Symmetry and Orbit Detection via Lie‐Algebra Voting

Abstract

Figure 1: Adjoint invariant distance to orbit. Our Lie algebra voting approach to symmetry and orbit detection maps SE(3) transformations into points in a logarithmic space composed of a rotation part ω ∈ R 3 and a translation part u ∈ R 3 . The rotational orbit of the church and the translational orbit of the side railing (a) are mapped into collinear blue and red spheres respectively (a few transformations within these two orbits are marked with circled numbers to enhance comprehension). When the scene is centered, the two lines are orthogonal to each other and easy to distinguish (b). However, after a rigid translation of the scene, the rotational orbit now has u-values near the translation orbit points, making it impossible to automatically distinguish these two orbits using a Euclidean distance (d), while our adjoint invariant distance for orbit shows no discernible difference in results as evidenced by a binning of detected orbit sizes for both situations (e). AbstractIn this paper, we formulate an automatic approach to the detection of partial, local, and global symmetries and orbits in arbitrary 3D datasets. We improve upon existing voting-based symmetry detection techniques by leveraging the Lie group structure of geometric transformations. In particular, we introduce a logarithmic mapping that ensures that orbits are mapped to linear subspaces, hence unifying and extending many existing mappings in a single Lie-algebra voting formulation. Compared to previous work, our resulting method offers significantly improved robustness as it guarantees that our symmetry detection of an input model is frame, scale, and reflection invariant. As a consequence, we demonstrate that our approach efficiently and reliably discovers symmetries and orbits of geometric datasets without requiring heavy parameter tuning.

A Variational Approach to Registration with Local Exponential Coordinates

“…Almost without exception, all the examples presented here required a dozen or fewer iterations to compute the optimal solution. Jacobian calculations are also useful for an entirely different reason — when singular, its eigenvectors corresponding to null eigenvalues reveal symmetries in data sets [SAD*16]. This may include translational periodicity, rotational invariances or a combination of both.…”

“…Indeed, numerous algorithms for constructing interpolants in Lie groups have been proposed, with applications ranging from synthesizing animations in computer graphics or path planning in robotics [ŽK98, Agr06], to designing integrators for ordinary differential equations [Mar99]. A recent application exploiting the recognition of rigid body motions as a Lie group to detect symmetries in data sets can be found in [SAD*16]. As a particular example of the potency of adopting canonical coordinates for problems posed in Lie groups, we highlight an analysis of the motion and structure recovery problem arising in computer vision [MKS01], where the parameterization of the essential manifold (closely related to SO 3 ) plays a key role.…”

Feature Curve Co‐Completion in Noisy Data

“…Symmetry detection based on transformation space clustering was proposed in [MGP06, PMW*08, SAD*16]. In these approaches the co‐occurring parts are detected in a three step procedure.…”

“…Some methods detect whether the extracted transformations align on a grid (cf. [PMW*08, SAD*16]) in a third step, which allows them to find orbits of multiple occurrences. These approaches strongly rely on the quality of the initial point matches.…”

“…As a final step of the model generation phase we detect orbits in the transformation space (i.e. identifying whether subsets of the reoccurring transformations can be described by a grid), as in [PMW*08, SAD*16].…”