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Computable Elastic Distances Between Shapes LAURENT YOUNES

Computable Elastic Distances Between Shapes LAURENT YOUNES. Presented by Kexue Liu 11/28/00. Overview. Introduction Comparison of plane curves Definition of distances with invariance restrictions The Case of closed curves Experiments. Introduction.

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Computable Elastic Distances Between Shapes LAURENT YOUNES

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  1. Computable Elastic Distances Between ShapesLAURENT YOUNES Presented by Kexue Liu 11/28/00

  2. Overview • Introduction • Comparison of plane curves • Definition of distances with invariance restrictions • The Case of closed curves • Experiments

  3. Introduction • Why need measure distances between shapes? Human eyes can easily figure out the differences and similarities between shapes, but in computer vision and recognition, need some measurements(distance!) • The definition of the distance is crucial. • Base the comparison on the whole outline, considered as a plane curve. • Related to snakes: Active contour model . • Minimal energy required to deform one curve into another! It is formally defined from a left invariant Riemannian distance on an infinite dimensional group acting on the curves, which can be explicitly,easily computed.

  4. IntroductionPrinciples of the approach • From group action cost to deformation cost (distance between two curves) • From distance on Gto group action cost • A suitable distance onG :which corresponds the energy to transform one curve to another.

  5. IntroductionPrinciples of the approach From group action cost to deformation cost • C : the object space, each object in it can be deformed into another. • The deformation is represented by a group action G x C-> C , (a,C)->a.C on C i.e. for all a,b  G, C  C, a.(b.C)=(ab).C, e.C=C • Transitive : forallC1, C2  C, there existaG, s.t a.C1=C2 . • (a,C) : the cost of the transformation C->a.C (1) d(C1, C2 ) = inf (a,C1) , C2 = a.C1, the smallest cost required to deform C1 to C2

  6. IntroductionPrinciples of the approach • Proposition 1:Assume G acts transitively on C and that  is a function defined on G x C , taking values in [0,+[, s.t. i) for all CC, (e,C)=0. ii) for all aG, CC, (a, C)=(a-1,a.C) iii) for all a,bG, CC, a.C1=C2 ,(ab,C)(b,C)+(a,b.C) then d defined by (1) is symmetric, satisfies the triangle inequality, and is such that d(C,C)=0 for C C

  7. IntroductionPrinciples of the approach From distance on Gto group action cost • From elementary group theory, C can be identified to a coset space on G, fix C0 C, H0={h G, h.C0= C0}, C can be identified to G/H0 through the well-defined correspondence a.H0  a.C0 Proposition 2:Let dG be a distance on G. Assume that there exists : G s.t (h)=1 if hH0 and ,for all a, b, c G , (2) dG (ca, cb)= (c)dG (a, b) For C C, with C=b.C0 (3) (a, C)=dG (e, a-1)/(b) then  satisfies i), ii), iii) of proposition 1. The obtained distance is, d(C, C’)= (b)-1inf dG (e, a), aC’= C, where C=b.C0

  8. IntroductionPrinciples of the approach A suitable distance onG . • If a = (a(t), t[0,1]) is such a path subject to suitable regularity conditions, we define the length L(a) and then set dG(a0,a1)=inf{L(a),a(0)=a0,a(1)=a1} the infimum being computed over some set of admissible paths, dG is symmetrical and satisfies the triangle inequality. • If a(t), t[0,1] is admissible, so is a(1-t), t[0,1] and they have the same length. • If a(.) andã(.) are admissible, so is their concatenation, equal to a(2t) fort[0,1/2] and to ã(2t-1) fort[1/2,1] and thelength of the concatenation is the sum of the lengths.

  9. IntroductionPrinciples of the approach • We define the length of the path by the formula • For some norm. Thinking of as a way to represent the portion of path between a(t) and a(t+dt), defining a norm corresponds to defining the cost of a small variation of a(t). We must have dG(a(t),a(t+dt))= (a(t)) dG(e, a(t)-1a(t+dt)) so the problem is to define  and the cost of a small variation from the identity.

