Invariants via Moving Frames: Computation and Applications Irina - PowerPoint PPT Presentation

Invariants via Moving Frames: Computation and Applications Irina Kogan North Carolina State University ∗ DART, October 27–30, 2010, Beijing, China ∗ This work was supported in part by NSF grant CCF-0728801 1

Outline: • Definitions and examples of invariants • Applications: – congruence problem for curves; – symmetry reduction of variational problems; • Structure theorems • Computation via moving frames (classical, generalized, inductive and algebraic methods) 2

Group actions and invariants: 3

Group actions An action of a group G on a set Z is a map Φ: G × Z − → Z such that i. Φ( e, z ) = z , ∀ z ∈ Z . ii. Φ ( g 1 , Φ( g 2 , z )) = Φ( g 1 g 2 , z ) , ∀ z ∈ Z and ∀ g 1 , g 2 ∈ G . Example: Let M ( n, K ) = { n × n matrices over a field K } . A group G L ( n, K ) = { A ∈ M ( n, K ) | det( A ) � = 0 } acts on K n by: Φ( A, z ) = A z , ∀ A ∈ G L ( n, K ) and z ∈ K n . Notation: G � Z and Φ( g, z ) = g · z . 4

We will consider • G – smooth Lie group or algebraic group over a field K • Z – smooth manifold or algebraic variety • Φ – smooth map or polynomial or rational map A local action of a topological group G on a topological set Z is a map Φ: Ω − → Z defined on some open subset Ω ⊂ G × Z containing e × Z , such that i. Φ( e, z ) = z , ∀ z ∈ Z . ii. Φ ( g 1 , Φ( g 2 , z )) = Φ( g 1 g 2 , z ) , ∀ g 1 , g 2 , z such that ( g 2 , z ) ∈ Ω and ( g 1 g 2 , z ) ∈ Ω . 4

Invariants: A function F on Z is invariant under G � Z if F ( g · z ) = F ( z ) , ∀ z ∈ Z and ∀ g ∈ G . A function F , defined on an open subset U of a topological set Z , is locally invariant under G � Z if F ( g · z ) = F ( z ) , ∀ ( g, z ) ∈ Ω . for some open subset Ω ⊂ G × Z such that e × U ⊂ Ω . 5

Invariants under rotations on R 2 : SO (2 , R ) � R 2 by rotations Invariants y • Any smooth invariant on R 2 − { (0 , 0) } is functions of � x 2 + y 2 . Oz r = z • Any polynomial invariant on R 2 is functions of r 2 = x 2 + y 2 . x Orbits are level sets of r . 6

Invariants under rotations and translations on R 2 : Action: SE (2 , R ) = SO (2 , R ) ⋉ R 2 � R 2 by rotations and translations. R 2 is a single orbit. Invariants: constant functions. 7

Differential invariants for planar curves γ ( t ) = ( x ( t ) , y ( t )) under rotations and translations SE (2 , R ) -action on R 2 induces an action on x ( t ) , y ( t ) , ˙ x ( t ) , ˙ y ( t ) , . . . (jet bundle of curves in R 2 ). y ds , dy � dx � • Unit tangent: = T , N ds | T | = 1 ⇒ Infinitesimal arc-length: ds = P � x 2 + ˙ y 2 dt ˙ T • Unit normal: N ⊥ T , | N | = 1 . enet equation: dT • The Fr´ ds = κN x • Generators of the differential algebra of invariants: κ and d ds , where d 1 d √ ds = dt is an invariant differential operator. x 2 + ˙ y 2 ˙ • Fundamental local diff. invariants: κ, κ s = dκ ds, κ ss , . . . 8

An integral invariant for planar curves γ ( t ) = ( x ( t ) , y ( t )) , t ∈ [ a, b ] Notation: X ( t ) = x ( t ) − x ( a ) , Y ( t ) = y ( t ) − y ( a ) , a Y ( τ ) dX ( τ ) − 1 I [0 , 1] ( t ) = � t 2 X ( t ) Y ( t ) I [0 , 1] represnets the signed area between the curve and a secant. It is invariant under SA (2 , R ) ⊃ SE (2 , R ) action. 9

