Kernels for Persistence Diagrams with Applications in Geometry Processing

Transcription

1 ACAT meeting July 2015 Kernels for Persistence Diagrams with Applications in Geometry Processing Steve Oudot Inria, Geometrica group M. Carrière, S. O., M. Ovsjanikov. Stable Topological Signatures for Points on 3D Shapes. Proc. Sympos. on Geometry Processing, 2015.

2 Persistence Diagrams as Signatures It has been known since a number of years now that persistence diagrams can finite metric space / basepoint Filtration / zigzag persistence diagram dimensional homology generators background reading: 1-dimensional homology generators M. d Amico, P. Frosini, C. Landi. Using matching distance in size theory: A survey. International Journal of Imaging Systems and Technology, 16(5), , S. O.. Persistence Theory: from Quiver Representations to Data Analysis, Chapters 7 and 8. AMS Mathematical Surveys and Monographs, November

3 Persistence Diagrams as Signatures It has been known since a number of years now that persistence diagrams can finite metric space / basepoint Filtration / zigzag persistence diagram Pros: 0-dimensional homology generators 1-dimensional homology generators What makes persistence diagrams a useful descriptor is, first of all, that it does c topological signatures carry complementary information More importantly, stability properties: PDs come d B (Dg with R(X), stability Dg R(Y properties, )) 2d GH(X, which Y ) are [Chazal pretty al. rare 09, in 13] the 1

4 Persistence Diagrams as Signatures It has been known since a number of years now that persistence diagrams can finite metric space / basepoint Filtration / zigzag persistence diagram Pros: 0-dimensional homology generators 1-dimensional homology generators What makes persistence diagrams a useful descriptor is, first of all, that it does c topological signatures carry complementary information More importantly, stability properties: PDs come d B (Dg with R(X), stability Dg R(Y properties, )) 2d GH(X, which Y ) are [Chazal pretty al. rare 09, in 13] the Cons: this implies the space that oflinear persistence classifiers diagrams cannotisbe not used a Hilbert PDspace have been mostly used in unsuper having signatures long construction are slow to times compute is fineand as long (more asimportantly) comparisonsto can compare be made fast, but here t 1

5 Persistence Diagrams as Signatures It has been known since a number of years now that persistence diagrams can finite metric space / basepoint Filtration / zigzag persistence diagram Pros: 0-dimensional homology generators 1-dimensional homology generators What makes persistence diagrams a useful descriptor is, first of all, that it does c topological signatures carry complementary information More importantly, stability properties: PDs come d B (Dg with R(X), stability Dg R(Y properties, )) 2d GH(X, which Y ) are [Chazal pretty al. rare 09, in 13] the Cons: this implies the space that oflinear persistence classifiers diagrams cannotisbe not used a Hilbert PDspace have been mostly used in unsuper define kernels on the space of diagrams having signatures long construction are slow to times compute is fineand as long (more asimportantly) comparisonsto can compare be made fast, but here t explicit mapping to feature space 1

6 Kernels for Persistence Diagrams View persistence diagrams as: landscapes [Bubenik 2012] empirical measures [Bauer et al. 2015] map to function space L 2 roots of polynomials [Di Fabio, Ferri 2015] metric spaces [Carrière, O., Ovsjanikov 2015] map to finite-dim. vector space 2

7 Feature Map for now I am finite ignoring metric space the diagonal, assuming that all the diagrams we are conside a c b 3

8 Feature Map for now I am finite ignoring metric space the diagonal, assuming that all the diagrams we are conside a 5 c 4 3 b φ 1 a b c distance matrix a b c

9 Feature Map for now I am finite ignoring metric space the diagonal, assuming that all the diagrams we are conside a 5 c 4 3 b φ 1 a b c distance matrix a b c φ distribution of distances 3

