Kernel metrics on normal cycles for the matching of geometrical structures.

Transcription

1 Kernel metrics on normal cycles for the matching of geometrical structures. Pierre Roussillon University Paris Descartes July 18, / 34

2 Overview Introduction Matching of geometrical structures Kernel metrics for the dissimilarity measure Dissimilarity measure with normal cycles Normal bundle and normal cycle Kernel metrics on normal cycles Discrete framework Inexact matching problem First results synthetic data Real data Future work 2 / 34

3 Introduction Overview Introduction Matching of geometrical structures Kernel metrics for the dissimilarity measure Dissimilarity measure with normal cycles Normal bundle and normal cycle Kernel metrics on normal cycles Discrete framework Inexact matching problem First results synthetic data Real data Future work 2 / 34

4 Introduction Matching of geometrical structures Matching of geometrical structures Matching as a variational problem : given two shapes X 0 and X 1, optimize on the deformation φ : R d R d a regularity + dissimilarity functional : J(φ) = E(φ) + A(φ.X 0, X 1 ) 3 / 34

5 Introduction Matching of geometrical structures Matching of geometrical structures Matching as a variational problem : given two shapes X 0 and X 1, optimize on the deformation φ : R d R d a regularity + dissimilarity functional : J(φ) = E(φ) + A(φ.X 0, X 1 ) 3 / 34

6 Introduction Matching of geometrical structures Matching of geometrical structures Matching as a variational problem : given two shapes X 0 and X 1, optimize on the deformation φ : R d R d a regularity + dissimilarity functional : J(φ) = E(φ) + A(φ.X 0, X 1 ) For the set of deformations, we use the LDDMM framework (seen on previous week). For A, we introduce a new model for dissimilarity which takes into account curvature information on the shape. 3 / 34

7 Introduction Kernel metrics for the dissimilarity measure Metric on currents : kernel metrics defined as integrals over the shapes. S τ x x τ y y [C], [S] W = k W (x, y) τ x, τ y dh 1 (x)dh 1 (y) S C C 4 / 34

8 Introduction Kernel metrics for the dissimilarity measure Metric on currents : kernel metrics defined as integrals over the shapes. S τ x x τ y y [C], [S] W = k W (x, y) τ x, τ y dh 1 (x)dh 1 (y) S C Metric on normal cycles : kernel metrics defined as integrals over the normal bundles of the shapes. [N C ], [N S ] W = k p (x, y)k n (u, v) τ (x,u), τ (y,v) N S N C dh 1 (x, u)dh 1 (y, v) C N C N S τ (x,u) x u y v τ (y,v) 4 / 34

9 Introduction Kernel metrics for the dissimilarity measure When ε 0, this curve vanishes in the space of currents. To overcome this problem Charon and Trouvé developed a new dissimilarity measure with varifolds 1. 1 Nicolas Charon and Alain Trouvé (2013). The Varifold Representation of Nonoriented Shapes for Diffeomorphic Registration. In: SIAM J. Imaging Sciences 6.4, pp / 34

10 Introduction Kernel metrics for the dissimilarity measure Examples of matchings with currents. Different matchings obtained with currents as dissimilarity measure. High spatial frequencies are not seen in the space of currents. 6 / 34

11 Dissimilarity measure with normal cycles Overview Introduction Matching of geometrical structures Kernel metrics for the dissimilarity measure Dissimilarity measure with normal cycles Normal bundle and normal cycle Kernel metrics on normal cycles Discrete framework Inexact matching problem First results synthetic data Real data Future work 6 / 34

12 Dissimilarity measure with normal cycles Currents Ω m 0 (Rd ) is the space of m-differential forms vanishing at infinity. Definition : The space of m-currents The space of m-currents of R d is the topological dual of Ω m 0 (Rd ) the space of m-differential forms vanishing at infinity : Ω m 0 (Rd ). if T Ω m 0 (Rd ), ω Ω m 0 (Rd ) : T (ω) C ω 7 / 34

