Geodesic Warps by Conformal Mappingsrmclachl/VISI2316.pdf · 2012. 7. 26. · metric geometry of the conformal embeddings, similar to what was done for metrics on planar curves by

Geodesic Warps by Conformal Mappings

Stephen Marsland!1, Robert I McLachlan!!2, Klas Modin! ! !2, and MatthewPerlmutter†2

1 School of Engineering and Advanced Technology (SEAT)Massey University, Palmerston North, New Zealand

2 Institute of Fundamental Sciences (IFS)Massey University, Palmerston North, New Zealand

Abstract. In recent years there has been considerable interest in meth-ods for di!eomorphic warping of images, with applications e.g. in medicalimaging and evolutionary biology. The original work generally cited isthat of the evolutionary biologist D’Arcy Wentworth Thompson, whodemonstrated warps to deform images of one species into another. How-ever, unlike the deformations in modern methods, which are drawn fromthe full set of di!eomorphism, he deliberately chose lower-dimensionalsets of transformations, such as planar conformal mappings.In this paper we study warps of such conformal mappings. The approachis to equip the infinite dimensional manifold of conformal embeddingswith a Riemannian metric, and then use the corresponding geodesic equa-tion in order to obtain di!eomorphic warps. After deriving the geodesicequation, a numerical discretisation method is developed. Several exam-ples of geodesic warps are then given. We also show that the equationadmits totally geodesic solutions corresponding to scaling and transla-tion, but not to a"ne transformations.

1 Introduction

The use of di!eomorphic transformations in both image registration and shapeanalysis is now common and utilised in many machine vision and image analysistasks. One image or shape is brought into alignment with another by deformingthe image until some similarity measure, such as sum-of-squares distance betweenpixels in the two images, reaches a minimum. The deformation is computed asa geodesic curve with respect to some metric on the di!eomorphism group.For a general treatment and an overview of the subject see the monograph byYounes [1] and references therein.

The standard approach to the deformation method is to first perform ana"ne registration (principally to remove translation and rotation), and then toseek a geodesic warp of the image in the full set of di!eomorphisms of a fixed

! [email protected]!! [email protected]

! ! ! [email protected]† [email protected]

!"#$%&'()*!"#$%&'()(&*+&,+-."+/,&0/.12$)#3*4&$+.5+)6/"7*(8&!"#$%&'()(&*+&,#(-&"#.%(/&0(1()(.$(2

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2 Marsland, McLachlan, Modin, Perlmutter

domain. Typically the setting is to use the right-invariant H1" metric, which

leads to the so-called EPDi! equation (see e.g. [2]). However, in what is arguablythe most influential demonstration of the application of warping methods – theevolutionary biologist D’Arcy Wentworth Thompson’s seminal book ‘On Growthand Form’ [3] – Thompson warps images of one biological species into anotherusing relatively simple types of transformations, so that the gross features of thetwo images match. In a recent review of his work, biologist Arthur Wallace says:

“This theory cries out for causal explanation, which is something the greatman eschewed. [. . .] His transformations suggest coordinated rather than piece-meal changes to development in the course of evolution, an issue which almostcompletely disappeared from view in the era of the ‘modern synthesis’ of evolu-tionary theory, but which is of central importance again in the era of evo-devo.[. . .] All the tools are now in place to examine the mechanistic basis of trans-formations. Not only do we have phylogenetic systematics and evo-devo, but,so obvious that it is easy to forget, we have computers, and especially, in thiscontext, advanced computer graphics. We owe it to the great man to put thesethree things together to investigate the mechanisms that produce the morpholog-ical changes that he captured so elegantly with little more than sheets of graphpaper and, of course, a brilliant mind.” [4]

Figure no. in [3] Transformation group

515 x !" ax, y !" y513.2 x !" ax, y !" by509, 510, 518 x !" ax, y !" cx+ dy (shears)521–22, 513.5 x !" ax+ by, y !" cx+ dy (a"ne)506, 508 x !" ax, y !" g(y)511 x !" f(x), y !" g(y)517–20, 523, 513.1, 513.3, 513.4, 513.6, 514, 525 conformal524 ‘peculiar’

Table 1. Transformation groups used in some transformations in Chapter XII,‘On the Theory of Transformations, or the Comparison of Related Forms’, of [3].

