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Computable Elastic Distances Between Shapes LAURENT YOUNES

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Overview

- Introduction
- Comparison of plane curves
- Definition of distances with invariance restrictions
- The Case of closed curves
- Experiments

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.

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.

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 existaG, 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

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 CC, (e,C)=0.

ii) for all aG, CC, (a, C)=(a-1,a.C)

iii) for all a,bG, CC, 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

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 hH0 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

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.

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.

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)

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

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.

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)=(12,g2 g1,r1.r2g1)

with identity eG=(1,Id,1) and inverse (-1,g-1, řg-1), řis the complex conjugate of r .

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)

Comparison of the plane curvesRigorous definition of G

- Consider Hilbert Space:

with norm:

- Define

(15)

- Product

(16)

- It is well defined on

(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

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

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.

Comparison of the plane curvesRigorous definition of G

thus is identified for

Furthermore, dG (e,a) for a close to e is theinfimum of

2Y-1over 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)

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:

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.

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

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-Y2 , X,YĞ,(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 ttX+(1-t)Y does not necessarily stay within Ğ, however, the length of this straight line is X-Y2 . So we always havedG(a,b)2min{X-Y2 , X,YĞ,(X)=a,(Y)=b }.
- Equality is true provided the minimum in the right hand term is attainted for some X, Y such that ttX+(1-t)Y stays in Ğ.

Comparison of the plane curvesInvariant distance associated with G

- Because , the min is attained for
- Theorem 1: One define a distance on G by

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].

Comparison of the plane curvesInvariant distance associated with G

- The following lemma guarantee it is a distance!
- Lemma 1: one has

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.

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

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].

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].

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

Distances with invariance restrictionsModulo similarities

- One defines a distance(modulo similarities) between two plane curves with normalized angle functions by:

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.

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).

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