1 Numerical geometry of non-rigid shapes A journey to non-rigid world objects Invariant correspondence and shape synthesis non-rigid Alexander Bronstein.

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Presentation transcript:

1 Numerical geometry of non-rigid shapes A journey to non-rigid world objects Invariant correspondence and shape synthesis non-rigid Alexander Bronstein Michael Bronstein Numerical geometry of

2 Numerical geometry of non-rigid shapes A journey to non-rigid world Analysis and synthesis Elephant image: courtesy M. Kilian and H. Pottmann SYNTHESISANALYSIS

3 Numerical geometry of non-rigid shapes A journey to non-rigid world “Natural” correspondence?

4 Numerical geometry of non-rigid shapes A journey to non-rigid world Correspondence accurate ‘‘ ‘‘ makes sense ‘‘ ‘‘ beautiful ‘‘ ‘‘ Geometric Semantic Aesthetic

5 Numerical geometry of non-rigid shapes A journey to non-rigid world Correspondence Correspondence is not a well-defined problem!  Chances to solve it with geometric tools are slim.  If objects are sufficiently similar, we have better chances. Correspondence between deformations of the same object.

6 Numerical geometry of non-rigid shapes A journey to non-rigid world Minimum distortion correspondence A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

7 Numerical geometry of non-rigid shapes A journey to non-rigid world Numerical examples A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

8 Numerical geometry of non-rigid shapes A journey to non-rigid world Partial correspondence

9 Numerical geometry of non-rigid shapes A journey to non-rigid world TIME ReferenceTransferred texture Texture transfer A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

10 Numerical geometry of non-rigid shapes A journey to non-rigid world Texture substitution I’m Alice.I’m Bob. I’m Alice’s texture on Bob’s geometry A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

11 Numerical geometry of non-rigid shapes A journey to non-rigid world Invariant correspondence MATLAB ® intermezzo

12 Numerical geometry of non-rigid shapes A journey to non-rigid world = How to add two dogs? C A L C U L U S O F S H A P E S

13 Numerical geometry of non-rigid shapes A journey to non-rigid world Affine calculus of shapes ? A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

14 Numerical geometry of non-rigid shapes A journey to non-rigid world Metamorphing 100% Alice 100% Bob 75% Alice 25% Bob 50% Alice 50% Bob 75% Alice 50% Bob A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

15 Numerical geometry of non-rigid shapes A journey to non-rigid world Face caricaturization Interpolation Extrapolation A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006

16 Numerical geometry of non-rigid shapes A journey to non-rigid world Affine calculus of shapes

17 Numerical geometry of non-rigid shapes A journey to non-rigid world What happened? SHAPE SPACE IS NON-EUCLIDEAN!

18 Numerical geometry of non-rigid shapes A journey to non-rigid world Shape space Shape space is an abstract manifold Deformation fields of a shape are vectors in tangent space Our affine calculus is valid only locally Global affine calculus can be constructed by defining trajectories confined to the manifold Addition Combination

19 Numerical geometry of non-rigid shapes A journey to non-rigid world Choice of trajectory Equip tangent space with an inner product Riemannian metric on Select to be a minimal geodesic Addition: initial value problem Combination: boundary value problem

20 Numerical geometry of non-rigid shapes A journey to non-rigid world Choice of metric Deformation field of is called Killing field if for every Infinitesimal displacement by Killing field is metric preserving and are isometric Congruence is always a Killing field Non-trivial Killing field may not exist

21 Numerical geometry of non-rigid shapes A journey to non-rigid world Choice of metric Inner product on Induces norm measures deviation of from Killing field – defined modulo congruence Add stiffening term

22 Numerical geometry of non-rigid shapes A journey to non-rigid world Minimum-distortion trajectory Geodesic trajectory Shapes along are as isometric as possible to Guaranteeing no self-intersections is an open problem