Finding Mirror Symmetry via Registration and Optimal Symmetric Pairwise Assignment of Curves

Abstract

We demonstrate that the problem of fitting a plane of mirror symmetry to data in any Euclidian space can be reduced to the problem of registering two datasets. The exactness of the resulting solution depends entirely on the registration accuracy. This new Mirror Symmetry via Registration (MSR) framework involves (1) data reflection with respect to an arbitrary plane, (2) registration of original and reflected datasets, and (3) calculation of the eigenvector of eigenvalue -1 for the transformation matrix representing the reflection and registration mappings. To support MSR, we also introduce a novel 2D registration method based on random sample consensus of an ensemble of normalized cross-correlation matches. With this as its registration back-end, MSR achieves state-of-the-art performance for symmetry line detection in two independent 2D testing databases. We further demonstrate the generality of MSR by testing it on a database of 3D shapes with an iterative closest point registration back-end. We finally explore its applicability to examining symmetry in natural systems by assessing the degree of symmetry present in myelinated axon reconstructions from a larval zebrafish. Using the MSR-computed plane of symmetry, we introduce techniques for the optimal symmetric pairwise assignment between axon reconstructions and provide visualizations illustrating how neighborhood relationships between nearby axon pairs compare with the relationships between their mirror-reflected counterparts along the anteroposterior axis.