Scale Normalization for Isometric Shape Matching

Scale Normalization for Isometric Shape Matching

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

Problem Definition & Scope2 / 22

Scale problem inherent to isometric shape correspondence.

Scale normalization tested by shape correspondence. One shape is an isometric part of the other up to an

arbitrary scale.

Yusuf Sahillioğlu & Yücel Yemez, Scale Normalization for Isometric Shape Matching, Pacific Graphics, 2012.

Motivation3 / 22

Shape interpolation, animation.

Shape registration.

Shape recognition.

Time-varying reconstruction.

Statistical shape analysis.

Mapping between two shapes initiates cool apps.

Attribute transfer.


Contributions

A novel scale-invariant isometric distortion function.

Embedding avoided.

Correspondences are partial and dense at the same time.

No restriction on topology and triangulation type.

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Isometry

Our method is purely isometric. Intrinsic global property.

Similar shapes have similar metric structures.

Metric: geodesic distance.

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Scale Normalization

Scale normalization to prepare geodesic distances for the upcoming isometric distortion computations.

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Complete shapes (scale by max geodesic)

Partial shapes (max geodesic based normalization fails)

Partial shapes (scale by trusted matches) Partial shapes (scale by Euclidean embedding, e.g., Möbius)


Isometric Distortion Given , measure its isometric distortion:

O(N2) for a map of size N.

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in the most general setting.

: normalized geodesic distance b/w two vertices.


Isometric Distortion8 / 22

g ggggg

g g

average for .

in action:


Scale-invariant Isometric Distortion

Given , measure its scale-inv. isometric distortion:

This measure based on raw geodesics provides few trusted matches to be used in scale normalization.

O(N3) for a map of size N.

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: raw geodesic distance b/w two vertices.


Scale-invariant Isometric Distortion10 / 22

in action:

average for .


Minimizing Isometric Distortion11 / 22

Optimization for completely isometric shapes.

Optimization for partially isometric shapes.

Covers complete shape matching naturally.


Shape Matching Algorithm

Combinatorial optimization of for initial coarse correspondence.

The most extreme M source vertices are matched w/ |T| target extremities in the guidance of scale-inv. isometric distortion measure.

computational complexity where we set M=5.

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Two isometric distortion measures in action. Scale-invariant isometric distortion .

Isometric distortion w/ normalized geodesics .

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Winner



Use initial coarse correspondence to bring the meshes to the same scale.

Scale the target mesh by

Dense sampling.

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100 here

Same radius



Dense matching.

Minimum-weight perfect matching on cost matrix C.

ci,j = cost of matching si to tj //generating is traversed by

(si, tj).

Symmetric flip caring: repeat above (scaling, sampling, matching) with K-1 more generating initial coarse correspondences that follow in sorted distortions list.

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Computational Complexity16 / 22

Initial coarse correspondence:

Extension to dense map of size U: which also requires dense sampling.

Initial coarse correspondence dominates all.


Isometric part matching.

Results17 / 22


Also for complete matching

and for pairs w/ incompatible max geodesics.

Results18 / 22

Comparison w/ Möbius Voting (MV).


MV: bad extremity matching, triangulation-sensitive.

Different choices of M.

Results19 / 22


Too small. computation less reliable.

Too large. High computational

load. Inconsistent joint

sampling.

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Limitation & Future Work

Presence of uncommon parts may fail this framework which forces to match M=5 most extremes as a whole. Embedding into a more sophisticated framework should

help as it handles arbitrary scaling of the similar parts to be matched.


Conclusion21 / 22

A novel scale normalization method that proves useful for isometric partial and complete correspondence.

Scale-invariant isometric distortion minimized in the original 3D Euclidean space wherein isometry is defined. So does dense isometric matching.

Correspondences are partial and dense at the same time. Care for symmetric flips. No restriction on topology and triangulation. Applicable in its current form to 3D part retrieval problem.


People

Assoc. Prof. Yücel Yemez, supervisor

Yusuf, PhD student

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Documents

Scale Normalization for Isometric Shape Matching