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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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Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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
g g
average for .
in action:
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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).
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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 =
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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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Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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)
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
5M
M
M
Patch-based Matching ( )
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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 ( )
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.
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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:
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 Complexity15 / 24
Saliency sorting
C2F sampling
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
Computational Complexity16 / 24
Patch-based combinatorial matching
Merging
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
Computational Complexity17 / 24
Overall
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
6K vs. 16K
Experimental Results19 / 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 Results20 / 2
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.
Experimental Results21 / 24
Comparisons
GMDS O(N2logN)[Bronstein et al.]
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
Spectral O(N2logN)[Jain et al.]
Nonrigid world dataset
Our method O(NlogN)
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:
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
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).
Yusuf Sahillioğlu & Yücel Yemez, Coarse-to-Fine Combinatorial Matching for Dense Isometric Shape Correspondence, SGP’11.
People
Assoc. Prof. Yücel Yemez, supervisor
Yusuf, PhD student
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