Moving Frames in Applications

Transcription

1 Moving Frames in Applications Peter J. Olver University of Minnesota olver Madrid, September 2009

2 Moving Frames Classical contributions: M. Bartels ( 1800), J. Serret, J. Frénet, G. Darboux, É. Cotton, Élie Cartan Modern developments: (1970 s) S.S. Chern, M. Green, P. Griffiths, G. Jensen, T. Ivey, J. Landsberg,... The equivariant approach: (1997 ) PJO, M. Fels, G. Marí Beffa, I. Kogan, J. Cheh, J. Pohjanpelto, P. Kim, M. Boutin, D. Lewis, E. Mansfield, E. Hubert, E. Shemyakova, O. Morozov, R. McLenaghan, R. Smirnov, J. Yue, A. Nikitin, J. Patera,...

3 Moving Frame Space Curves tangent normal binormal t = dz n = d2 z ds ds 2 b = t n s arc length t b n Frénet Serret equations dt ds = κ n dn ds = κ t + τ b κ curvature τ torsion db ds = τ n

4 Moving Frame Space Curves tangent normal binormal t = dz n = d2 z ds ds 2 b = t n s arc length z t b n Frénet Serret equations dt ds = κ n dn ds = κ t + τ b κ curvature τ torsion db ds = τ n

5 I did not quite understand how he [Cartan] does this in general, though in the examples he gives the procedure is clear. Nevertheless, I must admit I found the book, like most of Cartan s papers, hard reading. Hermann Weyl Cartan on groups and differential geometry Bull. Amer. Math. Soc. 44 (1938)

6 Applications of Moving Frames Differential geometry Equivalence Symmetry Differential invariants Rigidity Identities and syzygies Joint invariants and semi-differential invariants Invariant differential forms and tensors Integral invariants

7 Classical invariant theory Computer vision object recognition symmetry detection Invariant variational problems Invariant numerical methods Mechanics, including DNA Poisson geometry & solitons Killing tensors in relativity Invariants of Lie algebras in quantum mechanics Control theory Lie pseudo-groups

8 The Basic Equivalence Problem M smooth m-dimensional manifold. G transformation group acting on M finite-dimensional Lie group infinite-dimensional Lie pseudo-group

9 Equivalence: Determine when two p-dimensional submanifolds are congruent: N and N M N = g N for g G Symmetry: Find all symmetries, i.e., self-equivalences or self-congruences: N = g N

10 Classical Geometry F. Klein Euclidean group: G = SE(m) = SO(m) R m E(m) = O(m) R m z A z + b A SO(m) or O(m), b R m, z R m isometries: rotations, translations, (reflections) Equi-affine group: G = SA(m) = SL(m) R m A SL(m) volume-preserving Affine group: A GL(m) G = A(m) = GL(m) R m Projective group: G = PSL(m + 1) acting on R m RP m = Applications in computer vision

11 Tennis, Anyone?

12 Classical Invariant Theory Binary form: Q(x) = n k=0 ( ) n a k k x k Equivalence of polynomials (binary forms): Q(x) = (γx + δ) n Q ( ) αx + β γx + δ g = ( α β γ δ ) GL(2) multiplier representation of GL(2) modular forms

13 Q(x) = (γx + δ) n Q ( ) αx + β γx + δ Transformation group: g : (x, u) ( αx + β γx + δ, ) u (γx + δ) n Equivalence of functions equivalence of graphs Γ Q = { (x, u) = (x, Q(x)) } C 2

14 Moving Frames Definition. A moving frame is a G-equivariant map ρ : M G Equivariance: ρ(g z) = g ρ(z) ρ(z) g 1 left moving frame right moving frame ρ left (z) = ρ right (z) 1

15 The Main Result Theorem. A moving frame exists in a neighborhood of a point z M if and only if G acts freely and regularly near z.

16 Isotropy & Freeness Isotropy subgroup: G z = { g g z = z } for z M free the only group element g G which fixes one point z M is the identity: = G z = {e} for all z M. locally free the orbits all have the same dimension as G: = G z is a discrete subgroup of G. regular all orbits have the same dimension and intersect sufficiently small coordinate charts only once irrational flow on the torus effective the only group element which fixes every point in M is the identity: g z = z for all z M iff g = e: G M = \ z M G z = {e}