13 Dissimilarity measure with normal cycles Currents Ω m 0 (Rd ) is the space of m-differential forms vanishing at infinity. Definition : The space of m-currents The space of m-currents of R d is the topological dual of Ω m 0 (Rd ) the space of m-differential forms vanishing at infinity : Ω m 0 (Rd ). if T Ω m 0 (Rd ), ω Ω m 0 (Rd ) : T (ω) C ω Example : x R d, α Λ m. δ α x (ω) = ω(x) α. Then δ α x Ω m 0 (Rd ). 7 / 34

14 Dissimilarity measure with normal cycles Currents Ω m 0 (Rd ) is the space of m-differential forms vanishing at infinity. Definition : The space of m-currents The space of m-currents of R d is the topological dual of Ω m 0 (Rd ) the space of m-differential forms vanishing at infinity : Ω m 0 (Rd ). if T Ω m 0 (Rd ), ω Ω m 0 (Rd ) : T (ω) C ω Example : x R d, α Λ m. δx α (ω) = ω(x) α. Then δx α Ω m 0 (Rd ). Let S be a m-submanifold of R d, C 1, oriented. One can associate with S a m-current [S] : [S](ω) = S ω(x) τ S (x) dh m (x) = S δ τ S(x) x with τ S (x) = τ 1 (x) τ m (x) and (τ i (x)) 1 i m is a positively oriented orthonormal basis of T x S. This expression is parametrization free. 7 / 34

15 Dissimilarity measure with normal cycles Normal bundle and normal cycle Volume of the ε-tube : the beginning of normal cycles X ε = {x R d d(x, X) ε} and X ε = {x R d d(x, X) = ε}. 2 Herbert Federer (1959). Curvature measures. In: Trans. Amer. Maths. Soc / 34

16 Dissimilarity measure with normal cycles Normal bundle and normal cycle Volume of the ε-tube : the beginning of normal cycles X ε = {x R d d(x, X) ε} and X ε = {x R d d(x, X) = ε}. Federer 2 expresses the volume of the ε-tube around a set X : d H d ((X B) ε ) = α k C d k (X; B)ε k, B Borel set of R d k=0 where the C k can be interpreted as curvature measures. 2 Herbert Federer (1959). Curvature measures. In: Trans. Amer. Maths. Soc / 34

17 Dissimilarity measure with normal cycles Normal bundle and normal cycle Example for a C 2 surface S = M where M is a domain of R 3. Then : H 3 ((M B) ε ) = α 0 C 3 (M, B)+α 1 C 2 (M, B)ε+α 2 C 1 (M, B)ε 2 +α 3 C 0 (M, B)ε 3 with C 3 (M, B) = H 3 (M B), C 2 (M, B) = 1 A(S B) 2 C 1 (M, B) = 1 H(S B) 4π C 0 (M, B) = 3 K(S B) 4π 9 / 34

18 Dissimilarity measure with normal cycles Normal bundle and normal cycle Normal bundle Definition (Normal bundle) the tangent cone of X at point x is : Tan(X, x) = { v R d ε > 0, y X, c > 0 c(y x) v ε } the normal cone of X at point x is : Nor(X, x) = { u R d, v Tan(X, x), u, v < 0 } N X = {(x, n) X Nor(X, x) S d 1 } is the unit normal bundle of X. All these definitions work for set with positive reach 10 / 34

19 Dissimilarity measure with normal cycles Normal bundle and normal cycle Normal cycles Definition (Normal cycle) The normal cycle of a set X R d with positive reach is the current associated with its normal bundle : N(X) := [N X ] N(X) Ω d 1 0 (R d S d 1 ) 11 / 34

20 Dissimilarity measure with normal cycles Normal bundle and normal cycle Rewriting the volume of the ε-tube using normal cycle For a set X : d H d ((X B) ε ) = α k C d k (X; B)ε k k=0 3 M Zähle (1987). Curvatures and currents for unions of set with positive reach. In: Geometriae Dedicata 23, pp / 34

