Computing and Processing Correspondences with Functional Maps
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1 Computing and Processing Correspondences with Functional Maps Maks Ovsjanikov 1 Etienne Corman 2 Michael Bronstein 3,4,5 Emanuele Rodolà 3 Mirela Ben-Chen 6 Leonidas Guibas 7 Frédéric Chazal 8 Alexander Bronstein 6,3,4 1 Ecole Polytechnique 3 USI Lugano 4 Tel Aviv University 5 Intel 2 CMU 6 Technion 7 Stanford University 8 INRIA SIGGRAPH Course, Los Angeles, 30 July /52
2 Functional Maps by Simultaneous Diagonalization of Laplacians 2/52
3 Choice of the basis Functional correspondence matrix C expressed in the Laplacian eigenbases Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
4 Choice of the basis Functional correspondence matrix C expressed in the Laplacian eigenbases Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
5 Problem with Laplacian eigenbases M φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 N φ N 2 φ N 3 φ N 4 φ N 5 φ N 6 Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
6 Problem with Laplacian eigenbases M φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 N φ N 2 φ N 3 φ N 4 φ N 5 φ N 6 Isometric manifolds with simple spectrum: sign ambiguity T F φ M i = ±φ N i Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
7 Problem with Laplacian eigenbases M φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 N φ N 2 φ N 3 φ N 4 φ N 5 φ N 6 Isometric manifolds with simple spectrum: sign ambiguity T F φ M i = ±φ N i General spectrum: ambiguous rotation of eigenspace Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
8 Problem with Laplacian eigenbases M φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 N φ N 2 φ N 3 φ N 4 φ N 5 φ N 6 Isometric manifolds with simple spectrum: sign ambiguity T F φ M i = ±φ N i General spectrum: ambiguous rotation of eigenspace Non-isometric manifolds: eigenvectors can differ dramatically in order and form Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
9 Problem with Laplacian eigenbases M φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 N φ N 2 φ N 3 φ N 4 φ N 5 φ N 6 Isometric manifolds with simple spectrum: sign ambiguity T F φ M i = ±φ N i General spectrum: ambiguous rotation of eigenspace Non-isometric manifolds: eigenvectors can differ dramatically in order and form Incompatibilities tend to increase with frequency Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
10 Coupled bases M a 1 +a 2 + +a k f φ M 1 φ M 2 φ M k T F N b 1 + b 2 + +b k g φ N 1 φ N 2 φ N k Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
11 Coupled bases M a 1 +a 2 + +a k f T F φ M 1 φ M 2 φ M k a b N b 1 + b 2 + +b k g φ N 1 φ N 2 φ N k Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
12 Coupled bases M â 1 +â 2 + +â k f ˆφ M 1 ˆφ M 2 ˆφ M k T F â ˆb N ˆb 1 + ˆb 2 + +ˆb k g ˆφ N 1 ˆφ N 2 ˆφ N k Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
13 Coupled bases M â 1 +â 2 + +â k f ˆφ M 1 ˆφ M 2 ˆφ M k T F Ĉ I N ˆb 1 + ˆb 2 + +ˆb k g ˆφ N 1 ˆφ N 2 ˆφ N k Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
14 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
15 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
16 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: δ ij = ˆφ M i, ˆφ M j L2 (M) = k l,m=1 p li p mj φ M l,φ M m L2 (M) Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
17 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: δ ij = ˆφ M i, ˆφ M j L2 (M) = p li p lj k l=1 Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
18 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: δ ij = ˆφ M i, ˆφ M j L2 (M) = p li p lj = (P P) ij k l=1 Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
19 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: P P = I and Q Q = I Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