17 Geometric Construction z O z Normalization = choice of cross-section to the group orbits

18 Geometric Construction z O z k K Normalization = choice of cross-section to the group orbits

19 Geometric Construction z O z g = ρ left (z) k K Normalization = choice of cross-section to the group orbits

20 Geometric Construction z O z g = ρ right (z) k K Normalization = choice of cross-section to the group orbits

21 K cross-section to the group orbits O z orbit through z M k K O z unique point in the intersection k is the canonical or normal form of z the (nonconstant) coordinates of k are the fundamental invariants g G unique group element mapping k to z ρ(z) = g left moving frame ρ(h z) = h ρ(z) k = ρ 1 (z) z = ρ right (z) z = freeness

22 Algebraic Construction Coordinate cross-section r = dim G m = dim M K = { z 1 = c 1,..., z r = c r } left w(g, z) = g 1 z right w(g, z) = g z g = (g 1,..., g r ) group parameters z = (z 1,..., z m ) coordinates on M

23 Choose r = dim G components to normalize: w 1 (g, z)= c 1... w r (g, z)= c r Solve for the group parameters g = (g 1,..., g r ) = Implicit Function Theorem The solution g = ρ(z) is a (local) moving frame.

24 The Fundamental Invariants Substituting the moving frame formulae g = ρ(z) into the unnormalized components of w(g, z) produces the fundamental invariants I 1 (z) = w r+1 (ρ(z), z)... I m r (z) = w m (ρ(z), z) Theorem. Every invariant I(z) can be (locally) uniquely written as a function of the fundamental invariants: I(z) = H(I 1 (z),..., I m r (z))

25 Invariantization Definition. The invariantization of a function F : M R with respect to a right moving frame g = ρ(z) is the the invariant function I = ι(f ) defined by I(z) = F (ρ(z) z). ι(z 1 ) = c 1,... ι(z r ) = c r, cross-section variables phantom invariants ι(z r+1 ) = I 1 (z),... ι(z m ) = I m r (z). fundamental invariants ι [ F (z 1,..., z m ) ] = F (c 1,..., c r, I 1 (z),..., I m r (z))

26 Invariantization amounts to restricting F to the crosssection I K = F K and then requiring that I = ι(f ) be constant along the orbits. In particular, if I(z) is an invariant, then ι(i) = I. Invariantization defines a canonical projection ι : functions invariants

27 Prolongation Most interesting group actions (Euclidean, affine, projective, etc.) are not free! Freeness typically fails because the dimension of the underlying manifold is not large enough, i.e., m < r = dim G. Thus, to make the action free, we must increase the dimension of the space via some natural prolongation procedure. An effective action can usually be made free by:

28 Prolonging to derivatives (jet space) G (n) : J n (M, p) J n (M, p) = differential invariants Prolonging to Cartesian product actions G n : M M M M = joint invariants Prolonging to multi-space G (n) : M (n) M (n) = joint or semi-differential invariants = invariant numerical approximations

29 Prolonging to derivatives (jet space) G (n) : J n (M, p) J n (M, p) = differential invariants Prolonging to Cartesian product actions G n : M M M M = joint invariants Prolonging to multi-space G (n) : M (n) M (n) = joint or semi-differential invariants = invariant numerical approximations

30 Euclidean Plane Curves Special Euclidean group: G = SE(2) = SO(2) R 2 acts on M = R 2 via rigid motions: w = R z + b To obtain the classical (left) moving frame we invert the group transformations: y = cos φ (x a) + sin φ (u b) v = sin φ (x a) + cos φ (u b) w = R 1 (z b) Assume for simplicity the curve is (locally) a graph: C = {u = f(x)} = extensions to parametrized curves are straightforward

31 Prolong the action to J n via implicit differentiation: y = cos φ (x a) + sin φ (u b) v = sin φ (x a) + cos φ (u b) v y = sin φ + u x cos φ cos φ + u x sin φ u v yy = xx (cos φ + u x sin φ ) 3 v yyy = (cos φ + u x sin φ )u xxx 3u2 xx sin φ (cos φ + u x sin φ ) 5.

32 Prolong the action to J n via implicit differentiation: y = cos φ (x a) + sin φ (u b) v = sin φ (x a) + cos φ (u b) v y = sin φ + u x cos φ cos φ + u x sin φ u v yy = xx (cos φ + u x sin φ ) 3 v yyy = (cos φ + u x sin φ )u xxx 3u2 xx sin φ (cos φ + u x sin φ ) 5.

33 Normalization: r = dim G = 3 y = cos φ (x a) + sin φ (u b) = 0 v = sin φ (x a) + cos φ (u b) = 0 v y = sin φ + u x cos φ cos φ + u x sin φ = 0 v yy = u xx (cos φ + u x sin φ ) 3 v yyy = (cos φ + u x sin φ )u xxx 3u2 xx sin φ (cos φ + u x sin φ ) 5.

