A Part-aware Surface Metric for Shape Analysis Rong Liu 1, Hao Zhang 1, Ariel Shamir 2, and Daniel Cohen-Or 3 1 Simon Fraser University, Canada 2 The Interdisciplinary Center, Israel 3 Tel Aviv University, Israel

01/04/2009Part-aware Surface Metric2 Motivation – Shape Parts  Parts are useful to many geometry processing applications: Shape retrieval  [Berretti99, Dey03, Funkhouser06, Shalom08] Shape modeling  [Funkhouser04, Kraevoy07] Animation  [Katz03, Lien05]

01/04/2009Part-aware Surface Metric3 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

01/04/2009Part-aware Surface Metric4 Distances  Euclidean  Geodesic [Carmo76]  Isophotic [Pottmann04]  Diffusion distance [deGoes08] Surface based!

01/04/2009Part-aware Surface Metric5 Missing: Connection to Volume

01/04/2009Part-aware Surface Metric6 Shape Diameter Function [Shapira08]  Not a metric  Same value on different parts Distance(a,b) = SDF(a)-SDF(b) = 0 a b

01/04/2009Part-aware Surface Metric7 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

01/04/2009Part-aware Surface Metric8 Overview of Part-aware Metric  Derived as graph distance on primal/dual graph 1. Geodesic distance 2. Angular distance 3. VSI distance: Captures part information! Large distance between faces from different parts and vice versa = shortest graph distance between and distance (, )

01/04/2009Part-aware Surface Metric9 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)

01/04/2009Part-aware Surface Metric10 Geodesic Distance & Angular Distance  Geodesic distance: insensitive to parts  Angular distance: subject to leakage problem No angular difference!

01/04/2009Part-aware Surface Metric11 Volumetric Shape Image  Look at the object from inside

01/04/2009Part-aware Surface Metric12 Volume-based Distance Measure  Visibility can capture part information: Significant visibility changes across part boundaries

01/04/2009Part-aware Surface Metric13 VSI-distance: Step 1 Connect to surface: find reference points

01/04/2009Part-aware Surface Metric14 VSI-distance: Step 2 Sample visible regions from ref. points The Volumetric Shape Image (VSI) stores the normalized intersection points (|S|=100)

01/04/2009Part-aware Surface Metric15 VSI-distance: Step 3 Compute VSI difference Difference is based on the reach of local volume along sampling direction

01/04/2009Part-aware Surface Metric16 Moving Along a Path on the Surface

01/04/2009Part-aware Surface Metric17 Moving Along a Path on the Surface

01/04/2009Part-aware Surface Metric18 Moving Along a Path on the Surface

01/04/2009Part-aware Surface Metric19 Moving Along a Path on the Surface

01/04/2009Part-aware Surface Metric20 Moving Along a Path on the Surface

01/04/2009Part-aware Surface Metric21 VSI Differences Along the Path VSI Diff a b c d e

01/04/2009Part-aware Surface Metric22 No “Leakage” Problem Angular distance fields VSI distance fields

01/04/2009Part-aware Surface Metric23 Combined Distance Graph  Metric derived as the graph distance on a combined weighted graph geodesic graphangular graphVSI graph edge weight normalization = combined graph edge weight normalization

01/04/2009Part-aware Surface Metric24 Comparison with Other Metrics GeodesicDiffusion[deGoes08] Angular[Katz03] Part-aware

01/04/2009Part-aware Surface Metric25 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 + n 2 /3r 3 )=O(n 2 ) where r = 100 is grid resolution, n is number of triangles A mesh with 50K faces: 15 seconds

01/04/2009Part-aware Surface Metric26 Applications  Segmentation  Registration  Part-aware sampling  Retrieval

01/04/2009Part-aware Surface Metric27 Segmentation  Test algorithm: spectral clustering [Liu04] Use distances to derive a spectral embedding of input mesh faces Cluster in embedding space

01/04/2009Part-aware Surface Metric28 Segmentation (Cont’d) With geodesic+angular distance With part-aware distance

01/04/2009Part-aware Surface Metric29 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

01/04/2009Part-aware Surface Metric30 Embedding homer stretched geodesicpart-aware

01/04/2009Part-aware Surface Metric31 Registration geodesicpart-aware

01/04/2009Part-aware Surface Metric32 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

01/04/2009Part-aware Surface Metric33 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

01/04/2009Part-aware Surface Metric34 Retrieval (Cont’d) Geodesic distance: invariant to articulation Part-aware distance: stronger discriminating capability

01/04/2009Part-aware Surface Metric35 Retrieval (Cont’d) GeodesicD2 Part Aware

01/04/2009Part-aware Surface Metric36 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

01/04/2009Part-aware Surface Metric37 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

01/04/2009Part-aware Surface Metric38 Thank You!