Comparing persistence diagrams through complex vectors

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1 Comparing persistence diagrams through complex vectors Barbara Di Fabio ARCES, University of Bologna Final Project Meeting Vienna, July 9, 2015

2 B. Di Fabio, M. Ferri, Comparing persistence diagrams through complex vectors, Accepted for publication in the proceedings of ICIAP 2015: 18th International Conference on Image Analysis and Processing Main reference: Ferri, M., Landi, C.: Representing size functions by complex polynomials. In: Proc. Math. Met. in Pattern Recognition. vol. 9, pp (1999)

3 Outline Introductive notions The natural pseudo-distance Persistence diagrams Bottleneck distance Persistence diagrams vs complex vectors Persistence diagrams vs complex polynomials Complex polynomials vs complex vectors Complex vectors comparison Chosen transformations Chosen distances Comparing the bottleneck distance and our method Computational cost Performances 3 of 30

4 Introductive notions We model a shape as a pair (X,f) where X is a topological space f :X R a (continuous) function X (X,f 1 ) (X,f 2 ) (X,f 3 )... 4 of 30

5 Introductive notions Suppose we have a database D containing n objects... M 1 M 2 M 3 M 4... Model the shape of each object M i as a pair (X i,f i );... (X 1,f 1 ) (X 2,f 2 ) (X 3,f 3 ) (X 4,f 4 )... Model D ={M i,i =1,...,n} as a set X ={(X i,f i ),i =1,...,n}. 4 of 30

6 Our approach to shape comparison How can we compare two shapes (X 1,f 1 ), (X 2,f 2 )? d (X 1,f 1 ), (X 2,f 2 ) =? 5 of 30

7 The natural pseudo-distance Definition Let (X 1,f 1 ),(X 2,f 2 ) be two shapes. If H = H (X 1,X 2 ) denotes the set of homeomorphisms between X 1 and X 2, then the natural pseudo-distance is defined as { inf δ((x 1,f 1 ),(X 2,f 2 ))= max f 1 (p) f 2 (h(p)), H /0 h H p X 1 +, otherwise. In general, the natural pseudo-distance is only a pseudo-metric: δ((x 1,f 1 ),(X 2,f 2 ))=0 (X 1,f 1 )=(X 2,f 2 ). 6 of 30

8 The natural pseudo-distance vs persistence diagrams Persistent homology allows us to describe such pairs by means of suitable shape descriptors: the persistence diagrams Instead of comparing shapes, we can compare shape descriptors d 7 of (X 1,f 1 ), (X 2,f 2 ) δ (X 1,f 1 ), (X 2,f 2 )

9 Persistent homology Let (X,f) be a shape, and let us denote by X u ={x X :f(x) u} with u R. 8 of 30

10 Persistent homology Let (X,f) be a shape, and let us denote by X u ={x X :f(x) u} with u R. Definition For u v and k Z, let ι u,v k :H k (X u ) H k (X v ) be the map induced by inclusion of X u in X v. The kth-persistent homology group is imι u,v k. When this group is finitely generated, we denote by βf k(u,v) its rank and call it the associated persistent Betti number. 8 of 30

11 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 0 β 1 = 0 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

12 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 0 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

13 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 2 β 1 = 0 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

14 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 0 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

15 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 1 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

16 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 2 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

17 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 3 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

18 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 4 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

19 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 5 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

20 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 4 β 2 = 0 a 5 a 4 a 3 a 2 a 1 9 of 30

21 From shapes to filtrations: an example X f a 10 a 9 a 8 a 7 a 6 β 0 = 1 β 1 = 4 β 2 = 1 a 5 a 4 a 3 a 2 a 1 9 of 30

22 Persistence diagram (intuitively) It encodes the birth level ū and the death level v of a homology class by a point (ū, v); if the class doesn t die, by a line u =ū. X f a 10 a 9 a 8 a 7 a 6 a 5 a 4 a 3 a 2 a 1 10 of 30

23 The bottleneck distance v r b D v 1 r D v 2 rr b a 2 a a a 2 3 c c 01 c 01 c Definition r+a+b+c r +a +c u The bottleneck distance between D 1 and D 2 is given by u u d B (D 1,D 2 )=min max p σ(p) σ p D 1 where σ varies among all the bijections between D 1 and D of 30

1 0 0-0.1-0.1-0.2-0.2-0.3-0.3-0.4-0.4-0.4-0.3-0.2-0.1 0 0.1 0.2 0.