34 Solve for the group parameters: y = cos φ (x a) + sin φ (u b) = 0 v = sin φ (x a) + cos φ (u b) = 0 v y = sin φ + u x cos φ cos φ + u x sin φ = 0 = Left moving frame ρ : J 1 SE(2) a = x b = u φ = tan 1 u x

35 a = x b = u φ = tan 1 u x Differential invariants u v yy = xx (cos φ + u x sin φ ) 3 κ = u xx (1 + u 2 x )3/2 v yyy = v yyyy = dκ ds = (1 + u2 x )u xxx 3u x u2 xx (1 + u 2 x )3 d2 κ ds 2 3κ3 = Contact invariant one-form arc length dy = (cos φ + u x sin φ) dx ds = 1 + u 2 x dx

36 Dual invariant differential operator arc length derivative d dy = 1 cos φ + u x sin φ d dx d ds = u 2 x d dx Theorem. All differential invariants are functions of the derivatives of curvature with respect to arc length: dκ κ, ds, d 2 κ ds 2,

37 The Classical Picture: n t z Moving frame R = ( 1 1 ux 1 + u 2 u x x 1 ρ : (x, u, u x ) (R, a) SE(2) ) = ( t, n ) a = ( ) x u

38 Equi-affine Curves G = SA(2) z A z + b A SL(2), b R 2 Invert for left moving frame: y = δ (x a) β (u b) v = γ (x a) + α (u b) α δ β γ = 1 w = A 1 (z b) Prolong to J 3 via implicit differentiation dy = (δ β u x ) dx D y = 1 δ β u x D x

39 Prolongation: y = δ (x a) β (u b) v = γ (x a) + α (u b) v y = γ α u x δ β u x u v yy = xx (δ β u x ) 3 v yyy = (δ β u x ) u xxx + 3 β u2 xx (δ β u x ) 5 v yyyy = u xxxx (δ β u x ) β (δ β u x ) u xx u xxx + 15 β 2 u 3 xx (δ β u x ) 7 v yyyyy =...

40 Normalization: r = dim G = 5 y = δ (x a) β (u b) = 0 v = γ (x a) + α (u b) = 0 v y = γ α u x δ β u x = 0 v yy = u xx (δ β u x ) 3 = 1 v yyy = (δ β u x ) u xxx + 3 β u2 xx (δ β u x ) 5 = 0 v yyyy = u xxxx (δ β u x ) β (δ β u x ) u xx u xxx + 15 β 2 u 3 xx (δ β u x ) 7 v yyyyy =...

41 Equi-affine Moving Frame ρ : (x, u, u x, u xx, u xxx ) (A, b) SA(2) A = ( α β γ δ b = ) ( ) a b = = ( 3 uxx 1 3 u 5/3 xx u xxx 3 u x uxx u 1/3 xx 1 3 u 5/3 xx u xxx ) ( x u ) Nondegeneracy condition: u xx 0.

42 Equi-affine arc length dy = (δ β u x ) dx ds = 3 uxx dx Equi-affine curvature v yyyy κ = 5 u xx u xxxx 3 u2 xxx 9 u 8/3 xx v yyyyy dκ ds v yyyyyy d2 κ ds 2 5κ2

43 Equivalence & Invariants Equivalent submanifolds N N must have the same invariants: I = I. Constant invariants provide immediate information: e.g. κ = 2 κ = 2 Non-constant invariants are not useful in isolation, because an equivalence map can drastically alter the dependence on the submanifold parameters: e.g. κ = x 3 versus κ = sinh x

44 Syzygies However, a functional dependency or syzygy among the invariants is intrinsic: e.g. κ s = κ 3 1 κ s = κ 3 1 Universal syzygies Gauss Codazzi Distinguishing syzygies.

45 Equivalence & Syzygies Theorem. (Cartan) Two smooth submanifolds are (locally) equivalent if and only if they have identical syzygies among all their differential invariants. Proof : Cartan s technique of the graph: Construct the graph of the equivalence map as the solution to a (Frobenius) integrable differential system, which can be integrated by solving ordinary differential equations.

46 Finiteness of Generators and Syzygies There are, in general, an infinite number of differential invariants and hence an infinite number of syzygies must be compared to establish equivalence. But the higher order syzygies are all consequences of a finite number of low order syzygies!

