Metamorphoses for Pattern Matching

Transcription

1 Metamorphoses for Pattern Matching Laurent Younes Joint work with Darryl Holm and Alain Trouvé Johns Hopkins University July Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

2 Pattern Matching The issue addressed in Diffeomorphic Metric Matching is the comparison of deformable structures via the action of diffeomorphisms. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

3 Pattern Matching The issue addressed in Diffeomorphic Metric Matching is the comparison of deformable structures via the action of diffeomorphisms. In particular, important problems are: Finding correspondences between images, surfaces, curves... Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

4 Pattern Matching The issue addressed in Diffeomorphic Metric Matching is the comparison of deformable structures via the action of diffeomorphisms. In particular, important problems are: Finding correspondences between images, surfaces, curves... Comparing deformable structures (with distances). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

5 Pattern Matching The issue addressed in Diffeomorphic Metric Matching is the comparison of deformable structures via the action of diffeomorphisms. In particular, important problems are: Finding correspondences between images, surfaces, curves... Comparing deformable structures (with distances). Building Riemannian manifolds on shapes. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

6 Pattern Matching The issue addressed in Diffeomorphic Metric Matching is the comparison of deformable structures via the action of diffeomorphisms. In particular, important problems are: Finding correspondences between images, surfaces, curves... Comparing deformable structures (with distances). Building Riemannian manifolds on shapes. Important applications in medical imaging (computational anatomy). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

7 Deformable templates The deformable template paradigm (U. Grenander) can be transcribed in out context as follows: Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

8 Deformable templates The deformable template paradigm (U. Grenander) can be transcribed in out context as follows: 1. Equip the group of diffeomorphisms with a right-invariant Riemannian metric. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

9 Deformable templates The deformable template paradigm (U. Grenander) can be transcribed in out context as follows: 1. Equip the group of diffeomorphisms with a right-invariant Riemannian metric. 2. Assume that an object n temp is given (the template)and consider the orbit N = Diff.n temp. Equip the orbit with the Riemannian projection of the metric on Diff. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

10 Optimal matching with deformable templates Geodesics provide optimal correspondences: with fixed n 0 and n 1, minimize 1 u t 2 dt with constraints ṅ t u t.n t = 0. 0 Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

11 Optimal matching with deformable templates Geodesics provide optimal correspondences: with fixed n 0 and n 1, minimize 1 u t 2 dt with constraints ṅ t u t.n t = 0. Problem: restricted to an orbit. 0 Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

12 Optimal matching with deformable templates Geodesics provide optimal correspondences: with fixed n 0 and n 1, minimize 1 u t 2 dt with constraints ṅ t u t.n t = 0. Problem: restricted to an orbit. Inexact matching: minimize with ġ t u t g t = u t 2 dt + Error(n 1, g 1.n 0 ) Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

13 Optimal matching with deformable templates Geodesics provide optimal correspondences: with fixed n 0 and n 1, minimize 1 u t 2 dt with constraints ṅ t u t.n t = 0. Problem: restricted to an orbit. Inexact matching: minimize with ġ t u t g t = 0. 1 But the metric aspect is lost. 0 0 u t 2 dt + Error(n 1, g 1.n 0 ) Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

14 Metamorphoses Definition Let N be a manifold acted upon by a Lie group G. A metamorphosis is a pair of curves (g t, η t ) G N parameterized by t, with g 0 = id G. Its image is the curve n t N defined by the action n t = g t.η t. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

15 Invariant Lagrangian minimization Given n 0 and n 1, we will optimize, over metamorphoses (g t, η t ), the integrated Lagrangian 1 0 L(g t, ġ t, η t, η t )dt with end-point conditions g 0 = id G, η 0 = n 0 and g 1 η 1 = n 1. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

16 Let g denote the Lie algebra of G. We assume that L is invariant by the right action of G on G N defined by (g, η)h = (gh, h 1 η). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

17 Let g denote the Lie algebra of G. We assume that L is invariant by the right action of G on G N defined by (g, η)h = (gh, h 1 η). Thus, there exists a function l defined on g TN such that L(g, U g, η, ξ η ) = l(u g g 1, gη, gξ η ). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

