A Part-aware Surface Metric for Shape Analysis

A Part-aware Surface Metric for Shape Analysis Rong Liu1, Hao Zhang1, Ariel Shamir2, and Daniel Cohen-Or3 1Simon Fraser University, Canada 2The Interdisciplinary Center, Israel 3Tel Aviv University, Israel

Motivation – Shape Parts • Parts are useful to many geometry processing applications: • Shape retrieval • [Berretti99, Dey03, Funkhouser06, Shalom08] • Shape modeling • [Funkhouser04, Kraevoy07] • Animation • [Katz03, Lien05] Part-aware Surface Metric

Shape Analysis • Many problems involving the analysis and understanding of a 3D object utilize a metric, which prescribes a distance function between points on the boundary surface of the object… a b Part-aware Surface Metric

Distances • Euclidean • Geodesic [Carmo76] • Isophotic [Pottmann04] • Diffusion distance [deGoes08] Surface based! Part-aware Surface Metric

Missing: Connection to Volume Part-aware Surface Metric

a b Shape Diameter Function [Shapira08] • Not a metric • Same value on different parts • Distance(a,b) = SDF(a)-SDF(b) = 0 Part-aware Surface Metric

Contribution • A novel part-aware surface distance metric • Able to effectively capture part information of a shape • Based on volumetric considerations • Applications: • Segmentation • Shape registration • Part-aware sampling • Shape retrieval Part-aware Surface Metric

= shortest graph distance between and distance( , ) Overview of Part-aware Metric • Derived as graph distance on primal/dual graph • Geodesic distance • Angular distance • VSI distance: Captures part information! • Large distance between faces from different parts and vice versa Part-aware Surface Metric

Geodesic Distance & Angular Distance • Geodesic distance (approximate) • Distant faces tend to belong to different parts (Gestalt principle of proximity) • Angular distance • Faces separated by concave regions tend to belong to different parts (Minima rule) Part-aware Surface Metric

No angular difference! Geodesic Distance & Angular Distance • Geodesic distance: insensitive to parts • Angular distance: subject to leakage problem Part-aware Surface Metric

Volumetric Shape Image • Look at the object from inside Part-aware Surface Metric

Volume-based Distance Measure • Visibility can capture part information: Significant visibility changes across part boundaries Part-aware Surface Metric

VSI-distance: Step 1 Connect to surface: find reference points Part-aware Surface Metric

VSI-distance: Step 2 Sample visible regions from ref. points The Volumetric Shape Image (VSI) stores the normalized intersection points (|S|=100) Part-aware Surface Metric

VSI-distance: Step 3 Compute VSI difference Difference is based on the reach of local volume along sampling direction Part-aware Surface Metric

Moving Along a Path on the Surface Part-aware Surface Metric

VSI Differences Along the Path d VSI Diff c b e a Part-aware Surface Metric

No “Leakage” Problem Angular distance fields VSI distance fields Part-aware Surface Metric

Combined Distance Graph • Metric derived as the graph distance on a combined weighted graph geodesic graph angular graph VSI graph edge weight normalization edge weight normalization = combined graph Part-aware Surface Metric

Comparison with Other Metrics Geodesic Diffusion[deGoes08] Part-aware Angular[Katz03] Part-aware Surface Metric

Practical Issues • Able to handle open meshes with reasonably well-defined volume • Speed-up • Space voxelization (100*100*100) for ray-mesh intersection detection • Use Sampling of VSI and interpolation • Efficiency • Empirical complexity: O(rn + n2/3r3)=O(n2) where r = 100 is grid resolution, n is number of triangles • A mesh with 50K faces: 15 seconds Part-aware Surface Metric

Applications • Segmentation • Registration • Part-aware sampling • Retrieval Part-aware Surface Metric

Segmentation • Test algorithm: spectral clustering [Liu04] • Use distances to derive a spectral embedding of input mesh faces • Cluster in embedding space Part-aware Surface Metric

Segmentation (Cont’d) With geodesic+angular distance With part-aware distance Part-aware Surface Metric

Registration • Test algorithm: spectral embedding [Jain07] and iterative closest point [Besl92] alignment • Use distances to derive a spectral embedding • Register in the embedding • Geodesic distance is usually used as it is intrinsic (invariant to articulation) • Geodesic distance is not invariant to stretch! But VSI distance is (although not to articulations) • Part aware: best to combines the two Part-aware Surface Metric

Embedding geodesic part-aware homer stretched Part-aware Surface Metric

Registration geodesic part-aware Part-aware Surface Metric

Part-aware Sampling • Algorithm: max-min (farthest point) sampling • Add samples iteratively • Each sample maximizes the minimum distance to previously chosen samples With isophotic distance With part-aware distance Part-aware Surface Metric

Object Retrieval • Test algorithm: probability distribution of shape function [Osada02] • Use the histogram of a shape function as signature and chi-square to measure histogram distances histogram of pair-wise geodesic distances between vertices Part-aware Surface Metric

Retrieval (Cont’d) Geodesic distance: invariant to articulation Part-aware distance: stronger discriminating capability Part-aware Surface Metric

Retrieval (Cont’d) Part Aware Geodesic D2 Part-aware Surface Metric

Summary • Use the Volume! • A novel part-aware metric • Based on volumetric considerations • Able to capture part information effectively • Improved upon previous metrics • Demonstrated effectiveness for a variety of geometry processing and analysis applications • Mesh segmentation • Shape registration • Part-sensitive sampling • Shape Registration Part-aware Surface Metric

Future Work • Intelligent ways to tune the weights for geodesic, angular, and VSI distances • Application dependent? • Training? • More systematic test on shape retrieval • Expand database • Compare with more algorithms • More applications Part-aware Surface Metric

Thank You! Part-aware Surface Metric

A Part-aware Surface Metric for Shape Analysis