[Paper Review] Finding Mirror Symmetry via Registration

Marcelo Cicconet, David G. C. Hildebrand+1 more|arXiv (Cornell University)|Nov 18, 2016

Medical Image Segmentation Techniques21 references3 citations

TL;DR

This paper introduces Mirror Symmetry via Registration (MSR), a novel framework that formulates mirror symmetry detection in any Euclidean space as a registration problem. By reflecting data across an arbitrary plane and registering it to the original, MSR computes the symmetry plane as the eigenvector corresponding to eigenvalue -1 of the transformation matrix, achieving state-of-the-art performance in 2D symmetry detection and 86% accuracy in 3D shape symmetry detection using ICP registration.

ABSTRACT

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.

Motivation & Objective

• To develop a general, dimension-agnostic method for detecting mirror symmetry in Euclidean spaces using registration.
• To improve robustness and accuracy of symmetry detection in the presence of clutter and occlusion commonly found in real-world images.
• To enable symmetry detection in complex natural systems, such as neuronal morphologies, where symmetry may reflect functional or developmental principles.
• To provide a new benchmark for testing registration algorithms using symmetry detection as a downstream task.
• To extend symmetry detection beyond single symmetry lines to include full 3D shape and biological structure analysis.

Proposed method

• Reflect the input data across an arbitrary plane to generate a reflected dataset.
• Perform registration between the original and reflected datasets using a registration algorithm (e.g., ICP or RANSAC-based NCC registration).
• Compute the transformation matrix from the registration and reflection mappings.
• Extract the eigenvector corresponding to eigenvalue -1 of the composite transformation matrix to determine the symmetry plane.
• For 2D, use a RANSAC-based registration method over normalized cross-correlation (NXC) matches between image patches and the full image.
• For 3D, employ the Iterative Closest Point (ICP) algorithm as the registration back-end to align point clouds.

Experimental results

Research questions

• RQ1 Can mirror symmetry detection in R^n be reduced to a registration problem with mathematically exact solutions?
• RQ2 How does the MSR framework perform compared to state-of-the-art methods on 2D symmetry detection benchmarks?
• RQ3 Can MSR generalize to 3D symmetric shapes and achieve high accuracy despite the absence of standardized 3D symmetry databases?
• RQ4 To what extent can MSR detect symmetry in complex biological structures like myelinated axons in zebrafish?
• RQ5 Can MSR be used to identify symmetric pairs of neuronal skeletons and quantify morphological symmetry in neural circuits?

Key findings

• MSR achieves state-of-the-art performance on two independent 2D symmetry detection databases: the CVPR 2013 Real World Images competition and the NYU Symmetry database.
• On a curated 3D dataset of 203 symmetric shapes from the McGill 3D Shape Benchmark, MSR achieved 86% accuracy using ICP as the registration back-end.
• The method successfully computed a visually plausible symmetry plane for myelinated axon reconstructions in a larval zebrafish, with projections orthogonal to the computed plane showing strong symmetry.
• The application of the Munkres assignment algorithm to pairwise symmetry costs enabled globally optimal left-right skeleton matching in the zebrafish axon data.
• The framework is robust to initial plane choice, with the best result selected based on registration confidence across three initial reflections.
• The method is mathematically exact except for the registration step, meaning its accuracy is bounded only by the underlying registration algorithm’s performance.