20 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: P P = I and Q Q = I Approximate eigenbasis: approximately diagonalizes the Laplacian ˆφ M i, ˆφ M j L2 (M) = k l,m=1 p li p mj φ M l, φ M m L2 (M) Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
21 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: P P = I and Q Q = I Approximate eigenbasis: approximately diagonalizes the Laplacian ˆφ M i, ˆφ M j L2 (M) = k l,m=1 p li p mj λ m φ M l,φ M m L2 (M) Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
22 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: P P = I and Q Q = I Approximate eigenbasis: approximately diagonalizes the Laplacian ˆφ M i, ˆφ M j L2 (M) = p li p lj λ l k l=1 Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
23 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: P P = I and Q Q = I Approximate eigenbasis: approximately diagonalizes the Laplacian ˆφ M i, ˆφ M j L2 (M) = p li p lj λ l = (P Λ M,k P) ij k l=1 Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
24 Coupled bases Find a new pair of approximate orthonormal eigenbases ˆφ M i = p ji φ M j k ˆφ N i = q ji φ N j k j=1 j=1 parametrized by k k matrices P = (p ij ) and Q = (q ij ) i = 1,...,k Coupling: P A Q B Orthonormality: P P = I and Q Q = I Approximate eigenbasis: approximately diagonalizes the Laplacian ˆφ M i, ˆφ M j L2 (M) = p li p lj = (P Λ M,k P) ij 0, i j k l=1 Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
25 Joint diagonalization problem min P,Q off(p Λ M,k P)+off(Q Λ N,k Q)+µ P A Q B s.t. P P = I Q Q = I Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
26 Joint diagonalization problem min P,Q off(p Λ M,k P)+off(Q Λ N,k Q)+µ P A Q B s.t. P P = I Q Q = I Off-diagonal elements penalty off(x) = i j x2 ij Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
27 Joint diagonalization problem min P,Q off(p Λ M,k P)+off(Q Λ N,k Q)+µ P A Q B s.t. P P = I Q Q = I Off-diagonal elements penalty off(x) = i j x2 ij Dirichlet energy off(x) = trace(x) for k > k Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
28 Joint diagonalization problem min P,Q off(p Λ M,k P)+off(Q Λ N,k Q)+µ QP A B 2 F s.t. P P = I Q Q = I Off-diagonal elements penalty off(x) = i j x2 ij Dirichlet energy off(x) = trace(x) for k > k If Frobenius norm is used and k = k, due to rotation invariance C = QP is the functional correspondence matrix Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
29 Joint diagonalization problem min P,Q off(p Λ M,k P)+off(Q Λ N,k Q)+µ P A Q B 2,1 s.t. P P = I Q Q = I Off-diagonal elements penalty off(x) = i j x2 ij Dirichlet energy off(x) = trace(x) for k > k If Frobenius norm is used and k = k, due to rotation invariance C = QP is the functional correspondence matrix Robust norm X 2,1 = j x j 2 allows coping with outliers Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
30 Example of joint diagonalization Mesh with 8.5K vertices Mesh with 850 vertices Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
31 Example of joint diagonalization Mesh with 8.5K vertices Point cloud with 850 vertices Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
32 Choice of the basis Functional correspondence matrix C expressed in standard Laplacian eigenbases Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
33 Choice of the basis Functional correspondence matrix C expressed in coupled approximate eigenbases Kovnatsky, Bronstein 2, Glashoff, Kimmel /52
34 Multiple shapes C ij A i A j M j M i M p M 1 M 2 Kovnatsky, Bronstein 2, Glashoff, Kimmel 2013; Kovnatsky, Glashoff, Bronstein /52
35 Multiple shapes P i A i P j A j M j M i M p M 1 M 2 Kovnatsky, Bronstein 2, Glashoff, Kimmel 2013; Kovnatsky, Glashoff, Bronstein /52
36 Multiple shapes P i A i P j A j P j P i M j P p M i M p P 1 P 2 M 1 M 2 Kovnatsky, Bronstein 2, Glashoff, Kimmel 2013; Kovnatsky, Glashoff, Bronstein /52
