justin solomon, mirela ben-chen, adrian butscher, and leo guibas · 2011. 6. 9. · justin solomon,...

Justin Solomon, Mirela Ben-Chen, Adrian Butscher, and Leo Guibas

Stanford University

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Segmentation Part Discovery

Segmentation Part Discovery

Segmentation Part Discovery

Hierarchical Mesh Segmentation Based on Fitting Primitives Attene et al, Visual Computer 22.3 (2006)

Plane

Sphere

¼nX

i=1

How do you choose?

Plane

Sphere

?¼nX

i=1

Preserves Pairwise Distances

Preserves Pairwise Distances

^

^

Segmentation and part-finding invariant to

near-isometry

Segmentation and part-finding invariant to

near-isometry

Plane Sphere

Find symmetries using approximate Killing vector fields

Compute and cluster

isometry-invariant point signatures

<digression>

Killing

Vector

Fields

<digression>

Killing

Vector

Fields Wilhelm Killing

1847-1923

Distance-Preserving Self Map

Distance-Preserving Self Map ^

!(p) = tangent vector at p

Continuous self-isometry

Át : §! §

Continuous self-isometry

Át : §! §

Killing vector ¯elddÁtdt

: §! T§

Vector ¯eld: ! 2 REOperator (matrix) K measuring

deviation from isometry

Want to minimize kK!k2 subject to k!k = 1

mFind eigenvectors (\eigen¯elds") of K

</digression>

Eigenfields of K are localized on parts!

­i = component i

­i µ surface §iNi = neck i

* When the necks are small

* When the necks are small

** See supplemental document

E(!1; : : : ; !N) =X

i

X

j>i

E(!i; !j)

m£ k

…

k£ k

m£ k

…

=k£ k

m£ km£ k

… …

=k£ k

m£ km£ k

… …

Untangled vector fields

Compute signature for each point

K-means

Find AKVFs

Untangle

Shape Diameter [Shapira et al. 2008]

Intrinsic Primitives Randomized Cuts [Golovinskiy et al. 2008]

Randomized Cuts [Golovinskiy et al. 2008]

Shape Diameter [Shapira et al. 2008]

Intrinsic Primitives

1. Segmentation into intrinsically symmetric parts

2. KVFs of a composite come from KVFs of its parts

3. Untangle KVFs for better localization