An discrete invariants for quadratic forms The standard action of G L ( n, C ) on C n induces an action on the space V n d of homogeneous polynomials of degree d in n variables: � A − 1 x � , ∀ A ∈ G L ( n, C ) and x ∈ C n . A · P ( x ) = P There are well known canonical forms for G L ( n, C ) � V n 2 : x 2 1 + · · · + x 2 k , for k = 0 , . . . n. k is a discrete invariant for G L ( n, C ) � V n 2 . 10

Types of the invariants: • local smooth; • polynomial, rational, and algebraic; • differential; • integral; • integro-differential; • discrete; • . . . 11

Applications: • Equivalence (congruence) problems for – sub-manifolds (in particular curves and surfaces) – for polynomials – differential equations – . . . • Symmetry reduction of – differential equations – variational problems – algebraic equations • Invariant geometric flows • . . . 12

Equivalence problem for curves 13

curves Equivalence problem for curves in R n . • Problem: Given an action of a group G on R n and curves γ 1 : [ a, b ] → R n and γ 2 : [ c, d ] − → R n decide whether there exists g ∈ G such − that Image ( γ 1 ) = g · Image ( γ 2 ) . • If such g ∈ G exists then γ 1 and γ 2 are called G -equivalent, or G -congruent: γ 1 ∼ = γ 2 . 14

Transformations on R 2 commonly appearing in computer image processing: • Special Euclidean (orientation preserving rigid motions): X = cos( φ ) x − sin( φ ) y + a, Y = sin( φ ) x + cos( φ ) y + b. • Euclidean (rigid motions): X = cos( φ ) x − sin( φ ) y + a, Y = ǫ (sin( φ ) x + cos( φ ) y ) + b ǫ = ± 1 • similarity X = λ (cos( φ ) x − sin( φ ) y ) + a , Y = ǫλ (sin( φ ) x + cos( φ ) y ) + b , ǫ = ± 1 , λ � = 0 . • equi-affine (area and orientation preserving): X = α x + β y + a, Y = γ x + δ y + b, αδ − βγ = 1 • affine: X = α x + β y + a, Y = γ x + δ y + b αδ − β γ � = 0 � α β a � • projective: X = α x + β y + a ν x + µ y + c , Y = γ x + δ y + b ν x + µ y + c , det γ δ b � = 0 ν µ c 15

Euclidean and equi-affine frame Euclidean geometry in R 2 Equi-affine geometry in R 2 SE (2 , R ) = SO (2 , R ) ⋉ R 2 SA (2 , R ) = SL (2 , R ) ⋉ R 2 Moving Frame: y y N N ~ N ~ N P P ~ ~ P P T T ~ ~ T T x x ds , dy dα , dy � dx � � dx � N = dT T = , N ⊥ T, | N | = 1 T = , ds dα dα Infinitesimal arc-length: � det | TN | = 1 ⇒ dα = y 1 / 3 1 + y 2 | T | = 1 ⇒ ds = x dx xx dx Fundamental differential invariants: dT dN ds = κN dα = µT ⇓ ⇓ µ α = dµ κ s = dκ ds , κ ss , . . . dα , µ αα , . . . . 16

curves Differential invariants for planar curves Let G be an r -dim’l Lie group acting on the plane. For almost all actions ∃ • a local differential invariant ξ ( G -curvature) of differential order r − 1 ; • an G -arclength) invariants differential form ̟ (infinitesimal of differential order at most r − 2 and the dual invariant differential operator D ̟ . s.t. any other local differential invariant on an open subset of the jet space J ( R 2 , 1) is a smooth function of ξ, D ̟ ξ, D 2 ̟ ξ, . . . 17

