Multi-scale partial intrinsic symmetry detection

Abstract

Figure 1: Multi-scale partial intrinsic symmetry detection: five symmetry scales (large to small) are detected. Each symmetric region is shown in uniform color. Note the detection of inter-and intra-object symmetries, as well as cylindrical symmetry of the limbs. AbstractWe present an algorithm for multi-scale partial intrinsic symmetry detection over 2D and 3D shapes, where the scale of a symmetric region is defined by intrinsic distances between symmetric points over the region. To identify prominent symmetric regions which overlap and vary in form and scale, we decouple scale extraction and symmetry extraction by performing two levels of clustering. First, significant symmetry scales are identified by clustering sample point pairs from an input shape. Since different point pairs can share a common point, shape regions covered by points in different scale clusters can overlap. We introduce the symmetry scale matrix (SSM), where each entry estimates the likelihood two point pairs belong to symmetries at the same scale. The pair-to-pair symmetry affinity is computed based on a pair signature which encodes scales. We perform spectral clustering using the SSM to obtain the scale clusters. Then for all points belonging to the same scale cluster, we perform the second-level spectral clustering, based on a novel point-to-point symmetry affinity measure, to extract partial symmetries at that scale. We demonstrate our algorithm on complex shapes possessing rich symmetries at multiple scales. Links:DL PDF WEB DATA

Characterization of Partial Intrinsic Symmetries

“…As the complexity of the space of all partial symmetries is believed to be large [24], [28], existing methods proceed by reducing the search space after sampling the surface.…”

“…Xu et al [28] decouple the problem into the two problems of symmetric point pair detection and symmetry scale detection. The specific difficulty of this method is the preservation of all shortest geodesic paths after a partial mapping to a larger context.…”

“…The key challenge is that the space of partial intrinsic symmetries might be too large to systematically explore and report [24], [28]. Following Brunton et al [6], we define partial correspondences by sets that are mapped with preservation of the local metric.…”

“…Following Brunton et al [6], we define partial correspondences by sets that are mapped with preservation of the local metric. Unlike Xu et al [28], we do not require that the shortest geodesic paths to be maintained outside the mapped region, which leads to a much higher combinatorial complexity due to the global dependency on topology and boundaries of the subsets. In contrast, metric-preserving maps can be parametrized by only five degrees of freedom (two out of which do not even need fine sampling) [6], which makes an approximate enumeration of all isometries feasible in polynomial time.…”

“…While this approach reduces the complexity of the problem, it might miss some intrinsic r-symmetries. Further, existing methods have focused on either detecting partial infinitesimal symmetries [8], [26] or partial discrete symmetries [24], [28], [22], [13], [4], [12], [18], [23], [9], [15] but not both.…”