21 Dissimilarity measure with normal cycles Normal bundle and normal cycle Rewriting the volume of the ε-tube using normal cycle For a set X : can write 3 d H d ((X B) ε ) = α k C d k (X; B)ε k k=0 d 1 H d ((X B) ε ) = α k N(X)(ω d k P 1 X (B))εk + H d (X B) k=0 where the ω k are the Lipschitz-Killing universal differential forms, ω k Ω d 1 (R d S d 1 ) N(X) encodes curvature information on X. 3 M Zähle (1987). Curvatures and currents for unions of set with positive reach. In: Geometriae Dedicata 23, pp / 34

22 Dissimilarity measure with normal cycles Kernel metrics on normal cycles Spatial kernel, normal kernel We define a RKHS W of differential forms such that W Ω d 1 0 (R d S d 1 ) Thus, we have Ω d 1 0 (R d S d 1 ) W. 13 / 34

23 Dissimilarity measure with normal cycles Kernel metrics on normal cycles Spatial kernel, normal kernel We define a RKHS W of differential forms such that W Ω d 1 0 (R d S d 1 ) Thus, we have Ω d 1 0 (R d S d 1 ) W. For this, we choose K W : (R d S d 1 ) 2 L(Λ d 1 (R d R d )) : with K W ((x, u), (y, v)) = k p (x, y)k n (u, v)id Λ d 1 (R d R d ) A spatial kernel k p : k p (x, y) = exp( x y 2 ) σ 2 W A normal kernel k n : the reproducing kernel of some Sobolev space H s (S d 1 ), s > 0. Normal cycles are thus considered in a Hilbert space with an explicit scalar product. 13 / 34

24 Dissimilarity measure with normal cycles Kernel metrics on normal cycles Let S, C be two shapes : N(C), N(S) W = k p (x, y)k n (u, v) N C N S τ NC (x, u), τ NS (y, v) Λ d 1 (R d R d ) dhd 1 (x, u)dh d 1 (y, v) N S τ (x,u) y v τ (y,v) N C x u 14 / 34

25 Dissimilarity measure with normal cycles Kernel metrics on normal cycles Metric variation for normal cycles Theorem Let ϕ t be the flow of some smooth vector field v : R d R d and ω Ω 0 (R d S d 1 ). Then d dt t=0 N(ϕ t (X)) ω = dω(x, n) ṽ(x, n) τ NS (x, n) dh d 1 (x, n) N X ( ) v(x) where ṽ(x, n) = p n dv(x) T, ṽ : R.n d S d 1 R d R d The variation depends only of the orthogonal component of v 15 / 34

26 Dissimilarity measure with normal cycles Kernel metrics on normal cycles Metric variation for normal cycles Theorem Let ϕ t be the flow of some smooth vector field v : R d R d and ω Ω 0 (R d S d 1 ). Then d dt t=0 N(ϕ t (X)) ω = dω(x, n) ṽ(x, n) τ NS (x, n) dh d 1 (x, n) N X ( ) v(x) where ṽ(x, n) = p n dv(x) T, ṽ : R.n d S d 1 R d R d The variation depends only of the orthogonal component of v the gradient of the distance for the kernel metric is orthogonal to the shape. 15 / 34

27 Discrete framework Overview Introduction Matching of geometrical structures Kernel metrics for the dissimilarity measure Dissimilarity measure with normal cycles Normal bundle and normal cycle Kernel metrics on normal cycles Discrete framework Inexact matching problem First results synthetic data Real data Future work 15 / 34

28 Discrete framework Decomposition of the normal bundle for discrete curves Using additive properties for normal cycles, we decompose the normal bundle into elementary pieces : N(C 1 C n ) = n N N( C i ) + N({v j }) i=1 j=1 16 / 34

29 Discrete framework Decomposition of the normal bundle for discrete curves In red : cylindrical part. In green : spherical part. The normal bundle of a discrete curve has two components : A cylindrical component, N( C i ) cyl, which is the normal bundle associated with the interior of the segment. A spherical part : the normal bundle associated with vertices N({v j }) and N( C i ) sph 17 / 34