47 Example Plane Curves Thus, for Euclidean (or equi-affine or projective or... ) plane curves we need only know a single syzygy between κ and κ s in order to establish equivalence! If non-constant, both κ and κ s depend on a single parameter, and so, locally, are subject to a syzygy: κ s = H(κ) ( ) But then κ ss = d ds H(κ) = H (κ) κ s = H (κ) H(κ) and similarly for κ sss, etc. Consequently, all the higher order syzygies are generated by the fundamental first order syzygy ( ).

48 The Signature Map The generating syzygies are encoded by the signature map Σ : N S of the submanifold N, which is parametrized by the fundamental differential invariants: The image Σ(x) = (I 1 (x),..., I m (x)) S = Im Σ is the signature subset (or submanifold) of N.

49 Equivalence & Signature Theorem. equivalent Two smooth submanifolds are N = g N if and only if their signatures are identical S = S

50 Signature Curves Definition. The signature curve S R 2 of a curve C R 2 is parametrized by the two lowest order differential invariants S = { ( κ, dκ ds ) } R 2

51 Other Signatures Euclidean space curves: C R 3 S = { ( κ, κ s, τ ) } R 3 Euclidean surfaces: S R 3 (generic) S = { ( H, K, H,1, H,2, K,1, K,2 ) } κ curvature, τ torsion R 3 H mean curvature, K Gauss curvature Equi affine surfaces: S R 3 (generic) S = { ( P, P,1, P,2, P,11 ) } R 3 P Pick invariant

52 Equivalence & Signature Curves Theorem. Two smooth curves C and C are equivalent: C = g C if and only if their signature curves are identical: S = S = object recognition

53 Symmetry and Signature = Only totally singular submanifolds can have larger symmetry groups! Theorem. The dimension of the symmetry group G N = { g g N N } of a nonsingular submanifold N M equals the codimension of its signature: dim G N = dim N dim S Corollary. For a nonsingular submanifold N M, 0 dim G N dim N

54 Maximally Symmetric Submanifolds Theorem. The following are equivalent: The submanifold N has a p-dimensional symmetry group The signature S degenerates to a point: dim S = 0 The submanifold has all constant differential invariants N = H {z 0 } is the orbit of a p-dimensional subgroup H G = Euclidean geometry: circles, lines, helices, spheres, cylinders, planes,.. = Equi-affine plane geometry: conic sections. = Projective plane geometry: W curves (Lie & Klein)

55 Discrete Symmetries Definition. The index of a submanifold N equals the number of points in N which map to a generic point of its signature: ι N = min { # Σ 1 {w} } w S = Self intersections Theorem. The cardinality of the symmetry group of a submanifold N equals its index ι N. = Approximate symmetries

56 The Index Σ N S

57 The polar curve r = cos 3θ The Original Curve Euclidean Signature Numerical Signature

58 The Curve x = cos t cos2 t, y = sin t sin2 t The Original Curve Euclidean Signature Affine Signature

59 The Curve x = cos t cos2 t, y = 1 2 x + sin t sin2 t The Original Curve Euclidean Signature Affine Signature

60 Canine Left Ventricle Signature Original Canine Heart MRI Image Boundary of Left Ventricle

61 Smoothed Ventricle Signature

62

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65 Advantages of the Signature Curve Purely local no ambiguities Symmetries and approximate symmetries Extends to surfaces and higher dimensional submanifolds Occlusions and reconstruction Main disadvantage: Noise sensitivity due to dependence on high order derivatives.

66 Noise Reduction Strategy Use lower order invariants to construct a signature: joint invariants joint differential invariants integral invariants topological invariants...

67 Classical Invariant Theory M = R 2 \ {u = 0} G = GL(2) = { ( α β γ δ ) = α δ β γ 0 } (x, u) ( αx + β γx + δ, ) u (γx + δ) n n 0, 1

68 Prolongation: y = α x + β γ x + δ v = σ n u v y = σ u x n γ u σ n 1 v yy = σ2 u xx 2(n 1)γ σ u x + n(n 1)γ 2 u 2 σ n 2 v yyy = σ = γ x + δ = α δ β γ

69 Normalization: y = α x + β γ x + δ = 0 v = σ n u = 1 v y = σ u x n γ u σ n 1 = 0 σ = γ x + δ = α δ β γ v yy = σ2 u xx 2(n 1) γ σ u x + n(n 1)γ 2 u 2 σ n 2 = v yyy = 1 n(n 1)