  10. Comparison of the plane curvesInfinitesimal deformations Energy Functional • C = {m(s)=(x(s),y(s)), s[0,] } The parametrization is done by arc-length, so we have: V(s)={u(s),v(s) } considered as a vector starting at the point m(s) = { (s)=(x(s)+ u(s),y(s)+ v(s)} The field V (and its derivatives) is infinitely small. We define the energy of this deformation is : which associates s with the arc length in of the point m(s)+V(s)

  11. Comparison of the plane curvesInfinitesimal deformations • At order one, we have: (5) • Denote the angle between tangent to C at point m(s) and the x-axis. Let be the similar function defined for ,we have and(at order one) • Let , it is the normal component of. at order one, we may write

  12. Comparison of the plane curvesInfinitesimal deformations hence, (6) • Set • Set

  13. Comparison of the plane curvesInfinitesimal deformations • Taking the first order, we have: • The functional involves some action on the curve C, • Define , it is a function from [0,1] to the unit circle of (denote it as 1), then we may represent our set of objects as: • (7) C={( ,) , >0, :[0,1]->1,measurable},  is translation and scale invariant.

  14. Comparison of the plane curvesInfinitesimal deformations • The transformation which can naturally be associated to ,g,D: • Define the action: (9) (,g,r).(,)=( ,r. g) where  > 0, g is a diffeomorphism of [0,1] and r is a measurable function, defined on[0,1] and with values in 1. • Let G be the set composed with these 3-uples, it is embedded product (10) (1,g1,r1).(2,g2,r2)=(12,g2 g1,r1.r2g1) with identity eG=(1,Id,1) and inverse (-1,g-1, řg-1), řis the complex conjugate of r .

  15. Comparison of the plane curvesInfinitesimal deformations • If (,g,r) is close to (1,Id,1), then the energy functional is: • a = (,g,r) , we evaluate the cost of a small deformation C->a-1C by • Let C0=(1,1), b=(1/, Id, ), then C = (,)=b.C0 • With (13) (14)

  16. Comparison of the plane curvesRigorous definition of G • Consider Hilbert Space: with norm: • Define (15) • Product (16) • It is well defined on (17)

  17. Comparison of the plane curvesRigorous definition of G • Proposition 3 • Proof: (18) Let the inverse of , both of them are strictly increasing. Let ,then

  18. Comparison of the plane curvesRigorous definition of G thus • For • Denote the mapping • DEFINITION 1: Denote by G the set of 3-uples (,g,r)subject to the conditions: 1) ]0,+[ 2) g is continuous on [0,1] with values in  and such that • g(0)=0, g(1) = 1 • There exist a function q>0, a.e. on [0,1] such that 3) r is measurable, r :[0,1] ->1, 1 is the unit circle in 

  19. Comparison of the plane curvesRigorous definition of G • Proposition 4. • Remarks: Denoting by  the equivalence relation X2=Y2 then we can identity G with the quotient space The consistency of norm on 2 with formula (14). • Y=1+X, a small perturbation of 1 in 2, and assume |X(s)| is small for all s.

  20. Comparison of the plane curvesRigorous definition of G thus is identified for Furthermore, dG (e,a) for a close to e is theinfimum of 2Y-1over all Y s.t (Y) = a, whichis the quotient distance on • If and TXis linear and can be extended to all Y 2 and we have this means if , From (13)

  21. Comparison of the plane curvesAdmissible paths in G • Definition 2: • For all is differentiable in the general sense, and the derivative is • The total energy is finite:

  22. Comparison of the plane curvesAdmissible paths in G • The length of an admissible path is: • If a path is admissible in then it is admissible in • This definition satisfies the natural properties w.r.t. time reversal and concatenation.