10 Feature Map for now I am finite ignoring metric space the diagonal, assuming that all the diagrams we are conside a 5 c 4 3 b φ 1 a b c distance matrix a b c φ 2 (5, 4, 3, 0, 0, ) 2 1 sorted sequence with finite support (shape context) φ distribution of distances 3

11 Feature Map Φ = φ 4 φ 3 φ 2 φ 1 for now I am finite ignoring metric space the diagonal, assuming that all the diagrams we are conside a 5 c 4 3 b φ 1 (5, 4, 3, 0,, 0) a b c distance matrix a b c finite-dimensional vector φ 2 φ 4 (5, 4, 3, 0, 0, ) 2 1 sorted sequence with finite support (shape context) φ distribution of distances 3

12 In order to talk about stability, we first need to specify the metrics that will be u for now I am finite ignoring metric space the diagonal, P (R 2 ) assuming that all the diagrams we are conside the diagrams (with same cardinality) are compare using the Wa a 5 c 4 3 b φ 1 Finally, the vectors are viewed as elements of the usual space R (5, 4, 3, 0,, 0) Stability a b c (R D, l ) a b c Φ = φ 4 φ 3 φ 2 φ 1 distance matrix finite-dimensional vector φ 2 φ 4 Sequences are elements of the sequence space l (5, 4, 3, 0, 0, ) l 2 1 sorted sequence with finite support (shape context) φ distribution of distances P (R) Distance distributions are vie 3

13 In order to talk about stability, we first need to specify the metrics that will be u for now I am finite ignoring metric space the diagonal, P (R 2 ) assuming that all the diagrams we are conside the diagrams (with same cardinality) are compare using the Wa a ε a c 5 ε c 4 3 ε b b φ 1 a b c a 0 4 5? b c (5 ± 2ε, 4 ± 2ε, 3 ± 2ε, 0, 0) distance matrix Finally, the vectors are viewed as elements of the usual space R (5, 4, 3, 0,, 0) Stability (R D, l ) Φ = φ 4 φ 3 φ 2 φ 1 + [ εaa ε ab ε ac ] ε ba ε bb ε bc ε ca ε cb ε cc εxy [ 2ε, +2ε] finite-dimensional vector φ 2 φ 4 Sequences are elements of the sequence space l (5, 4, 3, 0, 0, ) sorted sequence with finite support (shape context) l (5 ± 2ε, 4 ± 2ε, 3 ± 2ε, 0, 0, )? φ 3 2 2ε distribution of distances P (R) Distance distributions are vie 3

14 In order to talk about stability, we first need to specify the metrics that will be u for now I am finite ignoring metric space the diagonal, P (R 2 ) assuming that all the diagrams we are conside the diagrams (with same cardinality) are compare using the Wa a ε a c 5 ε c 4 3 ε b b φ 1 a b c a 0 4 5? b c (5 ± 2ε, 4 ± 2ε) (further truncation) distance matrix Finally, the vectors are viewed as elements of the usual space R (5, 4, 3, 0,, 0) Stability (R D, l ) Φ = φ 4 φ 3 φ 2 φ 1 + [ εaa ε ab ε ac ] ε ba ε bb ε bc ε ca ε cb ε cc εxy [ 2ε, +2ε] finite-dimensional vector φ 2 φ 4 Sequences are elements of the sequence space l (5, 4, 3, 0, 0, ) sorted sequence with finite support (shape context) l (5 ± 2ε, 4 ± 2ε, 3 ± 2ε, 0, 0, )? φ 3 2 2ε distribution of distances P (R) Distance distributions are vie 3

15 Adding the diagonal diagonal has infinite multiplicity useful for when point clouds have different cardinalities 4

16 Adding the diagonal diagonal has infinite multiplicity useful for when point clouds have different cardinalities some points may prefer the diagonal to other points to reduce the cost of the matching 4

17 Adding the diagonal diagonal has infinite multiplicity useful for when point clouds have different cardinalities some points may prefer the diagonal to other points to reduce the cost of the matching replace W (X, Y ) with { d B = inf m:x Y max sup p matched p m(p), sup p unmatched p p } 4