37 Multiple shapes min P 1,...,P p p trace(p i Λ Mi P i )+µ P i A i P j A j i j i=1 s.t. P i P i = I Synchronization problem Matrices P 1,...,P p orthogonally align the p eigenbases Kovnatsky, Bronstein 2, Glashoff, Kimmel 2013; Kovnatsky, Glashoff, Bronstein /52
38 Computing Functional Maps with Manifold Optimization 13/52
39 min P trace(p ΛP)+µ PA B s.t. P P = I 14/52
40 min P trace(p ΛP)+µ PA B s.t. P P = I Optimization on the Stiefel manifold of orthogonal matrices 14/52
41 Manifold optimization toy example: eigenvalue problem min x R x Ax s.t. x x = 1 3 Minimization of a quadratic function on the sphere 15/52
42 Manifold optimization toy example: eigenvalue problem min x S(3,1) x Ax Minimization of a quadratic function on the sphere 15/52
43 Optimization on the manifold: main idea X (k) X (k+1) S Absil et al /52
44 Optimization on the manifold: main idea f(x (k) ) X (k) P X (k) S f(x (k) ) T X (k)s S Absil et al /52
45 Optimization on the manifold: main idea f(x (k) ) X (k) P X (k) α (k) S f(x (k) ) T X (k)s S Absil et al /52
46 Optimization on the manifold: main idea f(x (k) ) X (k) P X (k) α (k) S f(x (k) ) T X (k)s R X (k) X (k+1) S Absil et al /52
47 Optimization on the manifold repeat Compute extrinsic gradient f(x (k) ) Projection: S f(x (k) ) = P X (k)( f(x (k) )) Compute step size α (k) along the descent direction S f(x (k) ) Retraction: X (k+1) = R X (k)( α (k) S f(x (k) )) k k +1 until convergence; Absil et al. 2009; Boumal et al /52
48 Optimization on the manifold repeat Compute extrinsic gradient f(x (k) ) Projection: S f(x (k) ) = P X (k)( f(x (k) )) Compute step size α (k) along the descent direction S f(x (k) ) Retraction: X (k+1) = R X (k)( α (k) S f(x (k) )) k k +1 until convergence; Projection P and retraction R operators are manifold-dependent Absil et al. 2009; Boumal et al /52
49 Optimization on the manifold repeat Compute extrinsic gradient f(x (k) ) Projection: S f(x (k) ) = P X (k)( f(x (k) )) Compute step size α (k) along the descent direction S f(x (k) ) Retraction: X (k+1) = R X (k)( α (k) S f(x (k) )) k k +1 until convergence; Projection P and retraction R operators are manifold-dependent Typically expressed in closed form Absil et al. 2009; Boumal et al /52
50 Optimization on the manifold repeat Compute extrinsic gradient f(x (k) ) Projection: S f(x (k) ) = P X (k)( f(x (k) )) Compute step size α (k) along the descent direction S f(x (k) ) Retraction: X (k+1) = R X (k)( α (k) S f(x (k) )) k k +1 until convergence; Projection P and retraction R operators are manifold-dependent Typically expressed in closed form Black box : need to provide only f(x) and gradient f(x) Absil et al. 2009; Boumal et al /52
51 Partial Functional Maps 18/52
52 Partial Laplacian eigenvectors ζ 2 ζ 3 ζ 4 ζ 5 ζ 6 ζ 7 ζ 8 ζ 9 ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 ψ 8 ψ 9 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 Laplacian eigenvectors of a shape with missing parts (Neumann boundary conditions) Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
53 Partial Laplacian eigenvectors Functional correspondence matrix C Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
54 Perturbation analysis: intuition M M φ1 φ2 φ3 φ 1 φ 2 φ 3 λ 1 λ2 λ3 M λ 1 λ 2 λ 3 M M φ 1 φ 2 φ 3 M φ 1 φ2 φ3 λ 1 λ 1 λ 2 λ 3 λ2 λ3 Ignoring boundary interaction: disjoint parts (block-diagonal matrix) Eigenvectors = Mixture of eigenvectors of the parts Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
55 Perturbation analysis: eigenvalues M r k N eigenvalue number Slope r k M N (depends on the area of the cut) Consistent with Weyl s law Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
56 Perturbation analysis: details M M M M +td M te te M+tD M M Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
