algorithm emphasise symmetry in terms of boundary position only. We describe an algorithm in which mutual support (concordance) from symmetric boundaries is required to identify axes of symmetry. This algorithm combines evidence from the position and the relative strength of boundaries to identify true symmetry and avoid spurious responses. This Concordance-based Medial Axis Transform (CMAT) operates over scale and provides a description that is much closer to natural notions of symmetry. We demonstrate the performance of the CMAT algorithm with test figures used by other authors, and medical images that are relatively complex in structure and have edges not clearly defined. The CMAT algorithm is computationally less complex, and provides a better correspondence between symmetry and scale.