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