24 Bottleneck distance properties and drawbacks Stability with respect to the bottleneck distance: d B (D f,d g ) f g This implies resistance to noise. 12 of 30

25 Bottleneck distance properties and drawbacks Stability with respect to the bottleneck distance: optimality 1 : d B (D f,d g ) f g d B (D f,d g ) δ((x,f),(y,g)) and results to be the most discriminative distance among all the stable metrics. 1 d Amico, M., Frosini, P., Landi, C.: Natural pseudo-distance and optimal matching between reduced size functions. Acta. Appl. Math. 109, (2010) 12 of 30

26 Bottleneck distance properties and drawbacks Stability with respect to the bottleneck distance: optimality 1 : d B (D f,d g ) f g d B (D f,d g ) δ((x,f),(y,g)) and results to be the most discriminative distance among all the stable metrics. It suffers from combinatorial explosion. 1 d Amico, M., Frosini, P., Landi, C.: Natural pseudo-distance and optimal matching between reduced size functions. Acta. Appl. Math. 109, (2010) 12 of 30

27 Bottleneck distance properties and drawbacks Stability with respect to the bottleneck distance: optimality 1 : d B (D f,d g ) f g d B (D f,d g ) δ((x,f),(y,g)) and results to be the most discriminative distance among all the stable metrics. It suffers from combinatorial explosion: We propose a preprocessing algorithm to decrease such a cost. 1 d Amico, M., Frosini, P., Landi, C.: Natural pseudo-distance and optimal matching between reduced size functions. Acta. Appl. Math. 109, (2010) 12 of 30

28 Motivations In general, realizing that two shapes are very dissimilar just requires a rough estimation of their (dis)similarity (rough estimation faster procedure) of 30

29 Motivations... Highest levels of accuracy should be needed only for critical cases. 13 of 30

30 Motivations This motivates the introduction of a preprocessing (dis)similarity measure. 13 of 30

31 Ordinary Persistence diagrams 2 Main condition: v v u u 2 Cohen-Steiner, D., Edelsbrunner, H., Harer, J., Extending persistence using Poincar and Lefschetz duality. Found. Comput. Math. 9, of 30

32 Ordinary Persistence diagrams Let ={(u,v) R 2 :u =v}, + ={(u,v) R 2 :u v}, and + = +. Multiplicities of points Let (u,v) +. µ k f (u,v) = min ε>0 βk f β k f (u+ ε,v ε) βk f (u ε,v ε)+ (u+ ε,v+ ε)+βk f (u ε,v+ ε) Definition The k-persistence diagram is the set of all points with µ f (u,v)>0, counted with their multiplicity, augmented by adding a countable infinity of points of the diagonal. 15 of 30

33 Persistence diagrams vs complex vectors Persistence diagrams complex polynomials complex vectors points roots coefficients Theorem If the roots of two polynomials are close each other, then so their coefficients are. Or equivalently: If the coefficients of two polynomials are far from each other, then so their roots are. 16 of 30

34 Persistence diagrams vs complex polynomials We define a continuous map T : + C. 17 of 30

35 Persistence diagrams vs complex polynomials We define a continuous map T : + C. For every z Im(T), we define the multiplicity of z as µ(z)= µ(t 1 (z)). 17 of 30

36 Persistence diagrams vs complex polynomials We define a continuous map T : + C. For every z Im(T), we define the multiplicity of z as µ(z)= µ(t 1 (z)). Let D be a persistence diagram, and p 1 =(u 1,v 1 ),...,p s =(u s,v s ) its proper points with multiplicity r 1,...,r s, respectively. Let now the complex numbers z 1,...,z s be obtained from p 1,...,p s by T. 17 of 30

37 Persistence diagrams vs complex polynomials We define a continuous map T : + C. For every z Im(T), we define the multiplicity of z as µ(z)= µ(t 1 (z)). Let D be a persistence diagram, and p 1 =(u 1,v 1 ),...,p s =(u s,v s ) its proper points with multiplicity r 1,...,r s, respectively. Let now the complex numbers z 1,...,z s be obtained from p 1,...,p s by T. We associate to D the complex polynomial 17 of 30 f D (t)= s j=1 (t z j ) r j.