70 Moving frame: α = u (1 n)/n H γ = 1 n u(1 n)/n Hessian: β = x u (1 n)/n H δ = u 1/n 1 n xu(1 n)/n H = n(n 1)u u xx (n 1) 2 u 2 x 0 Note: H 0 if and only if Q(x) = (a x + b) n = Totally singular forms Differential invariants: J v yyy n 2 (n 1) κ v yyyy K + 3(n 2) n 3 (n 1) dκ ds

71 Absolute rational covariants: J 2 = T 2 H 3 K = U H 2 H = 1 2 (Q, Q)(2) = n(n 1)QQ (n 1) 2 Q 2 Q xx Q yy Q 2 xy T = (Q, H) (1) = (2n 4)Q H nqh Q x H y Q y H x U = (Q, T ) (1) = (3n 6)Q T nqt Q x T y Q y T x deg Q = n deg H = 2n 4 deg T = 3n 6 deg U = 4n 8

72 Signatures of Binary Forms Signature curve of a nonsingular binary form Q(x): S Q = { (J(x) 2, K(x)) = ( T (x) 2 H(x) 3, U(x) H(x) 2 )} Nonsingular: H(x) 0 and (J (x), K (x)) 0. Signature map: Σ : Γ Q S Q Σ(x) = (J(x) 2, K(x)) Theorem. Two nonsingular binary forms are equivalent if and only if their signature curves are identical.

73 Maximally Symmetric Binary Forms Theorem. If u = Q(x) is a polynomial, then the following are equivalent: Q(x) admits a one-parameter symmetry group T 2 is a constant multiple of H 3 Q(x) x k is complex-equivalent to a monomial the signature curve degenerates to a single point all the (absolute) differential invariants of Q are constant the graph of Q coincides with the orbit of a one-parameter subgroup

74 Symmetries of Binary Forms Theorem. The symmetry group of a nonzero binary form Q(x) 0 of degree n is: A two-parameter group if and only if H 0 if and only if Q is equivalent to a constant. = totally singular A one-parameter group if and only if H 0 and T 2 = c H 3 if and only if Q is complex-equivalent to a monomial x k, with k 0, n. = maximally symmetric In all other cases, a finite group whose cardinality equals the index of the signature curve, and is bounded by ι Q 6n 12 U = ch 2 4n 8 otherwise

75 Joint Invariants A joint invariant is an invariant of the k-fold Cartesian product action of G on M M: I(g z 1,..., g z k ) = I(z 1,..., z k ) A joint differential invariant or semi-differential invariant is an invariant depending on the derivatives at several points z 1,..., z k N on the submanifold: I(g z (n) 1,..., g z (n) k ) = I(z(n) 1,..., z (n) k )

76 Joint Euclidean Invariants Theorem. Every joint Euclidean invariant is a function of the interpoint distances d(z i, z j ) = z i z j z j z i

77 Joint Equi Affine Invariants Theorem. Every planar joint equi affine invariant is a function of the triangular areas [ i j k ] = 1 2 (z i z j ) (z i z k ) z j z i z k

78 Joint Projective Invariants Theorem. Every joint projective invariant is a function of the planar cross-ratios [ z i, z j, z k, z l, z m ] = A B C D A D C B

79 Three point projective joint differential invariant tangent triangle ratio: [ ] [ ] [ ] [ ] [ ] [ ] z 0 z 1 z 0 z 1 z 2 z 2

80 Joint Euclidean Signature z 0 a z 1 b e c d z 3 f z 2

81 Joint signature map: Σ : C 4 S R 6 a = z 0 z 1 b = z 0 z 2 c = z 0 z 3 d = z 1 z 2 e = z 1 z 3 f = z 2 z 3 Syzygies: = six functions of four variables Φ 1 (a, b, c, d, e, f) = 0 Φ 2 (a, b, c, d, e, f) = 0 Universal Cayley Menger syzygy C R 2 det 2a 2 a 2 + b 2 d 2 a 2 + c 2 e 2 a 2 + b 2 d 2 2b 2 b 2 + c 2 f 2 a 2 + c 2 e 2 b 2 + c 2 f 2 2c 2 = 0

82 Joint Equi Affine Signature Requires 7 triangular areas: [ ], [ ], [ ], [ ], [ ], [ ], [ ] z 1 z 2 z 0 z 5 z 3 z 4

83 Joint Invariant Signatures The joint invariant signature subsumes other signatures, but resides in a higher dimensional space and contains a lot of redundant information. Identification of landmarks can significantly reduce the redundancies (Boutin) It includes the differential invariant signature and semidifferential invariant signatures as its coalescent boundaries. Invariant numerical approximations to differential invariants and semi-differential invariants are constructed (using moving frames) near these coalescent boundaries.

