Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence

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Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape

Correspondence

Yusuf Sahillioğlu and Yücel YemezComputer Eng. Dept., Koç University, Istanbul, Turkey

Problem Definition & Apps2 / 24

Shape interpolation

Shape registration

Shape matching

Time-varying recon.

Statistical shape analysis

Goal: Find a mapping between two isometric shapes

Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.

Attribute transfer

Contributions Avoid embedding

C2F joint sampling of evenly-spaced salient vertices

geodesic curvatureintegral

Euclideanembedding

Non-Euclideanembedding

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O(NlogN) time complexity for dense correspondenceYusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.

Isometry Our method is purely isometric Intrinsic global property

Similar shapes have similar metric structures

Metric: geodesic distance

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Isometric Distortion Given , measure its isometric distortion:

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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 Distortion6 / 24

g ggggg

average for .

in action:

Minimizing Isometric Distortion

N = |S| = |T| for perfectly isometric shapes. N! different mappings; intractable.

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Solution: Patch-by-patch matching to reduce search space. Optimal mapping maps nearby vertices in source to

nearby vertices in target.

Recursively subdivide matched patches into smaller patches (C2F sampling) to be matched (combinatorial search).

Coarse-to-Fine Sampling : set of base vertices sampled from at level .

Sampling radii s.t. for k=0,1,..,K. at level defines patch : all vertices within a

distance from the base .

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greens inherited from level k−1blues are all vertices ( )patches being defined ( )

blacks + greens =

Correspondence Algorithm Correspondence at level k is obtained in two

steps: Match level k bases inside the patch pairs matched at level

k−1. Merge patch-based local correspondences into one global

correspondence over whole surface.

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Patch-based Matching ( ) Ensure base vertices fall into each patch to

allow combinatorial matching. Patch radius to select for such an :

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, area of the largest patch at level k−1.

M=5 samples with circular patches to cover blue area

(enlarge a bit to cover whites)

Patch-based Matching ( )

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Combinatorial matching

greens inherited from level k−1

blacks + greens =

Correspondence Merging ( ) Merge patch-to-patch correspondences into one

global correspondence that covers the whole surface.

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Multi-graph single graph. Also, diso values made available.1st pass over source samples to keep only one match per sample, the one

with the min diso.

2nd pass over target samples to assign one match per isolated sample, the

one with the min diso.

Trim matches with diso > 2Diso, i.e., outliers.

Insight to the Algorithm13 / 24

Conditions for the algorithm to work correctly 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)

Optimal mapping assigns si to tj which is as nearest to the ground-truth ti as possible

Inclusion assertion is then expected to apply:

Inclusion assertion (demonstration)14 / 24

Computational Complexity15 / 24

Saliency sorting

C2F sampling

Patch-based combinatorial matching

Merging

Overall

Experimental Results18 / 24

Details captured, smooth flow Many-to-one

Two meshes at different resolutions

red line: the worst match

w.r.t. isometric distortion

6K vs. 16K

red line: the worst match w.r.t. isometric distortion

for four more pairs:

red line: the worst match w.r.t. isometric distortiongreen line: the worst match w.r.t. ground-truth

distortionYusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.

Comparisons

GMDS O(N2logN)[Bronstein et al.]

Spectral O(N2logN)[Jain et al.]

Nonrigid world dataset

Our method O(NlogN)

Future Work22 / 24

Symmetric flip issue Purely isometry-based methods naturally fail at symmetric

inputs Not intrinsically symmetric only one optimal solution

Our method may still occasionally fail to find the optimum due to initial coarse sampling

Solution suggested

A solution for symmetric flips due to initial coarse sampling:

Conclusion23 / 24

Computationally efficient C2F dense isometric shape correspondence algorithm (O(NlogN)).

Isometric distortion minimized in the original 3D Euclidean space wherein isometry is defined.

Accurate for isometric and nearly isometric pairs. Different levels of detail thanks to the C2F joint

sampling. No restriction on topology. Symmetric flips may occasionally occur due to

initial coarse sampling (but can be healed as proposed).

People

Assoc. Prof. Yücel Yemez, supervisor

Yusuf, PhD student

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Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence

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