18 Let g denote the Lie algebra of G. We assume that L is invariant by the right action of G on G N defined by (g, η)h = (gh, h 1 η). Thus, there exists a function l defined on g TN such that L(g, U g, η, ξ η ) = l(u g g 1, gη, gξ η ). For a metamorphosis (g t, η t ), let u t = ġ t gt 1, n t = g t η t and ν t = g t η t, We have L(g t, ġ t, η t, η t ) = l(u t, n t, ν t ). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

19 Notation The pairing between a linear form p and a vector u is denoted ( p u ). Duality with respect to this pairing is denoted with a exponent: ( p Au ) = ( A p u ). When G acts on a manifold Ñ, define the operator on T Ñ Ñ, taking values in g by ( δ ñ u ) = ( δ uñ ). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

20 Euler Equations Compute the first variation for 1 0 l(u t, n t, ν t )dt with fixed boundary conditions n 0 and n 1. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

21 Euler Equations Compute the first variation for 1 0 l(u t, n t, ν t )dt with fixed boundary conditions n 0 and n 1. Consider a variation δu and ω = δn. From n = gη and ν = g η we get ṅ = ν + un and ω = δν + uω + δun. (Derivatives are assumed to be taken in a chart.) Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

22 Euler Equations Compute the first variation for 1 0 l(u t, n t, ν t )dt with fixed boundary conditions n 0 and n 1. Consider a variation δu and ω = δn. From n = gη and ν = g η we get ṅ = ν + un and ω = δν + uω + δun. (Derivatives are assumed to be taken in a chart.) Write 1 0 ( ( δl ) ( δl ) ( δl )) δu δu + δn ω + ω uω δu n dt = 0. δν Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

23 From the δu term δl δu + δl δν n = 0. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

24 From the δu term From the ω term: with δl δu + δl δν n = 0. δl t δν + u δl δν δl δn = 0 ( δl δν uω) = ( u δl δν ω). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

25 We therefore obtain the system of equations δl δu + δl δν n = 0 δl t δν + u δl δν = δl δn ṅ = ν + un Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

26 We therefore obtain the system of equations δl δu + δl δν n = 0 δl t δν + u δl δν = δl δn ṅ = ν + un Note that δl δν n associated to the invariance of the Lagrangian. The special form of the boundary conditions ensure that this momenum is zero. δu + δl Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

27 Euler-Poincaré reduction Make the variations ξ = δgg 1 and ϖ = gδη. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

28 Euler-Poincaré reduction Make the variations ξ = δgg 1 and ϖ = gδη. We have δu = ξ + [ξ, u] (EP reduction on the group, cf. Holm, Marsden, Ratiu). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

29 Euler-Poincaré reduction Make the variations ξ = δgg 1 and ϖ = gδη. We have δu = ξ + [ξ, u] (EP reduction on the group, cf. Holm, Marsden, Ratiu). We also have δn = δ(gη) = ϖ + ξn. Using ν = g η, yielding δν = gδ η + ξν and ϖ = gδη, yielding ϖ = uϖ + g η. we get δν = ϖ + ξν uϖ. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

30 Euler-Poincaré reduction Make the variations ξ = δgg 1 and ϖ = gδη. We have δu = ξ + [ξ, u] (EP reduction on the group, cf. Holm, Marsden, Ratiu). We also have δn = δ(gη) = ϖ + ξn. Using ν = g η, yielding δν = gδ η + ξν and ϖ = gδη, yielding ϖ = uϖ + g η. we get δν = ϖ + ξν uϖ. At t = 0, g 0 = id and n 0 = g 0 η 0 = cst so that ξ 0 = 0 and ϖ 0 = 0. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

31 Euler-Poincaré reduction Make the variations ξ = δgg 1 and ϖ = gδη. We have δu = ξ + [ξ, u] (EP reduction on the group, cf. Holm, Marsden, Ratiu). We also have δn = δ(gη) = ϖ + ξn. Using ν = g η, yielding δν = gδ η + ξν and ϖ = gδη, yielding ϖ = uϖ + g η. we get δν = ϖ + ξν uϖ. At t = 0, g 0 = id and n 0 = g 0 η 0 = cst so that ξ 0 = 0 and ϖ 0 = 0. At t = 1, g 1 η 1 = cst so that ξ 1 n 1 + ω 1 = 0. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