57 Perturbation analysis: boundary interaction strength 20 Value of f 10 Eigenvector perturbation depends on length and position of the boundary Perturbation strength c f(m)dm, where M n ( ) 2 φi (m)φ j (m) f(m) = i,j=1 j i λ i λ j Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
58 Partial functional maps Model shape M Query shape N Part M M isometric to N Data f 1,...,f q L 2 (N) g 1,...,g q L 2 (M) Part M T F Query N Partial functional map (T F f i )(m) g i (m), m M Model M Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
59 Partial functional maps Model shape M Query shape N Part M M isometric to N Data f 1,...,f q L 2 (N) g 1,...,g q L 2 (M) Part v T F Query N Partial functional map T F f i g i v, v : M [0,1] Model M Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
60 Partial functional maps Model shape M Query shape N Part M M isometric to N Data f 1,...,f q L 2 (N) g 1,...,g q L 2 (M) Part v C Query N Partial functional map CA B(v), v : M [0,1] A = ( φ N ) i,f j L 2 (N) B(v) = ( φ M ) i,g j v L2 (M) Model M Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
61 Partial functional maps Model shape M Query shape N Part M M isometric to N Data f 1,...,f q L 2 (N) g 1,...,g q L 2 (M) Part v C Query N Partial functional map CA B(v), v : M [0,1] A = ( φ N ) i,f j L 2 (N) B(v) = ( φ M ) i,g j v L2 (M) Model M Optimization problem w.r.t. correspondence C and part v min C,v CA B(v) 2,1 +ρ corr (C)+ρ part (v) Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
62 Partial functional maps min CA B(v) 2,1 +ρ corr (C)+ρ part (v) C,v Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
63 Partial functional maps min CA B(v) 2,1 +ρ corr (C)+ρ part (v) C,v Part regularization Area preservation v(m)dx N M Spatial regularity = small boundary length (Mumford-Shah) Rodolà, Cosmo, Bronstein, Torsello, Cremers 2016; Bronstein /52
64 Partial functional maps min CA B(v) 2,1 +ρ corr (C)+ρ part (v) C,v Part regularization Area preservation v(m)dx N M Spatial regularity = small boundary length (Mumford-Shah) Correspondence regularization Slanted diagonal structure Approximate ortho-projection (C C) i j 0 rank(c) r Rodolà, Cosmo, Bronstein, Torsello, Cremers 2016; Bronstein /52
65 Alternating minimization C-step: fix v, solve for correspondence C v-step: fix C, solve for part v min C CA B(v ) 2,1 +ρ corr (C) min v C A B(v) 2,1 +ρ part (v) Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
66 Alternating minimization C-step: fix v, solve for correspondence C v-step: fix C, solve for part v min C CA B(v ) 2,1 +ρ corr (C) min v C A B(v) 2,1 +ρ part (v) Iteration Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
67 Example of convergence Time (sec.) C-step v-step Energy Iteration Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
68 Examples of partial functional maps Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
69 Examples of partial functional maps Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
70 Examples of partial functional maps Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
71 Examples of partial functional maps Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
72 Partial functional maps vs Functional maps 100 % Correspondences PFM Func. maps Geodesic error Correspondence performance for different basis size k Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
73 Partial correspondence performance 100 Cuts Holes % Correspondences Geodesic Error Geodesic Error PFM RF IM EN GT SHREC 16 Partial Matching benchmark Rodolà et al. 2016; Methods: Rodolà, Cosmo, Bronstein, Torsello, Cremers 2016 (PFM); Sahillioğlu, Yemez 2012 (IM); Rodolà, Bronstein, Albarelli, Bergamasco, Torsello 2012 (GT); Rodolà et al (EN); Rodolà et al (RF) 31/52
74 Partial correspondence performance 1 Cuts Holes Mean geodesic error Partiality (%) Partiality (%) PFM RF IM EN GT SHREC 16 Partial Matching benchmark Rodolà et al. 2016; Methods: Rodolà, Cosmo, Bronstein, Torsello, Cremers 2016 (PFM); Sahillioğlu, Yemez 2012 (IM); Rodolà, Bronstein, Albarelli, Bergamasco, Torsello 2012 (GT); Rodolà et al (EN); Rodolà et al (RF) 32/52