18 Adding the diagonal diagonal has infinite multiplicity useful for when point clouds have different cardinalities some points may prefer the diagonal to other points to reduce the cost of the matching replace W (X, Y ) with { d B = inf m:x Y max sup p matched p m(p), sup p unmatched p p } Problem: generates instability in distance matrix (d B << W ) Solution: change the distance matrix... 4

19 Adding the diagonal Φ = φ 4 φ 3 φ 2 φ 1 x 3 distance matrix x 1 x 1 x 2 x 2 x 3 φ 1 (7.2, 7.2, 5.6, 4) (7.2, 7.2, 5.6, 4, 4, 4, 0,, 0) finite-dimensional vector D ij = min{ x i x j, x i x i, x j x j } φ 2 φ 4 (7.2, 7.2, 5.6, 4, 4, 4, 0, ) sorted sequence (finite support) φ distribution of distances 4

20 Stability Theorem: For any finite persistence diagrams X, Y, for any feature space dimension D, Φ(X) Φ(Y ) 2d B (X, Y ) 5

21 Stability Theorem: For any finite persistence diagrams X, Y, for any feature space dimension D, Φ(X) Φ(Y ) 2d B (X, Y ) p 1, Φ(X) Φ(Y ) p 2D 1/p d B (X, Y ) 5

22 Stability Theorem: For any finite persistence diagrams X, Y, for any feature space dimension D, Φ(X) Φ(Y ) 2d B (X, Y ) p 1, Φ(X) Φ(Y ) p 2D 1/p d B (X, Y ) case p = useful for retrieval and NN-classifiers (fast proximity queries) case p = 2 useful for linear / kernel-based classifiers (scalar product) 5

23 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects 6

24 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects feature space: L 2 (N R) L 2 (R 2 ) feature space: L 2 (R 2 ) feature space: l p (R D, l p ) 6

25 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects feature space: L 2 (N R) L 2 (R 2 ) feature space: L 2 (R 2 ) feature space: l p (R D, l p ) feature map: explicit (combi. construction) feature map: explicit (closed form solution) feature map: explicit (combi. construction) 6

26 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects feature space: L 2 (N R) L 2 (R 2 ) feature space: L 2 (R 2 ) feature space: l p (R D, l p ) feature map: explicit (combi. construction) feature map: explicit (closed form solution) feature map: explicit (combi. construction) kernel(s): linear, Gaussian, etc. kernel(s): kσ kernel(s): linear Gaussian, etc. 6

27 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects feature space: L 2 (N R) L 2 (R 2 ) feature space: L 2 (R 2 ) feature space: l p (R D, l p ) feature map: explicit (combi. construction) feature map: explicit (closed form solution) feature map: explicit (combi. construction) kernel(s): linear, Gaussian, etc. kernel(s): kσ kernel(s): linear Gaussian, etc. complexity on n-points diagrams: complexity on n-points diagrams: complexity on n-points diagrams: this - feature is for all map: the landscapes, O(n 2 ) for just one it is O(n- log feature n) map: N/A (or discretize...) same - kernel: here: O(n this is 2 for ) all the landscapes, for just one - kernel: it is O(n) 2 ) - feature map: O(n 2 ) - kernel: O(D) 6

28 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects feature space: L 2 (N R) L 2 (R 2 ) feature space: L 2 (R 2 ) feature space: l p (R D, l p ) feature map: explicit (combi. construction) feature map: explicit (closed form solution) feature map: explicit (combi. construction) kernel(s): linear, Gaussian, etc. kernel(s): kσ kernel(s): linear Gaussian, etc. complexity on n-points diagrams: complexity on n-points diagrams: complexity on n-points diagrams: this - feature is for all map: the landscapes, O(n 2 ) for just one it is O(n- log feature n) map: N/A (or discretize...) same - kernel: here: O(n this is 2 for ) all the landscapes, for just one - kernel: it is O(n) 2 ) - feature map: O(n 2 ) - kernel: O(D) Be careful stability: that these stability O(d results B ) hide some differences, stability: in particular 2 ino(d the constants. 1 B ( )) Basically, results involving stability: p = have O(d absolute B ) constants bu p O(Pers d B ( )) p O(D 1/p d B ( )) 6