32 Write 1 ( ( δl δu ξ ) ( δl ) ( δl )) ad u ξ + δn ϖ + ξn + ϖ + ξν uϖ dt = 0 δν 0 Using integration by parts we get the boundary equation ( ) ( ) δl δl + n 1 = 0. δu δν 1 1 Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

33 Evolution from ξ: δl t δu + δl ad u δu + δl δn n + δl δν ν = 0 Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

34 Evolution from ξ: δl t δu + δl ad u δu + δl δn n + δl δν ν = 0 Evolution from ϖ: δl t δν + u δl δν δl δn = 0. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

35 EPMorph We therefore obtain the system δl t δu + δl ad u δu + δl δn n + δl δν ν = 0 δl t δν + u δl δν δl δn = 0 ) ) ( δl δu 1 + ṅ = ν + un ( δl δν 1 n 1 = 0 Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

36 Special case 1: Riemannian metric Define an invariant Riemannian metric on G N, yielding l(u, n, ν) = (u, ν) 2 n. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

37 Special case 1: Riemannian metric Define an invariant Riemannian metric on G N, yielding l(u, n, ν) = (u, ν) 2 n. For example l(u, n, ν) = u 2 g + ν 2 n, for a given norm,. g, on g and a pre-existing Riemannian structure on N (Trouvé, Y.). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

38 Special case 1: Riemannian metric Define an invariant Riemannian metric on G N, yielding l(u, n, ν) = (u, ν) 2 n. For example l(u, n, ν) = u 2 g + ν 2 n, for a given norm,. g, on g and a pre-existing Riemannian structure on N (Trouvé, Y.). This in turn projects into a Riemannian metric on N. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

39 Special Case 2: Semi-direct product Assume that N is also a group and that g(nñ) = (gn)(gñ) (e.g., N is a vector space and the action of G is linear). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

40 Special Case 2: Semi-direct product Assume that N is also a group and that g(nñ) = (gn)(gñ) (e.g., N is a vector space and the action of G is linear). Consider the semi-direct product G SN with (g, n)( g, ñ) = (g g, (gñ)n) with a right-invariant metric specified by (idg,id N ) at the identity. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

41 Special Case 2: Semi-direct product Assume that N is also a group and that g(nñ) = (gn)(gñ) (e.g., N is a vector space and the action of G is linear). Consider the semi-direct product G SN with (g, n)( g, ñ) = (g g, (gñ)n) with a right-invariant metric specified by (idg,id N ) at the identity. Minimize the geodesic energy between (id G, n 0 ) and (g 1, n 1 ) with fixed n 0 and n 1. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

42 Special Case 2: Semi-direct product Assume that N is also a group and that g(nñ) = (gn)(gñ) (e.g., N is a vector space and the action of G is linear). Consider the semi-direct product G SN with (g, n)( g, ñ) = (g g, (gñ)n) with a right-invariant metric specified by (idg,id N ) at the identity. Minimize the geodesic energy between (id G, n 0 ) and (g 1, n 1 ) with fixed n 0 and n 1. Proposition: This is metamorphosis with reduced Lagrangian l(u, n, ν) = (u, n 1 ν) 2 (id G,id N ). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

43 Note that the evolution equations in this case correspond to the full conservation of momentum on the semi-direct product: ( δl δl ) (t), δu δν (t) ( δl = Ad(g,n) δl ) (t), 1 δu δν (t) Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

44 Note that the evolution equations in this case correspond to the full conservation of momentum on the semi-direct product: ( δl δl ) (t), δu δν (t) ( δl = Ad(g,n) δl ) (t), 1 δu δν (t) When N is a vector space, the Lagrangian does ot depend on n and the equations are δl δu + δl δν n = 0 δl t δν + u δl δν = 0 ṅ = ν + un Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

45 From now on, G is a group of diffeomorphisms over some open subset Ω R d. g is equipped with a Hilbert norm u t g with a continuous inclusion in C p 0 (Ω) (C p vector fields vanishing at infinity) with the supremum norm and p 1. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

