Numerical geometry of nonrigid shapes A journey to

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

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

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

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

Numerical geometry of non-rigid shapes A journey to non-rigid world 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. 5

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

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

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

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

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 10

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

Numerical geometry of non-rigid shapes A journey to non-rigid world How to add two dogs? 1 2 + = 1 2 CALCULUS OF SHAPES 12

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 13

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

15 Numerical geometry of non-rigid shapes A journey to non-rigid world Face caricaturization Extrapolation -0. 5 Interpolation 0 A. Bronstein, M. Bronstein, R. Kimmel, IEEE TVCG, 2006 0. 5 Extrapolation 1 1. 5

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

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

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

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

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

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

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