  Symmetry is prevalent in nature and a common theme in man-made designs. Both
the human visual system and computer vision algorithms can use symmetry to
facilitate object recognition and other tasks. Detecting mirror symmetry in
images and data is, therefore, useful for a number of applications. Here, 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. Finally, we explore its applicability to examining
symmetry in natural systems by assessing the degree of symmetry present in
myelinated axon reconstructions from a larval zebrafish.
