Fast and Accurate Intrinsic Symmetry Detection

In computer vision and graphics, various types of symmetries are extensively studied since symmetry present in objects is a fundamental cue for understanding the shape and the structure of objects. In this work, we detect the intrinsic reflective symmetry in triangle meshes where we have to find the intrinsically symmetric point for each point of the shape. We establish correspondences between functions defined on the shapes by extending the functional map framework and then recover the point-to-point correspondences. Previous approaches using the functional map for this task find the functional correspondences matrix by solving a non-linear optimization problem which makes them slow. In this work, we propose a closed form solution for this matrix which makes our approach faster. We find the closed-form solution based on our following results. If the given shape is intrinsically symmetric, then the shortest length geodesic between two intrinsically symmetric points is also intrinsically symmetric. If an eigenfunction of the Laplace-Beltrami operator for the given shape is an even (odd) function, then its restriction on the shortest length geodesic between two intrinsically symmetric points is also an even (odd) function. The sign of a low-frequency eigenfunction is the same on the neighboring points. Our method is invariant to the ordering of the eigenfunctions and has the least time complexity. We achieve the best performance on the SCAPE dataset and comparable performance with the state-of-the-art methods on the TOSCA dataset.

Rajendra Nagar and Shanmuganathan Raman, “Fast and Accurate Intrinsic Symmetry Detection”, in European Conference on Computer Vision (ECCV), Munich, Germany, Sep. 8-14, 2018. pdf, poster, code

Direct HKS matching vs. restricted HKS matching. (a)-(b) Pairs of intrinsically symmetric points using HKS similarity. It suffers from the fact that if two neighboring points are subjected to a same strength heat source, then their heat diffusions will be similar. (c) Pairs obtained by restricted HKS matching. We observe that the sign of low frequency eigenfunctions on two neighboring points is the same. Therefore, we assign a high cost for pairing two points having the same vectors of signs of eigenfunctions.

We propose a novel approach to determine the sign of an eigenfunction by showing that, if a manifold contains intrinsic symmetry and an eigenfunction is an even (odd) function, then its restriction to the shortest length geodesic between any two intrinsically symmetric points is an even (odd) function.

Visualization of the eigenfunction correction. Eigenfunction correction is a crucial step since the original eigenfunctions are not perfect self-isometry invariant, i.e., not perfect even or odd functions which degrade the accuracy of symmetry detection. We propose an optimization-based approach to transform original eigenfunctions to make them self-isometry invariant.

Our approach detects dense intrinsic symmetry, i.e., finds the intrinsically symmetric point for each point of the input shape, in real scans, in partial meshes, as well as in approximate settings.