75 Partial correspondence (part-to-full) Rodolà, Cosmo, Bronstein, Torsello, Cremers /52
76 Partial correspondence (part-to-part) Litany, Rodolà, Bronstein 2, Cremers /52
77 Key observation M N N C NN slant N N M C MM slant M M Litany, Rodolà, Bronstein 2, Cremers /52
78 Key observation M N N M C NM = C MM C NM C NN slant N M N M Litany, Rodolà, Bronstein 2, Cremers /52
79 Key observation M N N M Litany, Rodolà, Bronstein 2, Cremers 2016 C NM = C MM C NM C NN slant N M N M = M N 34/52
80 Partial correspondence (part-to-part) Litany, Rodolà, Bronstein 2, Cremers /52
81 Non-rigid puzzle (multi-part) Litany, Rodolà, Bronstein 2, Cremers /52
82 Litany, Bronstein
83 Non-rigid puzzles problem formulation Input Model M Parts N 1,...,N p Output Segmentation M i M Located parts N i N i Clutter Ni c Missing parts M 0 M 1 M 2 M 0 T F1 N N c 1 T 2 F2 N1 c N 2 Part N 1 Correspondences T Fi Model M Part N 2 Litany, Rodolà, Bronstein 2, Cremers /52
84 Non-rigid puzzles problem formulation Input Model M Parts N 1,...,N p Output Segmentation u i :M [0,1] Located parts v i :N i [0,1] Clutter 1 v i Missing parts u 0 u 1 u 2 u 0 C 1 C 2 v 2 v 1 Part N 1 Correspondences C i Model M Part N 2 Litany, Rodolà, Bronstein 2, Cremers /52
85 Non-rigid puzzles problem formulation min C i,u i,v i s.t. p p p C i A i (v i ) B(u i ) 2,1 + ρ part (u i,v i )+ ρ corr (C i ) i=1 p u i = 1 i=0 i=0 i=1 Litany, Rodolà, Bronstein 2, Cremers /52
86 Convergence example Outer iteration 1 Litany, Rodolà, Bronstein 2, Cremers /52
87 Convergence example Outer iteration 2 Litany, Rodolà, Bronstein 2, Cremers /52
88 Convergence example Outer iteration 3 Litany, Rodolà, Bronstein 2, Cremers /52
89 Convergence example Time (sec) Iteration number Litany, Rodolà, Bronstein 2, Cremers /52
90 Example: Perfect puzzle Model/Part Synthetic (TOSCA) Transformation Isometric Clutter No Missing part No Data term Dense (SHOT) Litany, Rodolà, Bronstein 2, Cremers /52
91 Example: Perfect puzzle Model/Part Synthetic (TOSCA) Transformation Isometric Clutter No Missing part No Data term Dense (SHOT) Segmentation Litany, Rodolà, Bronstein 2, Cremers /52
92 Example: Perfect puzzle Model/Part Synthetic (TOSCA) Transformation Isometric Clutter No Missing part No Data term Dense (SHOT) Correspondence Litany, Rodolà, Bronstein 2, Cremers /52
93 Example: Overlapping parts Model/Part Synthetic (FAUST) Transformation Near-isometric Clutter Yes (overlap) Missing part No Data term Dense (SHOT) Segmentation Litany, Rodolà, Bronstein 2, Cremers /52
94 Example: Overlapping parts Model/Part Synthetic (FAUST) Transformation Near-isometric Clutter Yes (overlap) Missing part No Data term Dense (SHOT) Correspondence Litany, Rodolà, Bronstein 2, Cremers /52
95 Example: Missing parts Model/Part Synthetic (TOSCA) Transformation Isometric Clutter Yes (extra part) Missing part Yes Data term Dense (SHOT) Litany, Rodolà, Bronstein 2, Cremers /52
96 Example: Missing parts Model/Part Synthetic (TOSCA) Transformation Isometric Clutter Yes (extra part) Missing part Yes Data term Dense (SHOT) Litany, Rodolà, Bronstein 2, Cremers 2016 Segmentation 43/52
97 Example: Missing parts Model/Part Synthetic (TOSCA) Transformation Isometric Clutter Yes (extra part) Missing part Yes Data term Dense (SHOT) Litany, Rodolà, Bronstein 2, Cremers 2016 Correspondence 43/52
98 Partial functional correspondence with spatial part model M φ M 1 φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 T F N N φ N 1 φ N 2 φ N 3 φ N 4 φ N 5 φ N 6 Slanted diagonal: T F φ M i,v φ N j L 2 (N) ±δ i,πj Complicated alternating optimization w.r.t. v and C Explicit spatial model v of the part O(n) complexity! Litany, Rodolà, Bronstein π j j N M 44/52