30 Discrete framework Decomposition of the normal bundle for discrete curves In red : cylindrical part. In green : spherical part. The normal bundle of a discrete curve has two components : A cylindrical component, N( C i ) cyl, which is the normal bundle associated with the interior of the segment. A spherical part : the normal bundle associated with vertices N({v j }) and N( C i ) sph These two components are orthogonal for the kernel metric. 17 / 34

31 Discrete framework Discrete scalar products N(C) cyl, N(S) cyl W = n C n S N(Ci ) cyl, N(S j ) cyl W = n C n S i j n C n S i=1 j=1 where θ ij = arccos segments. C i (S d 1 Ci ) i j S j (S d 1 S j ) τ NCi (x, u), τ NSj (y, v) k p (x, y)k n (u, v) dh 1 (x, u)dh 1 (y, v) k p (c i, d j ) f i, g j m 0 a m cos((2m + 1)θ ij ) ( ) fi f i, g j, c i and d j are the middle of the g j 18 / 34

32 Discrete framework N(C) sph, N(S) sph N C N S = N({x W k }), N({y l }) W i=k l=1 N C n S + N( C i ) sph, N( S j ) sph W i=1 j=1 N C N S = + k=1 l=1 n C n S i=1 j=1 a,b=1 k p (x f a i, y g b j ) ( ) fi where θ ij = arccos f i, g j g j ( k p (x k, y l ) 1 n x k + n yl 2 ) β 2 b 0 + ( 1) a+b b m cos((2m + 1)θ ij ) m 0 19 / 34

33 Inexact matching problem Overview Introduction Matching of geometrical structures Kernel metrics for the dissimilarity measure Dissimilarity measure with normal cycles Normal bundle and normal cycle Kernel metrics on normal cycles Discrete framework Inexact matching problem First results synthetic data Real data Future work 19 / 34

34 Inexact matching problem LDDMM The group { G of deformation in our framework is G := ϕ v v t 2 V }, dt < + with ϕ t t (x) = v t ϕ t (x) ϕ 0 = Id and (V,. V ) a Hilbert space of vector fields of R d such that V C 1 (R d, R d ). We minimize the functional J(v) = γ 1 0 v t 2 V dt + N(ϕ 1 v(x 0 )) N(X 1 ) 2 W 20 / 34

35 Inexact matching problem The inexact matching problem with kernel metric on normal cycles is well posed Theorem Suppose W Ω 0 1,0 (Rd S d 1 ) and V C 3 0 (Rd, R d ). Then there exists a solution for the problem : min γ v L 2 V 1 0 v t 2 V dt + N(ϕv 1.X 0 ) N(X 1 ) 2 W 21 / 34

36 Inexact matching problem Algorithm for curve matching Implemented in Matlab We use a geodesic shooting procedure with a L-BFGS algorithm. We iterate until convergence, with a tolerance of We use a Runge-Kunta (4,5) to integrate geodesic equations (ode45 in Matlab). k p Gaussian kernel, of width σ W. k n is a Sobolev kernel, associated with the operator L = (Id ) 3. k V is a Cauchy kernel of size σ V. 22 / 34

37 First results Overview Introduction Matching of geometrical structures Kernel metrics for the dissimilarity measure Dissimilarity measure with normal cycles Normal bundle and normal cycle Kernel metrics on normal cycles Discrete framework Inexact matching problem First results synthetic data Real data Future work 22 / 34

38 First results synthetic data Matching of a blue circle to a red circle with a peak, using normal cycles 23 / 34

39 First results synthetic data Matching of two disconnected circles 24 / 34

40 First results synthetic data Matching of two disconnected circles (a) Normal cycles. σ V = 2, σ W = 1 (b) Varifolds. σ V = 0.2, σ W = 0.1 Figure : Result of the matching between the two circles with normal cycles and varifolds. 25 / 34

41 First results synthetic data Matching of two disconnected circles (more difficult) 26 / 34

42 First results synthetic data Matching of two disconnected circles (more difficult) (a) Normal cycles. σ V = 2, σ W = 2 (b) Normal cycles. σ V = 5, σ W = 2 Figure : Result of the matching between the two circles with normal cycles and different σ V. 27 / 34