46 From now on, G is a group of diffeomorphisms over some open subset Ω R d. g is equipped with a Hilbert norm u t g with a continuous inclusion in C p 0 (Ω) (C p vector fields vanishing at infinity) with the supremum norm and p 1. Under this assumption, the flows associated to time-dependent v.f. that satisfy 1 0 u t 2 gdt < for a subgroup of Diff(Ω) (Trouvé; Dupuis et al.). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

47 Notation We will write the inner product in g under the form u, v m g = ( L g u v ) where L g is the duality operator from g to g. Its inverse, a kernel operator, is denoted K g. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

48 Landmarks and Peakons Take the space of configurations of Q landmarks N = Ω Q. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

49 Landmarks and Peakons Take the space of configurations of Q landmarks N = Ω Q. Elements η, ν N are Q-tuples of points in Ω, with tangent vectors being Q-tuples of d-dimensional vectors. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

50 Landmarks and Peakons Take the space of configurations of Q landmarks N = Ω Q. Elements η, ν N are Q-tuples of points in Ω, with tangent vectors being Q-tuples of d-dimensional vectors. Take l(u, n, ν) = u 2 g + 1 σ 2 Q k=1 ν k 2. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

51 Landmarks and Peakons Take the space of configurations of Q landmarks N = Ω Q. Elements η, ν N are Q-tuples of points in Ω, with tangent vectors being Q-tuples of d-dimensional vectors. Take l(u, n, ν) = u 2 g + 1 σ 2 Q k=1 ν k 2. We have δl/δu = 2L g u, and (δl/δν) = (2/σ 2 )(ν 1,..., ν N ). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

52 Let n = (q 1,..., q N ). We have δl Q δν n = 2 ν k σ 2 δ q k. k=1 Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

53 Let n = (q 1,..., q N ). We have δl Q δν n = 2 ν k σ 2 δ q k. k=1 Notation: if f is a vector field on R d and µ a measure on R d, ( f µ w ) = R d f (x) T w(x)dµ. (1) Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

54 Euler equations for landmark metamorphosis (with p k = ν k /σ 2 ) Q L g u = p k δ qk. k=1 dp k dt + Du(q k) T p k = 0, q k = u(q k ) + σ 2 p k, k = 1,..., Q k = 1,..., Q The case σ 2 = 0 gives the singular solutions to EPDiff. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

55 EPMorph/EPDiff Comparison with two 1-D peakons p= p= p= p= p= p= p= p= p= p= Figure: Head-on collision between two peakons. The plots show the evolution of q = q 2 q 1 over time for several values of p = p 2 p 1, for σ 2 = 0 (left) and σ 2 = 10 4 (right) Rem: Many simulations for peakons with EPdiff, e.g.: Holm and Staley, McLachlan and Marsland... Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

56 Head-on collisions Head-on collision of two peakons; left: EPDIFF; right: metamorphoses Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

57 p= p= p= p= p= p= p= p= p= p= Figure: One peakon overtaking another. The plots show the evolution of q = q 2 q 1 over time for several values of p = p 2 p 1, for σ 2 = 0 (left) and σ 2 = 0.05 (right) Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

58 Back Collision Left: EPDIFF; center: metamorphoses, low difference of energy; right: metamorphoses, high difference of energy. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

59 Image Metamorphosis Let H be a space of smooth functions from Ω to R, with g.n = n g 1. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

60 Image Metamorphosis Let H be a space of smooth functions from Ω to R, with g.n = n g 1. Take l(u, ν) = u 2 g + 1 σ 2 ν 2 L 2. Denote z = ν/σ 2. The Euler equations are L g u = z n ż + div(zu) = 0 ṅ + n T u = σ 2 z Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

61 Remark: H 1 α in 1-D Let m = L g u = (1 2 x )u, and ρ = σz: t m + u x m + 2m x u = ρ x ρ with t ρ + x (ρu) = 0 This is equivalent to the compatibility for dλ/dt = 0 of ( x 2 ψ mλ + ρ2 λ 2) ψ = 0 ( 1 ) t ψ = 2λ + u x ψ ψ xu Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

62 This model of image metamorphoses is a semi-direct product model. The conserved momentum is (L g u, z), yielding the equations L g u t + z t n t = cst and z t = det(dgt 1 )z 0 gt 1. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

