Metrics on the space of shapes

Transcription

1 Metrics on the space of shapes IPAM, July 4 David Mumford Division of Applied Math Brown University Collaborators Peter Michor Eitan Sharon

2 What is the space of shapes? S = set of all smooth connected plane curves, no self-intersections ( simple closed curves ) Infinite dimensional! Not a vector space BUT, locally linear, i.e. a manifold ψ () s = φ a () s + a (). s φ () s S U = Uφ, φ φ = a ψ smooth { } a ( v.sp.of fcns. a)

3 There are many other spaces of shapes The core space S has many completions, adding non-smooth limits, giving a zoo of Banach manifolds And S has higher dimensional versions: S n = smooth (n-1)-spheres in R n, i.e. bdries of blobs D n R n or even, fixing an ambient manifold N n and submanifold type M m, φ: M N φ an embedding { } Globally, S and S 3 are known to be top. trivial (contractible) but S n for n 7? is not For n =, can contract with the geometric heat equation.

4 The set of ellipses sits in S as a surface: Similarity between shapes can mean many things, A,B,C,D and E are all similar to the central shape in different ways:

5 A Road Map to the many metrics C -curves, the core space ( C-curves) 1 C -curves, w. θ( s) abs.cont. hence curvature κ( s) exists a.e. (can ask κ bnded, κ ds <, κ < ) curves of fin.length 1 w. fin.amount of bending, C -curves, i.e. i.e. θ( s) exists a.e. and is BV θ( s) continuous Michor Riemannian metric ( curves of finite length, ) i.e. rectifiable curves closed integral currents: cntble continuous unions of Lipschitz maps images of circle I R of finite total length "Frechet curves" subsets S R meas. w. fin. area C C, p W p= 1,, 1 1 BV C 1, W? 1,1 W C 1 L

6 Two simple metrics L 1 -metric leading to set of meas. subsets S R : a = a() s ds infinitesimally: C 1 leads to path path { Ct} = a( s, t) ds length: Ct = area swept out leads to global metric: d( S1, S) = area( S1 S) Frechet metric (like Hausdorff metric) on cont. maps f:s 1 R : infinitesimally: a = sup s C a( s) 1 leads to path path{ Ct} = sup a( s, t) dt s C length: t max. dist. moved leads to global metric: d( f, f ) = inf sup f ( x) f ( hx ( )) 1 1 diffeo h i x S Neither metric has good geodesics balls are like boxes, but they stack well, can measure volume (K.Leonard, using ε-entropy)

7 The Michor metric the simplest Riemannian metric Infinitesimally: ( a = a().1 s + Aκ ()) C s ds Globally: C { } If A=, get geodesic spray, positive curvature, but infimum of path lengths is zero If A>, the metric controls C and gives interesting geodesics not always unique. 1 path C a( s, t) ds. dt t = C t

8 A geodesic triangle Consider an ellipse rotated through, 6 and 1 degrees. These 3 ellipses form a triangle in S. Using the metrics with A=1.,.1 and.1, we join them with 3 geodesics. The path in S forming one of these edges is shown in the first row for the 3 metrics. The second row shows the whole triangle of shapes:

9 Axes: the royal road to shape description Humans perceive shapes as having parts, linked in a combinatorial pattern. The axis gives this (and even bit length compression, Leonard 4).

10 Axes in three dimensions Axes in 3D are trickier: Yan Cao s definition: Given a shape S, or even an arbitrary measure m with support S, consider the functional on potential axes: p E( Γ ) = dist( x, Γ ) µ ( dx) + α.length( Γ) An anatomical example: S

11 Shape analysis via complex analysis basic constructions from cx. analysis: D C, φ : D, conformal αz + β up to φ A, A( z) = (called SL ( R)) βz + α 3 surfaces S R, diffeo. to S, φ : C S conformal, up to az + b φ A, A( z) = (called SL ( C)) cz + d Use the first construction: : ( ) φ ( ), Int Int C φ : Ext( ) { } Ext( C) { }, with φ ( ) =, φ ( ) = pos.real

12 Two examples An ellipse and a kidney shaped object, with the conformal parametrization of their interiors and exteriors marked. The interior map has been chosen to carry to, but it may take to any other interior point. The interior and exterior parametrizations can be compared on their common boundary, defining the fingerprint: ψ φ φ ψ 1 ( z) = ( ( z)), up to A, A SL( R)

13 Constructing the fingerprint ψ + ψ 1 Φ + Φ Γ + Γ Γ The fingerprint for the ellipse and the kidney

14 Decoding the fingerprint Minima of ψ correspond (roughly) to points on C nearest to φ (). M A = arg min ( ψ A) def { φ } complex axis( C) = ( A()) # M > 1 A Combinatorial structure of the axis leads to a natural cell decomposition of S.

15 ψ Does the fingerprint determine C? YES!! via welding. Read the diagram backwards: + ψ Φ + Φ First glue on the left via ψ, then use the second basic construction to get the conformal φ s, hence the image curve C. / / Γ + Γ 1 Diff ( S ) SL( R) S transl.+scaling ( H) curves / / 1 Diff ( S ) rotations plus basept. x C Int( C) Γ H

16 S is a homogeneous metric space Welding allows the group Diff(S 1 ) to act on S : 1. Start with C and ψ.. Put angles θ on C via exterior map. 3. Cut open C along C, 4. Reglue with twist ψ (using angles θ). 5. Find new conformal isomorphism with and thus get C ψ. * IMPLEMENTED BY EITAN SHARON * C There is a unique Riemannian metric on Diff(S 1 ) /SL (R), left invariant by Diff(S 1 )! the Weil-Petersen metric. Need norm on tgt. sp. to Diff(S 1 ) at e, in SL (R) directions, invariant by conjugation by SL (R). Tangent space at e = lie algebra of Diff(S 1 ), vector flds to S 1, so: inθ X = ae θ, a = a n n n 3 = ( ) n n X n n a

17 A geodesic: ellipse to square Geodesics are expected always to exist and to be unique, because sectional curvature is negative.