84 Statistical Sampling Idea: Replace high dimensional joint invariant signatures by increasingly dense point clouds obtained by multiply sampling the original submanifold. The equivalence problem requires direct comparison of signature point clouds. Continuous symmetry detection relies on determining the underlying dimension of the signature point clouds. Discrete symmetry detection relies on determining densities of the signature point clouds.

85 Symmetry Preserving Numerical Methods Invariant numerical approximations to differential invariants. Invariantization of numerical integration methods. = Structure-preserving algorithms

86 Numerical approximation to curvature B A a c b C Heron s formula κ(a, B, C) = 4 abc = 4 s(s a)(s b)(s c) s = a + b + c 2 abc semi-perimeter

87 Invariantization of Numerical Schemes = Pilwon Kim, Martin Welk Suppose we are given a numerical scheme for integrating a differential equation, e.g., a Runge Kutta Method for ordinary differential equations, or the Crank Nicolson method for parabolic partial differential equations. If G is a symmetry group of the differential equation, then one can use an appropriate moving frame to invariantize the numerical scheme, leading to an invariant numerical scheme that preserves the symmetry group. In challenging regimes, the resulting invariantized numerical scheme can, with an inspired choice of moving frame, perform significantly better than its progenitor.

88 The group G acts on M the space of independent and dependent variables, and hence on the joint space M (n+1) = { z i z j for all i j } M (n+1) = M M All finite difference numerical schemes are prescribed by suitable functions F : M (n+1) R An invariantized scheme is simply obtained by invariantizing the functions I = ι(f ), which is, thus, automatically a function of the joint invariants of G.

89 The moving frame and hence the induced invariantization process depends on the choice of cross-section K M (n+1) to the group orbits. Intelligent choice of cross-section requires some understanding of the error terms in the numerical schemes. For ordinary differential equations, if the scheme is of order n in the underlying step size h, so that the local truncation error term is O(h n+1 ) one may, in very favorable circumstances, be able to choose the cross-section to eliminate all the lowest order error terms and thereby produce an invariantized scheme of order n + 1.

90 A more common scenario is that, because of the large number of terms forming the local error, one cannot eliminate all the low order terms, but one may be able to eliminate a large fraction of them by choice of cross-section. For instance, many terms in the error formula depend on u x, and so a good choice of cross-section would be to set (among other normalizations) u x = 0 and thereby eliminate all such terms in the error. The result is a scheme of the same order, but one in which the error tends be be less. And in many case, the result is a much better scheme.

91 The choice of cross-section is crucial. In some cases, we have used adaptively invariantized schemes, where the crosssection depends on the local behavior of the solution, to facilitate the construction & improve the accuracy. An invariantized scheme is easy to program! One merely replaces all stored quantities by their invariantizations and then runs the original non-invariant program on the invariantized data.

92 Invariantized Runge Kutta Schemes u xx + x u x (x + 1)u = sin x, u(0) = u x (0) = 1. Symmetry group: (x, u) (x, u + λ e x )

93 Invariantization for Driven Oscillator u xx + u = sin x α, α =.99 (x, u) (x, u + λ cos x + µ sin x)

94 Comparison of symmetry reduction and invariantization u xx + x u x (x + 1)u = sin x, u(0) = u x (0) = 1.

95 Adaptive Invariantization of Crank Nicolson for Burgers Equation (t, x, u) u t = ε u xx + u u x (t, x + λ t, u λ) ( Galilean boost t 1 µ t, x, (1 µ t)u µ x 1 µ t ) Non-invariant Invariantized.

96 Mathematical Morphology u(x) B/W image Dilation : Erosion : u S(x) = max y S u S(x) = min y S u(x + y) u(x + y) e.g. S = disk

97 Morphological PDEs Hamilton Jacobi partial differential equation: u t = ± u Symmetry Group: u ϕ(u) Here, we focus on the one-parameter subgroup u λu 1 + (λ 1)u

98 Upwind scheme: Invariantization of 1D Morphology u k+1 i u t = u x = u k i + t x max{uk i+1 uk i, uk i 1 uk i, 0} original theoretical upwind without invariantization upwind with invariantization D dilation of a single peak, 20 iterations, t = x = 0.5, without and with invariantization.

99 Invariantization of 2D Morphology Non-invariant upwind scheme: Invariantized upwind scheme:

100 Dilation with invariantized upwind scheme