38 Complex polynomials vs complex vectors We write f D (t)=t n a 1 t n 1 + +( 1) i a i t n i + +( 1) n a n, where a n k = z i1 z i2... z ik, 1 i 1 i 2... i k i.e., the (n k)-th coefficient a n k is equal to the sum of all possible sub-products of roots, taken k-at-a-time. 18 of 30

39 Complex polynomials vs complex vectors We write f D (t)=t n a 1 t n 1 + +( 1) i a i t n i + +( 1) n a n, where a n k = z i1 z i2... z ik, 1 i 1 i 2... i k i.e., the (n k)-th coefficient a n k is equal to the sum of all possible sub-products of roots, taken k-at-a-time. We associate to D the complex vector v =(a 1,...,a n ). 18 of 30

40 Complex vectors comparison Let v =(a 1,...,a n ) and u =(b 1,...,b m ) be associated with the persistence diagrams D, D, and suppose m<n. 19 of 30

41 Complex vectors comparison Let v =(a 1,...,a n ) and u =(b 1,...,b m ) be associated with the persistence diagrams D, D, and suppose m<n. We replace u by v =(a 1,...,a n), where v is the complex vector obtained by f D (t) t n m. 19 of 30

42 Complex vectors comparison Let v =(a 1,...,a n ) and u =(b 1,...,b m ) be associated with the persistence diagrams D, D, and suppose m<n. We replace u by v =(a 1,...,a n), where v is the complex vector obtained by f D (t) t n m. We define distances between the vectors (a 1,...,a k ) and (a 1,...,a k ), k {1,...,n}. 19 of 30

43 Complex vectors comparison Let v =(a 1,...,a n ) and u =(b 1,...,b m ) be associated with the persistence diagrams D, D, and suppose m<n. We replace u by v =(a 1,...,a n), where v is the complex vector obtained by f D (t) t n m. We define distances between the vectors (a 1,...,a k ) and (a 1,...,a k ), k {1,...,n}. Note that each coefficient a i encodes information of all the points of D!!! 19 of 30

44 Chosen transformations: T 1 T 1 : + C, with T 1 (u,v)=u+iv. 20 of 30

45 Chosen transformations: T 2 v u T 2 : + C, T 2 (u,v)= α (u+iv), if(u,v) (0,0) 2 (0, 0), otherwise where α = u 2 +v 2., pj T2(pj) 21 of 30

46 Chosen transformations: T 2 v u T 2 : + C, T 2 (u,v)= α (u+iv), if(u,v) (0,0) 2 (0, 0), otherwise where α = u 2 +v 2., pj T2(pj) 21 of 30

47 Chosen transformations: T 3 T 3 : + C, with T 3 (u,v)= v u (cosα sinα+i(cosα+sinα)), where 2 α = u 2 +v 2. pj T3(pj) 22 of 30

48 Chosen transformations: T 3 T 3 : + C, with T 3 (u,v)= v u (cosα sinα+i(cosα+sinα)), where 2 α = u 2 +v 2. pj T3(pj) 22 of 30

49 Chosen distances Let v k =(a 1,...,a k ) and v k =(a 1,...,a k ), k {1,...,n}, be associated with the persistence diagrams D, D. 23 of 30

50 Chosen distances Let v k =(a 1,...,a k ) and v k =(a 1,...,a k ), k {1,...,n}, be associated with the persistence diagrams D, D. d 1 (v,v )= k a j a j. j=1 d 2 (v,v )= k a j a j. j=1 j d 3 (v,v )= k a j a j 1/j. 23 of 30 j=1

51 Computational cost Algorithm 1: ComplexLists Input: List A of proper points of a persistence diagram D, M =max A {A:A databasedb} Output: List B of complex numbers associated with D 1: for each (u,v) A 4: if B <M 2: replace (u,v) by T(u,v) 5: append M B zeros to B 3: end for 6: end if C =O(M Db ) 24 of 30

52 Computational cost Algorithm 2: ComplexVectors Input: M, B =list(z 1,...,z M ) associated with D, k [0,M] Output: Complex vector V k associated with D 1: set V k =list() 2: for j {1,...,k} 3: compute c j (z 1,...,z M )= z i1 z i2... z ij 1 i 1 <i 2 <...<i j M 4: append c j to V k 5: end for C =O(M Db +(2k 2 +k M) Db ) 24 of 30

