An Optimization Approach to Improving Collections of Shape Maps Andy Nguyen, Mirela Ben-Chen, Katarzyna Welnicka, Yinyu Ye, Leonidas Guibas Computer Science Dept. Stanford University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A AA A A A A AAA

Introduction 2

Introduction 3

Introduction Quality of maps is a property of the collection 4

Introduction Corollary: Individual maps cannot be evaluated in isolation 5

Problem Statement Input A collection of related shapes A collection of maps between all pairs of shapes A distance measure on each shape 6

Problem Statement Output: A collection of maps between pairs of shapes that is Accurate Consistent 7

Graph Representation 8

Approach Cycle consistency tells us something about accuracy Remove the inaccuracies we find Repeat using the better collection 9

Related Work Learning Shape Metrics based on Deformations and Transport, Charpiat, NORDIA09 Dynamic time warping + matching energy for good shortest paths Disambiguating Visual Relations Using Loop Constraints, Zach et al, CVPR2010 Use cycle consistency to remove incorrect correspondences Pairwise mapping methods Möbius Voting for Surface Correspondence, Lipman et al, SIGGRAPH 2009 One Point Isometric Matching with the Heat Kernel, Ovsjanikov et al, SGP 2010 Blended Intrinsic Maps, Kim et al, SIGGRAPH

Definitions Accuracy error: Consistency error: 11

Relating Cycles to Edges Call low error “good,” high error “bad” Good and bad edges cause good and bad cycles If we can only evaluate the cycles, what can we say about the edges? 12

Relating Cycles to Edges Accuracy error of a path ° = { i 1, …, i n } is bounded*: 13 *If ground-truth maps preserve the distortion measure

Proposal – Linear Program For each 3-cycle ° in the graph, compute the distortion C ° Solve the following linear program to find weights for the edges: Minimize Subject to Where 14

Are 3-Cycles Sufficient? 15 AB CD

Proposal LP gives us a weighted graph Weights give us shortest-path map compositions But these are just like our input Run the LP again? 16

Are 3-Cycles Sufficient? 17 AB CD AB CD ABDBAC

Proposal - Complete Repeat the following: Solve LP => obtain edge-weighted graph Replace edges with shortest paths Until one of the following is true: No edge replacements happen, or No more 3-cycles are bad 18

Convergence - Experimental 19 Map Type LP WeightsFinal accuracy

Convergence - Theoretical “Almost-accurate” collection: Each 3-cycle has at most 1 bad map Every cycle’s distortion is either 0 or equal to the inaccuracy of the 1 bad map LP weights are exactly the map accuracy errors Guarantees consistency and accuracy after replacing maps with shortest paths 20

Results – 2D (DTW) 21 Max consistency error Fraction of maps Max accuracy error Fraction of cycles

Results – 2D (DTW) 22 Max consistency error Fraction of maps Max accuracy error Fraction of cycles

Results – 3D (Möbius Voting) 23

Results – 3D (Heat Kernel) 24 Max consistency error Fraction of maps Max accuracy error Fraction of cycles

Results – 3D (Blended Maps) 25 Animals

Results – 3D (Blended Maps) 26 Hands

Results – 3D (Blended Maps) 27 Humans

Results – 3D (Blended Maps) 28 Teddies

Future Work Prove convergence in more general cases Allow for multiple maps between a given pair of shapes Discover the structure of the collection using consistency information 29

Conclusions Collections contain information that allow us to better evaluate maps Cycle consistency can be used to identify and remove bad maps Using an LP with 3-cycle constraints lets us do this efficiently Repeating the process lets us incorporate longer cycles 30

Thank You