29 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects feature space: L 2 (N R) L 2 (R 2 ) feature space: L 2 (R 2 ) feature space: l p (R D, l p ) feature map: explicit (combi. construction) feature map: explicit (closed form solution) feature map: explicit (combi. construction) kernel(s): linear, Gaussian, etc. kernel(s): kσ kernel(s): linear Gaussian, etc. complexity on n-points diagrams: complexity on n-points diagrams: complexity on n-points diagrams: this - feature is for all map: the landscapes, O(n 2 ) for just one it is O(n- log feature n) map: N/A (or discretize...) same - kernel: here: O(n this is 2 for ) all the landscapes, for just one - kernel: it is O(n) 2 ) - feature map: O(n 2 ) - kernel: O(D) Be careful stability: that these stability O(d results B ) hide some differences, stability: in particular 2 ino(d the constants. 1 B ( )) Basically, results involving stability: p = have O(d absolute B ) constants bu p O(Pers d B ( )) p O(D 1/p d B ( )) injective feature map injective feature map non-injective feature map pd kernel? pd kernel? psd kernel 6

30 Side-by-side Comparison persistence landscapes heat diffusion shape context [Bubenik 2012] [Bauer et al. 2015] [Carrière, O., Ovsjanikov 2015] this reference [Bubenik, is ford lotko the algorithmic 2015] aspects feature space: L 2 (N R) L 2 (R 2 ) feature space: L 2 (R 2 ) feature space: l p (R D, l p ) feature map: explicit (combi. construction) feature map: explicit (closed form solution) feature map: explicit (combi. construction) kernel(s): linear, Gaussian, etc. kernel(s): kσ kernel(s): linear Gaussian, etc. complexity on n-points diagrams: complexity on n-points diagrams: complexity on n-points diagrams: this - feature is for all map: the landscapes, O(n 2 ) for just one it is O(n- log feature n) map: N/A (or discretize...) same - kernel: here: O(n this is 2 for ) all the landscapes, for just one - kernel: it is O(n) 2 ) - feature map: O(n 2 ) - kernel: O(D) Be careful stability: that these stability O(d results B ) hide some differences, stability: in particular 2 ino(d the constants. 1 B ( )) Basically, results involving stability: p = have O(d absolute B ) constants bu p O(Pers d B ( )) p O(D 1/p d B ( )) injective feature map injective feature map non-injective feature map pd kernel? kernels on diagrams are not additive pd kernel? kernel is additive psd kernel kernels on diagrams are not additive 6

31 Applications in Geometry Processing Bottomline: use feature vectors as signatures to compare shapes or parts thereof 7

32 Applications in Geometry Processing shape: compact metric space distance between shapes: Gromov-Hausdorff (GH) distance signature: feature vector from persistence diagram (choose filtration) - multi-scale: reflects the structure of the shape across scales - global/local: attached to the whole shape / to a (set of) base point(s) - stable: variations with GH-distance and base point location are controlled 7

33 Why Compare Shapes Comparisons between shapes or parts thereof occur in: shape classification (organizing large databases of shapes) TOSCA Shape Benchmark 8

34 Why Compare Shapes Comparisons between shapes or parts thereof occur in: shape classification (organizing large databases of shapes) shape retrieval (searching in databases of shapes) 8

35 Why Compare Shapes Comparisons between shapes or parts thereof occur in: shape classification (organizing large databases of shapes) shape retrieval (searching in databases of shapes) partial/global shape matching (finding the best mapping between shapes) 8