53 Computational cost Algorithm 3: VectorsComparison Input: L={V k :V k complex vector associated withdfor eachd Db} Output: Matrix of distances d(v k,v k ) 1: set M =(0 ij ), i,j =1,..., L 4: replace 0 ij,0 ji by d(i,j) 2: for each i {1,..., L } 5: end for 3: for each j {i,..., L } 6: end for C =O(M Db +(2k 2 +k M) Db +k Db 2 ) 24 of 30

54 Computational cost comparison Let if A,A are the subsets of proper points of two persistence diagrams D,D with A =r, A =r. 3 Efrat, A., Itai, A., Katz, M.J.: Geometry helps in bottleneck matching and 25 of 30 related problems. Algorithmica 31, 128 (2001).

55 Computational cost comparison Let if A,A are the subsets of proper points of two persistence diagrams D,D with A =r, A =r. The computational cost of the bottleneck distance 3 d B (D,D ) is ( ) C B =O (r +r ) 3/2 log(r +r ) 3 Efrat, A., Itai, A., Katz, M.J.: Geometry helps in bottleneck matching and 25 of 30 related problems. Algorithmica 31, 128 (2001).

56 Computational cost comparison Let if A,A are the subsets of proper points of two persistence diagrams D,D with A =r, A =r. The computational cost of the bottleneck distance 3 d B (D,D ) is ( ) C B =O (r +r ) 3/2 log(r +r ) Using our scheme, with Db =2 and M =max(r,r ), we have C =O(max(r,r )+2k 2 +k max(r,r )+2k) O(k (max(r,r )+k)). 3 Efrat, A., Itai, A., Katz, M.J.: Geometry helps in bottleneck matching and 25 of 30 related problems. Algorithmica 31, 128 (2001).

57 Computational cost comparison Let if A,A are the subsets of proper points of two persistence diagrams D,D with A =r, A =r. The computational cost of the bottleneck distance 3 d B (D,D ) is ( ) C B =O (r +r ) 3/2 log(r +r ) Using our scheme, with Db =2 and M =max(r,r ), we have C =O(max(r,r )+2k 2 +k max(r,r )+2k) O(k (max(r,r )+k)). Since k max(r,r ), in the worst case, we have C =O ( (max(r,r )) 2) which is higher than C B, but for pre-processing we may choose a favorable k (e.g. max(r,r k = ) ). 3 Efrat, A., Itai, A., Katz, M.J.: Geometry helps in bottleneck matching and 25 of 30 related problems. Algorithmica 31, 128 (2001).

58 Database (I) Database: triangle meshes derived from the Non-rigid world Benchmark 4 ; 228 models divided into 12 classes. 4 Bronstein, A., Bronstein, M., Kimmel, R.: Numerical Geometry of Non-Rigid Shapes. Springer Publishing Company, Incorporated, 1 edn. (2008) 26 of 30

59 Database (II) Each class contains a null model together with 18 deformations. 27 of 30 Victoria0 strength lev. 3strength lev. 2strength lev. 1 Def. 1 Def. 2 Def. 3 Def. 4 Def. 5 Def. 6

60 Filtering functions We define the following two filtering functions: f L = normalized distance from the line L; f P = normalized distance from the plane P; P L w B where w = n i=1 (v i B) v i B n i=1 v i B 2, {v 1,...,v n } being the vertices of a triangle mesh and B its center of mass. 28 of 30

61 Experimental results T1 T2 precision precision recall recall T3 precision Function: f L Distance: d 1 29 of 30 recall

62 Experimental results T1 T2 precision precision recall recall T3 precision Function: f L Distance: d 2 29 of 30 recall

63 Experimental results T1 T2 precision precision recall recall T3 precision Function: f L Distance: d 3 29 of 30 recall

64 Experimental results T1 T2 precision precision recall recall T3 precision Function: f P Distance: d 1 29 of 30 recall

65 Experimental results T1 T2 precision precision recall recall T3 precision Function: f P Distance: d 2 29 of 30 recall

66 Experimental results T1 T2 precision precision recall recall T3 precision Function: f P Distance: d 3 29 of 30 recall

67 30 of 30 Thank you for your attention!

Metric shape comparison via multidimensional persistent homology

Metric shape comparison via multidimensional persistent homology Patrizio Frosini 1,2 1 Department of Mathematics, University of Bologna, Italy 2 ARCES - Vision Mathematics Group, University of Bologna,