rock, paper, and scissors joint extrinsic and intrinsic similarity of non-rigid shapes
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Rock, paper, and scissors Joint extrinsic and intrinsic similarity of non-rigid shapes. Alex Bronstein, Michael Bronstein, Ron Kimmel. Department of Computer Science Technion – Israel Institute of Technology. Extrinsic vs intrinsic similarity. - PowerPoint PPT PresentationTRANSCRIPT
1Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Rock, paper, and scissors
Joint extrinsic and intrinsic similarity of non-rigid shapes
Alex Bronstein, Michael Bronstein, Ron Kimmel
Department of Computer ScienceTechnion – Israel Institute of Technology
2Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Extrinsic vs intrinsic similarity
Intrinsic similarity
Are the shapes congruent? Do the shapes have the
same
metric structure?
Extrinsic similarity
Rock, paper, and scissors: is the hand similar to a rock? Is it similar to
another posture of a hand?
The answer depends on the definition of similarity.
3Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Extrinsic similarity
Can be expressed as a distance between two shapes and
Find a rigid motion bringing the shapes into best alignment
Misalignment is quantified using the Hausdorff distance
or some of its variants
Computed using ICP algorithms
4Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Extrinsic similarity – limitations
Extrinsically similar Extrinsically dissimilar
Suitable for nearly rigid shapes Unsuitable for non-rigid
shapes
5Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Intrinsic similarity
Compare the intrinsic geometries of two shapes
Intrinsic geometry is expressed in terms of geodesic distances
Geodesic distances are computed using Dijkstra’s shortest path
algorithm or fast marching
Euclidean distance
Geodesic distance
6Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Intrinsic similarity – canonical forms
Embed intrinsic geometries of and into a common metric space
Minimum-distortion embeddings and computed using
multidimensional scaling (MDS) algorithms
Compare the images and as rigid shapes
A. Elad, R. Kimmel, CVPR 2001
7Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Intrinsic similarity – GMDS
Find the minimum distortion embedding of one shape into the other
The minimum distortion is the measure of intrinsic dissimilarity
Computed using the generalized MDS
BBK, PNAS 2006
8Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Intrinsic similarity – limitations
Intrinsically dissimilar
Intrinsically similar
Suitable for near-isometric
shape deformations
Unsuitable for deformations
modifying shape topology
9Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Extrinsically dissimilarIntrinsically similar
Extrinsically similarIntrinsically dissimilar
Extrinsically dissimilarIntrinsically dissimilar
THIS IS THE SAME SHAPE!
Desired result:
10Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Joint extrinsic and intrinsic similarity
Combine intrinsic and extrinsic similarities into a single criterion
Find a deformation of whose intrinsic geometry is similar to
and extrinsic geometry is more similar to
defines the relative importance of intrinsic and extrinsic criteria
is a collection of optimal tradeoffs between intrinsic and
extrinsic criteria
Can be formalized using the notion of Pareto optimality
11Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Intrinsic similarity
Ext
rinsi
c si
mila
rity
12Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Computation of joint similarity
Hybridization of ICP and GMDS in L2 formulation for robustness
Fix correspondence between and for intrinsic similarity
where is precomputed and
are computed at each iteration
Closest-point distance for extrinsic similarity
where are the closest points to in
More details in the paper
13Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – dataset
= topology changeData: tosca.cs.technion.ac.il
14Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – tradeoff curves
Dissi
mila
r
Simila
r
15Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – intrinsic similarity
= topology-preserving no topology changes
16Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – intrinsic similarity
= topology change= topology-preserving
17Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – extrinsic similarity
= topology change= topology-preserving
18Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – joint similarity
= topology change= topology-preserving
19Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – ROC curves
20Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Numerical example – shape morphing
Stronger intrinsic similarity (smaller λ)
Stronger extrinsicsimilarity (larger λ)
21Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
Conclusion
Extrinsic similarity is insensitive to topology changes, but sensitive to
non-rigid deformations
Intrinsic similarity is insensitive to nearly-isometric non-rigid
deformations, but sensitive to topology changes
Joint similarity is insensitive to both non-rigid deformations and topology
changes
Can be used to produce near-isometric morphs
22Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
References
A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel, Analysis of two-dimensional non-rigid shapes, IJCV, to appear.
A. M. Bronstein, M. M. Bronstein, R. Kimmel, Rock, Paper, and Scissors: extrinsic vs. intrinsic similarity of non-rigid shapes, Proc. ICCV, (2007).
I. Eckstein, J. P. Pons, Y. Tong, C. C. J. Kuo, and M. Desbrun, Generalized surface flows for mesh processing, Proc. SGP, (2007).
M. Kilian, N. J. Mitra, and H. Pottmann, Geometric modeling in shape space, Proc. SIGGRAPH, vol. 26, (2007).
A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel, Paretian similarity for partial comparison of non-rigid objects, Proc. SSVM, pp. 264-275, 2007.
A. M. Bronstein, M. M. Bronstein, R. Kimmel, Calculus of non-rigid surfaces for geometry and texture manipulation, IEEE TVCG, Vol. 13/5, pp. 902-913, (2007).
A. M. Bronstein, M. M. Bronstein, R. Kimmel, Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching, PNAS, Vol. 103/5, pp. 1168-1172, (2006).
23Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
References
F. Mémoli and G. Sapiro, A theoretical and computational framework for isometry invariant recognition of point cloud data, Foundations of Computational Mathematics 5 (2005), 313-346.
N. J. Mitra, N. Gelfand, H. Pottmann, and L. Guibas, Registration of point cloud data from a geometric optimization perspective, Proc. SGP, (2004), pp. 23-32.
A. Elad, R. Kimmel, On bending invariant signatures for surfaces, Trans. PAMI 25 (2003), no. 10, 1285-1295.
P. J. Besl and N. D. McKay, A method for registration of 3D shapes, Trans. PAMI 14 (1992), 239-256.
Y. Chen and G. Medioni, Object modeling by registration of multiple range images, Proc. Conf. Robotics and Automation, (1991).
E. L. Schwartz, A. Shaw, and E. Wolfson, A numerical solution to the generalized mapmaker's problem: flattening nonconvex polyhedral surfaces, Trans. PAMI 11 (1989), 1005-1008.
24Bronstein, Bronstein, and Kimmel Joint extrinsic and intrinsic similarity of non-rigid shapes
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