curves Relations between invariants of a group and its subgroup ∗ � • special Eucl.: κ = (¨ y ˙ x − ¨ x ˙ y ) x 2 + ˙ y 2 dt, d 1 d √ , ds = ˙ d s = 3 d t x 2 + ˙ y 2 ˙ x 2 + ˙ y 2 ) ( ˙ 2 • equi-affine: µ = 3 κ ( κ ss +3 κ 3 ) − 5 κ 2 d 1 d , dα = κ 1 / 3 ds, s d α = d s 9 κ 8 / 3 κ 1 / 3 • projective: η = 6 µ ααα µ α − 7 µ 2 αα − 9 µ 2 , dρ = µ 1 / 3 α µ dα, d 1 d d ρ = d α . α 6 µ 8 / 3 µ 1 / 3 α α Definition: Curves for which G -curvature or G -arclength are undefined are called G -exceptional. ∗ see (Kogan 2001, 2003) for a general method of deriving invariants of a group in terms of invariants of its subgroup 18

curves Congruence criteria for curves with specified initial point • Theorem: Two non G -exceptional curves are G -congruent iff their G - curvatures as functions of G -arclength coincide. → R 2 and γ 2 ( τ ) , τ ∈ [ c, d ] − → R 2 : For γ 1 ( t ) , t ∈ [ a, b ] − ∃ g ∈ G s. t. g · γ 1 ( a ) = γ 2 ( c ) and Image ( γ 1 ) = g · Image ( γ 2 ) � � t � τ ξ | γ 1 ( s 1 ) = ξ | γ 2 ( s 2 ) , where s 1 ( t ) = a ̟ | γ 1 and s 2 ( τ ) = ̟ | γ 2 c • Applicable only if: – initial point is specified – arc-length reparametrization is feasible in practice 19

G -curvature under reparametrization (¨ y ˙ x − ¨ x ˙ y ) Euclidean example: κ = : 2 x 2 + ˙ y 2 ) ( ˙ 3 γ ( τ ) = ( √ τ, cos √ τ ) , τ ∈ [0 , π 2 ] γ ( t ) = ( t, cos t ) , t ∈ [0 , π ] ˜ cos( √ τ ) cos( t ) κ | γ ( φ ( τ )) = − κ | ˜ γ ( τ ) = − (1+sin 2 ( √ τ )) 3 / 2 (1+sin 2 ( t )) 3 / 2 γ ( τ ) where t = φ ( τ ) = √ τ. κ | γ ( φ ( τ )) = κ | ¯ 20

curves Differential signature for planar curves (Calabi et al. (1998)) • Let ξ be G -curvature, ̟ -infinitesimal G -arclength and ξ ̟ = D ̟ ξ . • Definition: The G -signature of a non-exceptional curve γ ( t ) = � � x ( t ) , y ( t ) ∈ [ a, b ] , t is the image of a parametric curve � � ξ | γ ( t ) , ξ ̟ | γ ( t ) : � � S γ ( t ) = { ξ | γ ( t ) , ξ ̟ | γ ( t ) | t ∈ [ a, b ] } . • G -congruence criterion for non-exceptional curves ∼ γ 1 = γ 2 ⇓ ⇑ under certain conditions S γ 1 = S γ 2 21

curves Example 1 of Euclidean differential signature: √ √ γ ( t ) = ( 3 5 t − 4 5 cos t, 4 5 t + 3 γ ( t ) = ( t, cos t ) , ˜ 5 cos t ) , t ∈ [0 , π 2 ] t ∈ [0 , π ] Signatures ( κ 2 , κ 2 γ in R 2 Images of γ and ˜ s ) for γ and ˜ γ 22

Example 2 of Euclidean differential signature: γ ( t ) = ( t, cos t ) , t ∈ [0 , π ] ˜ γ ( t ) = ( t, cos t ) , t ∈ [0 , 2 π ] γ in R 2 Signatures ( κ 2 , κ 2 Images of γ and ˜ s ) for γ and ˜ γ Images of signatures of γ and ˜ γ coincide due to reflection symmetry of ˜ γ Signature for γ is traced 2 times when t ∈ [0 , π ] due to symmetry under rotations by π around the point ( π 2 , 0) . Signature for ˜ γ is traced 4 times when t ∈ [0 , 2 π ] ! 23