43 First results synthetic data With σ W >> 1 Figure : Matching of the blue circle to the red circle with normal cycles and σ W = 1000, σ V = 5. The green circle is the deformed circle. 28 / 34

44 First results synthetic data Matching of branching curves (normal cycles) Matching of two 3D curves. The initial curve is in black, the target curve in red. The deformed curve is in green. The trajectories along the flow are in blue. σ V = 0.2 et σ W = / 34

45 First results synthetic data Matching of branching curves (normal cycles) Matching of two 3D curves. The initial curve is in black, the target curve in red. The deformed curve is in green. The trajectories along the flow are in blue. σ V = 0.2 et σ W = / 34

46 First results synthetic data Matching of branching curves (normal cycles) Matching of two 3D curves. The initial curve is in black, the target curve in red. The deformed curve is in green. The trajectories along the flow are in blue. σ V = 0.2 et σ W = / 34

47 First results synthetic data Matching of branching curves (normal cycles) Matching of two 3D curves. The initial curve is in black, the target curve in red. The deformed curve is in green. The trajectories along the flow are in blue. σ V = 0.2 et σ W = / 34

48 First results synthetic data Matching of branching curves (normal cycles) (a) Normal cycles, view 1 (b) Normal cycles, view 2 Figure : Matching of two 3D curves. The initial curve is in black, the target curve in red. The deformed curve is in green. The trajectories along the flow are in blue. σ V = 0.2 et σ W = 0.3. The algorithm run in 811 seconds. 29 / 34

49 First results synthetic data Matching of branching curves (currents) (a) Currents, view 1 (b) Currents, view 2 Figure : Matching of two 3D curves. The initial curve is in black, the target curve in red. The deformed curve is in green. The trajectories along the flow are in blue. σ V = 0.2 et σ W = 0.3. The algorithm run in 178 seconds. 30 / 34

50 First results synthetic data Matching of branching curves (varifolds) (a) Varifolds, view 1 (b) Varifolds, view 2 Figure : Matching of two 3D curves. The initial curve is in black, the target curve in red. The deformed curve is in green. The trajectories along the flow are in blue. σ V = 0.2 et σ W = 0.3. The algorithm run in 316 seconds. 31 / 34

51 First results Real data Matching of brain sulci (Normal cycles) (a) t=0 Figure : Matching of brain sulci of two subjects, with normal cycles. σ V = 10 and σ W = 7. The size of the kernel is in mm. The algorithm run in 29 minutes. 32 / 34

52 First results Real data Matching of brain sulci (Normal cycles) (a) t=0.5 Figure : Matching of brain sulci of two subjects, with normal cycles. σ V = 10 and σ W = 7. The size of the kernel is in mm. The algorithm run in 29 minutes. 32 / 34

53 First results Real data Matching of brain sulci (Normal cycles) (a) t = 0.75 Figure : Matching of brain sulci of two subjects, with normal cycles. σ V = 10 and σ W = 7. The size of the kernel is in mm. The algorithm run in 29 minutes. 32 / 34

54 First results Real data Matching of brain sulci (Normal cycles) (a) t = 1 Figure : Matching of brain sulci of two subjects, with normal cycles. σ V = 10 and σ W = 7. The size of the kernel is in mm. The algorithm run in 29 minutes. 32 / 34

55 First results Real data Matching of brain sulci (Normal cycles) (a) Zoom Figure : Matching of brain sulci of two subjects, with normal cycles. σ V = 10 and σ W = 7. The size of the kernel is in mm. The algorithm run in 29 minutes. 33 / 34

56 Future work Overview Introduction Matching of geometrical structures Kernel metrics for the dissimilarity measure Dissimilarity measure with normal cycles Normal bundle and normal cycle Kernel metrics on normal cycles Discrete framework Inexact matching problem First results synthetic data Real data Future work 33 / 34

57 Future work Future work Algorithm for surfaces. Link between normal cycles and varifolds? Work on real data. 34 / 34