  23. Comparison of the plane curvesAdmissible paths in G • Definition 3: A path a(t), t[0,1] is admissible in G iff there exists a path X(t,.), t[0,1] which is admissible in and such that for all t, (X(t,.))=a(t). We now define the length of a path a in G acting on C=(,)(denoted Ll[a]) as 2  times the length of a corresponding path in . • Proposition 5: If two admissible paths in 2,X(t,.) and Y(t,.), satisfy: X(t,.)2 = Y(t,.)2 for all t, then

  24. Comparison of the plane curvesInvariant distance associated with G • Compute the distance between two elements in G as the length of the shortest admissible path in G joining them. • dG(a,b)=2 Inf{X-Y2 , X,YĞ,(X)=a,(Y)=b }, the Inf is over all the shortest paths in Ğ joining X and Y. • Paths of shortest length in 2 are straight lines but if X,Y Ğ , the straight line ttX+(1-t)Y does not necessarily stay within Ğ, however, the length of this straight line is X-Y2 . So we always havedG(a,b)2min{X-Y2 , X,YĞ,(X)=a,(Y)=b }. • Equality is true provided the minimum in the right hand term is attainted for some X, Y such that ttX+(1-t)Y stays in Ğ.

  25. Comparison of the plane curvesInvariant distance associated with G • Because , the min is attained for • Theorem 1: One define a distance on G by

  26. Comparison of the plane curvesInvariant distance associated with G • One defines a distance between two plane curves by the supremum being taken over functions  which are increasing diffeomorphisms of [0,1].

  27. Comparison of the plane curvesInvariant distance associated with G • The following lemma guarantee it is a distance! • Lemma 1: one has

  28. Distances with invariance restrictionsDefinition of invariant • Invariant: A distance d on C is said to be invariant by a group of transformations  acting on C, if for all   , for all C1, C2 ,we have d(C1,C2)= d(C1,C2). • Weakly invariant: if there exists a function  q() s.t. for all , C1, C2 , we have d(C1,C2)= q()d(C1,C2). • Defined module : If for all , C we have d(C,C)= d(C,C). In the literature, the term ‘invariant’ is often used for the last definition.

  29. Distances with invariance restrictionsScale invariance • By using appropriate energy definition and subspace of 2 , we get a scale invariant distant. • Theorem 2: One defines a distance on G by

  30. Distances with invariance restrictionsScale invariance • One defines a scale invariant distance between two plane curves by the infimum being taken over functions  which are strictly increasing diffeomorphisms of [0,1].

  31. Distances with invariance restrictionsModulo translations and scales • Theorem 3: One defines a distance (modulo translations and scales) between two plane curves with normalized angle functions by: the infimum being taken over functions  which are strictly increasing diffeomorphisms of [0,1].

  32. Distances with invariance restrictionsModulo similarities • Rotation invariance, the above distance is rotation invariant but not defined modulo rotation. Since rotations merely translate the angle functions. • Theorem 4: one defines a distance by

  33. Distances with invariance restrictionsModulo similarities • One defines a distance(modulo similarities) between two plane curves with normalized angle functions by:

  34. Case of closed curves • Object set consists of all sufficiently smooth plane curves, including closed curves. So the distance we have used need to be valid for comparing closed curves, but modification is required. • In case of open cures, there are only two choices for the starting points to parameterize the curves. But in closed curve case, the starting point maybe anywhere. • Define u.:[0,1]-> s.t. u.(s)= (s+u). for C1=(1,1) and C2=(2,2), dc(C1,C2) =inf d(u.C1,C2) where inf is over all u.

  35. Experiments • Database composed with eight outlines of airplanes for four types of airplanes. • The shapes have been extracted from 3- dimentional synthesis images under two slightly different view angles for each airplane. • Apply some stochastic noise to the outlines in order to obtain variants of the same shape which look more realistic. • The lengths of the curves have been computed after smoothing(using cubic spline representation).

  36. Experiments

  37. Experiments

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