63 Examples Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

64 Examples Interpolation between two face images (different persons). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

65 Densities Use the action g.n = det D(g 1 ) n g 1. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

66 Densities Use the action g.n = det D(g 1 ) n g 1. Reduced Lagrangian: l(u, ν) = 1 2 u 2 g + 1 2σ 2 ν 2 L 2. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

67 Densities Use the action g.n = det D(g 1 ) n g 1. Reduced Lagrangian: l(u, ν) = 1 2 u 2 g + 1 2σ 2 ν 2 L 2. The equations are L g u = n z ż + z T u = 0 ṅ + div(nu) = σ 2 z Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

68 The conservation of momentum gives L g u + n z = cst and z = z 0 g 1. The constant in the first equation vanishes for horizontal geodesics in GSN/G. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

69 Comparing measures In many instances, deformable objects are singular and should be compared as such. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

70 Comparing measures In many instances, deformable objects are singular and should be compared as such. The simplest example is the case of unlabelled discrete point sets. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

71 Comparing measures In many instances, deformable objects are singular and should be compared as such. The simplest example is the case of unlabelled discrete point sets. Such structures are conveniently represented as measures, e.g., (x i ) i I i I δ xi. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

72 Comparing measures In many instances, deformable objects are singular and should be compared as such. The simplest example is the case of unlabelled discrete point sets. Such structures are conveniently represented as measures, e.g., (x i ) i I i I δ xi. To apply (Riemannian) metamorphoses to measures, one needs to embed them in a Hilbert space. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

73 Measure Metamorphoses Let H be a Hilbert space of functions over R d and N = H. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

74 Measure Metamorphoses Let H be a Hilbert space of functions over R d and N = H. We assume that H C 0 so that for all x R d, K x H such that, for all f H, Kx, f H = f (x). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

75 Measure Metamorphoses Let H be a Hilbert space of functions over R d and N = H. We assume that H C 0 so that for all x R d, K x H such that, for all f H, Kx, f H = f (x). The kernel K H (x, y) := K x (y) satisfies the equation Kx, K y H = K H(x, y). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

76 K H provides an isometry between N and H via the relation η K H η with K H η, f H = ( η f ). In particular, the dual inner product on N is given by η, η N = ( η K H η ). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

77 Define the action of G Diff on N by ( gη f ) = ( η f g ), with g G, η N and f H. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

78 Define the action of G Diff on N by ( gη f ) = ( η f g ), with g G, η N and f H. For η N and w g, and for all f H, ( wη f ) = ( η f T w ). Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

79 Define the action of G Diff on N by ( gη f ) = ( η f g ), with g G, η N and f H. For η N and w g, and for all f H, ( wη f ) = ( η f T w ). We use the semi-direct product model and define the Lagrangian l(u, ν) = 1 2 u 2 g + 1 2σ 2 ν 2 N. Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

80 E-L Equations Let f = δl/δν = (1/σ 2 )K H ν The E-L Equations are: L g u = f u ḟ t + f T u = 0 ṅ u.n = σ 2 L H f Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

81 Integrated form In integrated form, this yields u = K g ( f n) with ġ t = u t g t. n t = g t n 0 + σ 2 g t t 0 g 1 s K 1 H (f 0 g 1 )ds s Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

82 Initial Value Problem Theorem Under some conditions on H, f 0, n 0, for all T > 0, there exists a unique solution to the previous system over [0, T ] with initial conditions n 0 and f 0. (The conditions essentially imply the loss of one derivative for n 0 and f 0 : if H H p, then f 0 H p+1 and n 0 H p+1.) Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

83 Boundary Value Problem The BVP requires to minimize, with fixed n 0 and n 1 E(u, n) := 1 0 u t 2 gdt σ 2 ṅ t u t n t 2 N dt. Theorem Under some conditions on H, for any given n 0, n 1 H, there exists a minimizer for the BVP. 0 Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43

84 Open issues Numerical schemes for measure metamorphoses. How smooth are geodesics? Extension of VPM methods to EPMorph? Happy Birthday Darryl ( ) Metamorphoses for Pattern Matching July / 43