CoarsetoFine Combinatorial Matching for Dense Isometric Shape Correspondence

Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence Yusuf Sahillioğlu and Yücel Yemez Computer Eng. Dept. , Koç University, Istanbul, Turkey

Problem Definition & Apps 2 / 24 Goal: Find a mapping between two isometric shapes ü Shape interpolation ü Attribute transfer ü Shape registration ü Time-varying recon. ü Shape matching ü Statistical shape analysis Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Contributions 3 / 24 ü Avoid embedding Euclidean embedding Non-Euclidean embedding ü C 2 F joint sampling of evenly-spaced salient vertices geodesic integral curvature ü O(Nlog. N) time complexity for dense correspondence Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Isometry 4 / 24 ü Our method is purely isometric ü Intrinsic global property ü Similar shapes have similar metric structures ü Metric: geodesic distance Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Isometric Distortion 5 / 24 ü Given , measure its isometric distortion: in the most general setting. : normalized geodesic distance b/w two vertices. Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Isometric Distortion 6 / 24 in action: average for g gg g g . Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Minimizing Isometric Distortion 7 / 24 ü N = |S| = |T| for perfectly isometric shapes. ü N! different mappings; intractable. ü Solution: Patch-by-patch matching to reduce search space. ü Optimal mapping maps nearby vertices in source to nearby vertices in target. ü Recursively subdivide matched patches into smaller patches (C 2 F sampling) to be matched (combinatorial search). Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Coarse-to-Fine Sampling 8 / 24 : set of base vertices sampled from ü at level . ü ü Sampling radii ü s. t. at level defines patch from the base. for k=0, 1, . . , K. : all vertices within a distance blues all vertices ( ( k− 1 ) ) greens inherited from level blacks +are greens = patches being defined Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Correspondence Algorithm 9 / 24 ü Correspondence at level k is obtained in two steps: ü Match level k bases inside the patch pairs matched at level k− 1. ü Merge patch-based local correspondences into one global correspondence over whole surface. ü Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Patch-based Matching ( ) 10 / 24 ü Ensure base vertices fall into each patch to allow combinatorial matching. ü Patch radius to select for such an : , area of the largest patch at level k− 1. M=5 samples with circular (enlarge a bit to cover whites) patches to cover blue area Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Patch-based Matching ( ) 11 / 24 ü Combinatorial matching greens inherited from level k− 1 blacks + greens = Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Correspondence Merging ( ) 12 / 24 ü Merge patch-to-patch correspondences into one global correspondence that covers the whole surface. st pass Trim Multi-graph matches with single dsamples graph. > 2 DAlso, i. e. , disooutliers. values made available. 1 nd 2 pass over source target to, assign keep only oneone match per isolated sample, the one the iso with one the min diso. Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Insight to the Algorithm 13 / 24 ü Conditions for the algorithm to work correctly ü ü High-resolution sampling on two perfectly isometric surfaces Evenly-spaced sampling s. t. every vertex is in at least one patch Distortion is a slowly changing convex function around optimum One optimal solution (no symmetric flips) ü Optimal mapping assigns si to tj which is as nearest to the ground -truth ti as possible ü Inclusion assertion is then expected to apply: Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Inclusion assertion (demonstration) 14 / 24 Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Computational Complexity 15 / 24 ü Saliency sorting ü C 2 F sampling Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Computational Complexity 16 / 24 ü Patch-based combinatorial matching ü Merging Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Computational Complexity 17 / 24 ü Overall Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Experimental Results 18 / 24 ü Details captured, smooth flow red line: the worst match w. r. t. isometric distortion ü Many-to-one 6 K vs. 16 K ü Two meshes at different resolutions Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Experimental Results 19 / 24 red line: the worst match w. r. t. isometric distortion Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Experimental Results 20 / 2 ü for four more pairs: red line: the worst match w. r. t. isometric distortion green line: the worst match w. r. t. ground-truth distortion Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Experimental Results 21 / 24 ü Comparisons Nonrigid world dataset GMDS O(N 2 log. N) [Bronstein et al. ] Spectral O(N 2 log. N) Our method O(Nlog. N) [Jain et al. ] Our method O(Nlog. N) Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Future Work 22 / 24 A solutionflip forissue symmetric flips due to initial coarse sampling: üü Symmetric ü Purely isometry-based methods naturally fail at symmetric inputs ü Not intrinsically symmetric only one optimal solution ü Our method may still occasionally fail to find the optimum due to initial coarse sampling ü Solution suggested Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

Conclusion 23 / 24 ü Computationally efficient C 2 F dense isometric shape correspondence algorithm (O(Nlog. N)). ü Isometric distortion minimized in the original 3 D Euclidean space wherein isometry is defined. ü Accurate for isometric and nearly isometric pairs. ü Different levels of detail thanks to the C 2 F joint sampling. ü No restriction on topology. ü Symmetric flips may occasionally occur due to initial coarse sampling (but can be healed as proposed). Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’ 11.

People 24 / 24 Yusuf, Ph. D student Assoc. Prof. Yücel Yemez, supervisor