computable elastic distances between shapes laurent younes

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Computable Elastic Distances Between ShapesLAURENT YOUNES

Presented by Kexue Liu11/28/00

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Overview Introduction Comparison of plane curves Definition of distances with

invariance restrictions The Case of closed curves Experiments

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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.

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IntroductionIntroduction

Principles of the approach

From group action cost to deformation cost (distance between two curves)

From distance on G to group action cost

A suitable distance on G : which corresponds the energy to transform one curve to another.

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IntroductionIntroduction


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 : for all C1, C2 C, there exist a 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

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Introduction Principles 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

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Introduction


From distance on G to 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

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Introduction

Principles of the approach A suitable distance on G . 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(.) and ã(.) are admissible, so is their concatenation, equal to a(2t) for t[0,1/2] and to ã(2t-1) for t[1/2,1] and the length of the concatenation is the sum of the lengths.

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Introduction

Principles 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.

1

0(a) a ( )tL t dt

a ( )t t

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Comparison of the plane curves

Infinitesimal 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)

122 ss yx

c~ m~

dssVVEl

s

2

0

)3( )()(

]~

,0[],0[:* llg s~

c~

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Infinitesimal 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

sssss yvxug 1*

)(* s)~(

~* s c~

ss yx ) sin( , )cos( **

)()1)((~

sin

)()1)((~

cos**

**

ssssssssssss

ssssssssssss

vxuyxyyvxuvyg

vxuyyxyvxuuxg

ssss vxuyD *sV

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Infinitesimal deformations

hence, (6) Set

Set

* * * *

* * * *

cos cos( )

sin sin( )

g D

g D

* * * *s s s sg D y u x v

(3) * 2 * * * 2

0 0( 1) ( ( ) ( ))l l

sE g ds g s s ds * *( ) ( ) / , / , ( ) ( ) and ( ) ( )g s g ls l l l s ls s ls

(3) 2 2

0 0( 1) ( ( ) ( )) (True Equality)l l

sE l g ds l g s s ds

( ) ( ) ( )D s g s s

(3) 2 2

0( , , , ) [( 1) ]

l

sE g D l l g D ds

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Infinitesimal 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.

(3) 2 2 2

0( , , , ) ( 1) [( 1) ]

l

sE g D l l l g D ds

ons. translati toup curvea zescharacteri (.)),( lζ( ) ( ) ( )s ss x s iy s

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Infinitesimal 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, ř g-1), ř is the complex conjugate of r .

1 1( ,θ) ( ,θ ) ( ,θ)l l g D g l

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Infinitesimal 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)

2(3) 2 2

0( , , , ) ( 1) [( 1) 1 ]

with first order approximation 1

l

s

iD

E g D l l l g r ds

e D

21 2 2 2

0( , ) ( 1) [( 1) 1 ]

l

sa C l l g r ds

1 2 2 2( , ) ( , ) / ( )Ga C d e a b

lb /1)( 22 2 2

0( , ) ( 1) [( 1) 1 ]

l

G sd e a g r ds

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Rigorous definition of G Consider Hilbert Space: with norm: Define (15) Product (16) It is well defined on

(17)

2 2([0, 1], )LL C12

2 0: ( )X X s ds

dssXdssXsgsX 21

0

2

0)()()(

)()())(*( sgYsXsYX X

}everywherealmost 0,{~ 2 XXG L

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Rigorous definition of G Proposition 3 Proof: (18)

Let the inverse of , both of them are strictly increasing.

Let ,then

* operation for the groupa is ~G

dsYdsXdsYX21

0

21

0

21

0*

h is Xg

dsvXsh

dudvvXuXuXdvvhXshs

21

0

21

0

22)(

0

2

0

)()(

)()()()(

)/(1: hXY duuXshdvvYs 21

0

2

0)()()(

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Rigorous 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

1** and XYYXgh Y

2221

0,)(,

~XXrdssXGX XX

),,(: XXX rgX

dqsgs

)()( 2

0

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Rigorous 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.

sm.homomorphi groupa is ~

: GG

22X iff )()( YYX

/~G

21

0

221

0

2Y

22

1

0

21

0

41)(1)-(

that so),(211 and

)(2)(211),(21

Xr

XYYr

XXYX

Y

Y

YY

g

g

Ys

Ys

J

RRR

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Rigorous definition of G

thus is identified for Furthermore, dG (e,a) for a close to e is the infimum of

2Y-1over all Y s.t (Y) = a, which is the quotient distance

on If and TX is linear and can be extended to all Y 2 and we

have

this means if , From (13)

))(,( YedG 212 to1 YY

/~G

.~

on lationleft trans thebe *:let ,~

GYXYTGX X

21

0

2221

0

2

2)( YdsYdsYXYT XXX

X g

YXYbXa with)(),(

22

11 2*12),()(),(

and )(

YXYXbaedabad

a

XGG

X

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Admissible paths in G Definition 2:

For all is differentiable in the general sense, and the derivative is

The total energy is finite:

A path ( ( ,.), [0,1]) is said to admissible in ( ( ,.)

for all ) if there exists a path, denoted ( ( ,.), [0,1]),s.t.t

X t t X t

t X t t

2 2L L

dssstXt )(),( functionscalar the,1

0 2L

1

0 ( , ) ( )tt X t s s ds

1 1 2

0 0( , )tX t s dsdt

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Admissible 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.

1/ 21 1 2

0 0( ) ( , )tL X X t s ds dt

tGstX allfor ~

)),((t and 2LG~

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Admissible 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

G~

G~

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Invariant 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 have dG(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.

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Invariant distance associated with G Because , the min is attained for

Theorem 1: One define a distance on G by

)(21

0

2

2YXYX YX R

( ) 0XY R

)( b ), ( a for

2

~cos22),(

~

2/11

0

)3(

ΔiiΔ

ssG

,h,eλ,g,e

dshgbad

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Invariant 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].

1/ 21(3)

0

( ) ( )( , ) 2 sup cos

2

for C ( , ), C ( , ).

G s

i i

g s sd C C l l ll g ds

l e l e

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Invariant distance associated with G The following lemma guarantee it is a distance!

Lemma 1: one has ~

and ~

0)~

,()3( llCCd

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Distances with invariance restrictions

Definition 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.

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Scale 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 1/ 22

12(2)

0( , ) log log 4 arccos cos

2

for a ( ), b ( )

G s s

iΔ iΔ

d a b g h ds

λ,g,e ,h,e

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Scale 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].

1/ 2212(2)

0

( ) ( )( , ) log log 4 Inf arccos ( ) cos

2

for C ( , ), C ( , ).

G s

i i

g s sd C C l l g s ds

l e l e

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Modulo 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].

1(1)

0

( ) ( )( , ) 2Inf arccos ( ) cos

2

for C ( , ), C ( , ).

G s

i i

g s sd C C g s ds

l e l e

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Modulo 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

1(0)

0( , ) 2 min arccos cos

2

for a ( ), b ( ), the min is over ]- , ]

G s s

iΔ iΔ

cd a b g h ds

g,e h,e c

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Modulo similarities One defines a distance(modulo similarities) between

two plane curves with normalized angle functions by:

1(0)

0

( ) ( )( , ) 2Inf min arccos ( ) cos

2

for C ( , ), C ( , ),with inf over ,and min over c ]- , ].

G s

i i

g s s cd C C g s ds

l e l e g

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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.

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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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Experiments

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Experiments

computable elastic distances between shapes laurent younes

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