36 Why Compare Shapes Comparisons between shapes or parts thereof occur in: shape classification (organizing large databases of shapes) shape retrieval (searching in databases of shapes) partial/global shape matching (finding the best mapping between shapes) shape segmentation and labelling 8

37 Why Compare Shapes Comparisons between shapes or parts thereof occur in: shape classification (organizing large databases of shapes) shape retrieval (searching in databases of shapes) partial/global shape matching (finding the best mapping between shapes) shape segmentation and labelling 8

38 Approach Associate PD / FV to every point Train linear classifiers on a database of FVs 9

39 Approach Associate PD / FV to every point choice of filtration(s)? stability guarantees? Train linear classifiers on a database of FVs 9

40 Filtrations from Functions Input: a compact metric space (X, d X ) and a Lipschitz function f : X R Signature: Φ(Dg f) 10

41 Filtrations from Functions Input: a compact metric space (X, d X ) and a Lipschitz function f : X R Signature: Φ(Dg f) Examples: distance to a base point x 0 X: f x0 (x) = d X (x, x 0 ) is 1-Lipschitz x 0 x 0 10

42 Filtrations from Functions Input: a compact metric space (X, d X ) and a Lipschitz function f : X R Signature: Φ(Dg f) Examples: Sun, Chen, Funkhauser, SGP 2010 fuzzy geodesic [Sun et al. 2010] of a pair of base points x 0, x 1 X: ( ) f x0,x 1 (x) = exp d X (x,x 0 )+d X (x,x 1 ) d X (x 0,x 1 ) is 2 σ σ -Lipschitz, x 0 x 0 x 1 x 1 10

43 Filtrations from Functions Input: a compact metric space (X, d X ) and a Lipschitz function f : X R Signature: Φ(Dg f) Examples: intersection configuration [Sun et al. 2010] of a quadruple of base points: f x0,x 1,x 2,x 3 (x) = f x0,x 1 (x) f x2,x 3 (x) is 4 σ -Lipschitz. x 0 x 0 x 1 x 1 10

44 Stability Results Desired stability result: This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). 11

45 Stability Results Desired stability result: This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). Definitions: - correspondence Y C X Y s.t. π X (C) = X and π Y (C) = Y y x X 11

46 Stability Results Desired stability result: This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). Definitions: - correspondence - distortion dist m (C) = sup (x,y),(x,y ) C d X(x, x ) d Y (y, y ) dist f (C) = sup (x,y) C f(x) g(y) 11

47 Stability Results Desired stability result: This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). Definitions: - correspondence - distortion - Gromov-Hausdorff distance d GH (X, Y ) = 1 2 inf C C(X,Y ) dist m (C) 11

48 Stability Results Desired stability result: This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). The result is not true in such generality: Indeed, - d B (Dg thef, finiteness Dg g) < of the (X, bottleneck d X ) and distance (Y, d Y ) implies are homologically that f and gequivalent have the sam this - dist means m (C) that andthe distgiven f (C) upper are finite bound regardless cannotof hold homological when X and types Y of arex, not Y homolo The idea Restrict behind thethis focus restriction to a class is that of sufficiently we want to regular ensure metric homological spaces equivalence bet 11

49 Stability Results Obtained stability result: length spaces with positive convexity radius This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), such that dist m (C) < 1 min{ϱ(x), ϱ(y )}, d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). 19c dist m (C) + dist f (C) 10 11

50 Stability Results Obtained stability result: length spaces with positive convexity radius This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), such that dist m (C) < 1 min{ϱ(x), ϱ(y )}, d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). 19c dist m (C) + dist f (C) 10 Prerequisite: d GH (X, Y ) < 1 20 min{ϱ(x), ϱ(y )} X = Y = d GH (X, Y ) < = ϱ(y ) d B (Dg f, Dg g) = 11