99 Spectral partial functional correspondence M φ M 1 φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 T F N ˆφ N 1 ˆφ N 2 ˆφ N 3 ˆφ N 4 ˆφ N 5 ˆφ N 6 Find a new basis {ˆφ N i }k i=1 such that T Fφ M i, ˆφ N j L 2 (N) δ ij Litany, Rodolà, Bronstein /52
100 Spectral partial functional correspondence M φ M 1 φ M 2 φ M 3 φ M 4 φ M 5 φ M 6 T F N ˆφ N 1 ˆφ N 2 ˆφ N 3 ˆφ N 4 ˆφ N 5 ˆφ N 6 Find a new basis {ˆφ N i }k i=1 such that T Fφ M i, k l=1 q ljφ N l L 2 (N) δ ij New basis functions {ˆφ N i }k i=1 are localized on N Optimization over coefficients Q = (q ij ) O(k 2 ) complexity! Litany, Rodolà, Bronstein /52
101 Spectral partial functional correspondence A k q Π k k B k q Litany, Rodolà, Bronstein /52
102 Spectral partial functional correspondence A r q Πr k B k q Π is k r partial permutation with elements (π i,i) = ±1 and r k M N Litany, Rodolà, Bronstein /52
103 Spectral partial functional correspondence A r q Πr k B k q Π is k r partial permutation with elements (π i,i) = ±1 and r k M N Relax Π Q s.t. Q Q = I (k r ortho-projection) min Q trace(q Λ N,k Q)+µ A r Q B k 2,1 s.t. Q Q = I Litany, Rodolà, Bronstein ; Kovnatsky, Glashoff, Bronstein 2, Kimmel 2013 (Joint diag) 46/52
104 Spectral partial functional correspondence A r q Πr k B k q Π is k r partial permutation with elements (π i,i) = ±1 and r k M N Relax Π Q s.t. Q Q = I (k r ortho-projection) min Q trace(q Λ N,k Q)+µ A r Q B k 2,1 s.t. Q Q = I Optimization on the Stiefel manifold with k 2 variables Litany, Rodolà, Bronstein ; Kovnatsky, Glashoff, Bronstein 2, Kimmel 2013 (Joint diag) 46/52
105 Spectral partial functional correspondence A r q Πr k B k q Π is k r partial permutation with elements (π i,i) = ±1 and r k M N Relax Π Q s.t. Q Q = I (k r ortho-projection) min Q trace(q Λ N,k Q)+µ A r Q B k 2,1 s.t. Q Q = I Non-smooth optimization on the Stiefel manifold with k 2 variables Litany, Rodolà, Bronstein ; Kovnatsky, Glashoff, Bronstein 2, Kimmel 2013 (Joint diag); Kovnatsky, Glashoff, Bronstein 2016 (MADMM) 46/52
106 Spectral partial functional correspondence A r q Πr k B k q Π is k r partial permutation with elements (π i,i) = ±1 and r k M N Relax Π Q s.t. Q Q = I (k r ortho-projection) min Q trace(q Λ N,k Q)+µ A r Q B k 2,1 s.t. Q Q = I Non-smooth optimization on the Stiefel manifold with k 2 variables Non-rigid alignment of eigenfunctions Litany, Rodolà, Bronstein ; Kovnatsky, Glashoff, Bronstein 2, Kimmel 2013 (Joint diag); Kovnatsky, Glashoff, Bronstein 2016 (MADMM) 46/52
107 Geometric interpretation Full shape N φ M 2,φM 3 and φ N 2,φN 3 Laplacian eigenbasis Part M φ M 2,φM 3 and ˆφ N 2, ˆφ N 3 New basis Litany, Rodolà, Bronstein /52
108 Convergence example Initialization Litany, Rodolà, Bronstein /52
109 Increasing partiality SPFM PFM rank = 36 rank = 23 rank = 7 Litany, Rodolà, Bronstein /52
110 SHREC 16 Partiality 100 cuts holes % Correspondences Geodesic Error Geodesic Error SPFM JAD RF PFM IM EN GT SHREC 16 Partial Matching benchmark: Rodolà et al. 2016; Methods: Unpublished work (SPFM); Rodolà, Cosmo, Bronstein, Torsello, Cremers 2016 (PFM); Sahillioğlu, Yemez 2012 (IM); Rodolà, Bronstein, Albarelli, Bergamasco, Torsello 2012 (GT); Rodolà et al (EN); Rodolà et al (RF) 50/52
111 Runtime Mean time per iteration (sec) SPFM PFM Number of vertices ( 10 4 ) Litany, Rodolà, Bronstein /52
112 Correspondence examples: topological noise Litany, Rodolà, Bronstein ; data: Bogo et al (FAUST) 52/52
113 Correspondence examples: topological noise Litany, Rodolà, Bronstein ; data: Bogo et al (FAUST) 52/52
114 Correspondence examples: topological noise Litany, Rodolà, Bronstein ; data: Rodola et al (SHREC) 52/52
115 Correspondence examples: topological noise Litany, Rodolà, Bronstein ; data: Rodola et al (SHREC) 52/52
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