51 Stability Results Obtained stability result: length spaces with positive convexity radius This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), such that dist m (C) < 1 min{ϱ(x), ϱ(y )}, d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). 19c dist m (C) + dist f (C) 10 f = d X (x, ) g = d Y (y, ) (x, y) C d B (Dg f, Dg g) 20dist m (C) 11

52 Stability Results Obtained stability result: length spaces with positive convexity radius This stability Theorem result (Stability): is in thelet same (X, vein d X ) asand the(y, previous d Y ) beone, two modulo compacthe metric resort spaces to corresp equipped with c-lipschitz functions f : X R and g : Y R. Then, for any correspondence C C(X, Y ), such that dist m (C) < 1 min{ϱ(x), ϱ(y )}, d B (Dg f, Dg g) O(c dist m (C) + dist f (C)). 19c dist m (C) + dist f (C) 10 f = d X (x, ) g = d Y (y, ) (x, y) C d B (Dg f, Dg g) 20dist m (C) 1 p Φ(Dg f), Φ(Dg g) p 40D 1/p dist m (C) 11

53 Stability Results f x ( ) = d X (x, ) Φ(Dg 1 (f)) log-log scale 12

54 Stability Results f x ( ) = d X (x, ) Φ(Dg 1 (f)) log-log scale 12

55 Computation on triangulated surfaces d(x, ) computed on 1-skeleton graph of mesh using Dijkstra s algorithm compute Dg 0 ( d(x, )) using union-find data structure use symmetry theorem [Cohen-Steiner, Edelsbrunner, Harer 2009] obtain ordinary part of Dg 1 (d(x, )) complexity: O(n 2 log n + n log n + nd) per shape (3-5 mins / 10k-15k vertices) 13

56 Unsupervised segmentation f x ( ) = d X (x, ) Φ(Dg 1 (f)) log-log scale map feature space (R D, l ) to R 3 via MDS k-means in R 3 use of instead of d B (before: 1 day / after: minutes) speeds up the process 14

57 Unsupervised segmentation f x ( ) = d X (x, ) Φ(Dg 1 (f)) log-log scale map feature space (R D, l ) to R 3 via MDS k-means in R 3 use of instead of d B (before: 1 day / after: minutes) speeds up the process 14

58 Supervised segmentation Approach 1: use k-nn classifier in feature space (R D, l ) 15

59 Supervised segmentation Approach 1: use k-nn classifier in feature space (R D, l ) Approach 2: use linear classifier (SVM) in feature space (R D, l 2 ) this + is graph to smooth cut [Kalogerakis the segments et al. boundaries 2010] 15

60 Supervised segmentation Approach 1: use k-nn classifier in feature space (R D, l ) Approach 2: use linear classifier (SVM) in feature space (R D, l 2 ) this + is graph to smooth cut [Kalogerakis the segments et al. boundaries 2010] results are shown for small training sets, for which the addition of our signatures to the pool of o SB5 SB5+PDs SB5 SB5+PDs Human Plier Cup Fish Glasses Bird Airplane Armadillo Ant Bust Chair Mech Octopus Bearing Table Vase Teddy FourLeg Hand percentage rand index of mislabelling = fraction(100 rand of pairs ofindex) points in the s 15

61 Non-Rigid Shape matching Approach: use framework of functional maps [Ovsjanikov et al. 2012] Given two shapes X, Y : - compute a linear map L2 (X) L2 (Y ) that best preserves a set of signatures - derive a point-to-point correspondence from this map - evaluate the quality of the correspondence 16

62 Non-Rigid Shape matching proportion of the points each Approach: plot represents results onofone particularmaps category of shapes.etfrom left to right and top usethe framework functional [Ovsjanikov al. 2012] distance to ground-truth correspondence 16

63 Non-Rigid Shape matching Approach: use framework of functional maps [Ovsjanikov et al. 2012] correspondences in flat regions are improved by topological signatures 16