We draw attention to two key aspects of Thompson’s examples: (i) the trans-formations are as simple as possible to achieve what he considers a good enoughmatch (see Table 1); and (ii) the classes of transformations that he considers allforms groups (or pseudogroups), either finite or infinite dimensional. Mostly, heuses conformal transformations, a constraint he is reluctant to give up:

“It is true that, in a mathematical sense, it is not a perfectly satisfactoryor perfectly regular deformation, for the system is no longer isogonal; but [. . .]approaches to an isogonal system under certain conditions of friction or con-straint.” [3, p. 1064]

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Geodesic Warps by Conformal Mappings 3

“ [. . .] it will perhaps be noticed that the correspondence is not always quite ac-curate in small details. It could easily have been made much more accurate by giv-ing a slightly sinuous curvature to certain of the coordinates. But as they stand,the correspondence indicated is very close, and the simplicity of the figures illus-trates all the better the general character of the transformation.” [ibid., p. 1074]

For applications in image registration we therefore suggest to vary the groupof transformations from which warps are drawn. If a low dimensional group givesa close match, then it should be preferred over a similar match from a higher-dimensional group. If necessary, local deformations from the full di!eomorphismgroup can be added later to account for fine details. In this paper we considerthe case of conformal transformations. More precisely, we consider the problemof formulating and solving a geodesic equation on the space of conformal map-pings. This is a fundamental sub-task in the framework of large deformationdi!eomorphic metric mapping (LDDMM) [5, 6, 7, 8, 9, 10, 11, 12], which isthe standard setup for di!eomorphic image registration. Based on the geodesicequation derived in this paper, the full conformal image registration problemwill be considered in future work.

Although the composition of two conformal maps is conformal, it need notbe invertible: we need to restrict the domain. The invertible conformal mapsfrom the disk to itself do form a group, the disk-preserving Mobius group, but itis only 3 dimensional. We are therefore led to consider the infinite dimensionalconfiguration space Con(U,R2) of conformal embeddings of a simply connectedcompact domain U ! R2 into the plane. This set is not a group, but it is apseudogroup.

In [13] the authors study a geodesic equation using an L2–metric on theinfinite dimensional manifold of conformal embeddings and a numerical methodis developed for the initial value problem, based on the reproducing Bergmankernel. Using numerical examples, it is shown that the geodesic equation is ill-conditioned as an initial value problem, and that cusps are developed in finitetime which leads to a break-down of the dynamics.

In this paper we continue the study of geodesics on the manifold of confor-mal embeddings, but now with respect to a more general class of Sobolev typeH1

" metrics. Furthermore, we develop a new numerical algorithm for solving theequations. The new method is based on a discrete variational principle, and di-rectly solves the two point boundary value problem. Our new numerical methodbehaves well, i.e., converges fast, for all the examples we tried as long as thedistance between the initial and final point on the manifold is not too large.From this observation we expect that the initial value problem with ! > 0 iswell-posed in the Hs Banach space topology, which would imply that the Rie-mann exponential is a local di!eomorphism (see [14, 15, 16]). This question willbe investigated in detail in future work, as it is out of the scope of the currentpaper. The experimental results in this paper (Section 6) are limiting to con-firming that the discrete Lagrangian method can reproduce the known geodesicsconsisting of linear conformal maps and can also calculate non-linear geodesicswith moderately large deformations. This is a first step towards exploring the

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metric geometry of the conformal embeddings, similar to what was done formetrics on planar curves by Michor and Mumford [17].

For analysis of 2D shapes, a setting using conformal mappings is devel-oped in [18]. There it is shown that the space of planar shapes is isomorphicto the quotient space Di!(S1)/PSL(2,R), where PSL(2,R) acts on Di!(S1)by right composition of its corresponding disk preserving Mobius transforma-tion restricted to S1. Furthermore it is shown that there is a natural metric onDi!(S1)/PSL(2,R), the Weil-Peterson metric, which has non-positive sectionalcurvature. The setting in this paper is related but di!erent: rather than study-ing planar shapes we study conformal transformations between planar domainsand we think of the manifold of conformal embeddings as a submanifold of thefull space of planar embeddings. The equation we obtain can be seen as a gen-eralisation and a restriction of the EPDi! equation. First, a generalisation bygoing from the group of di!eomorphisms of a fixed domain to the manifold ofembeddings from a planar domain into the entire plane.3 4 Second, a restrictionby restricting to conformal embeddings. The approach we take is similar to (andmuch influenced by) the recent paper [19], in which a geometric framework formoving boundary continuum equations in physics is developed.

2 Mathematical Setting and Choice of Metric

The linear space of smooth maps U " R2 is denoted C!(U,R2). Recall thatthis space is a Frechet space, i.e., it has a topology defined by a countableset of semi-norms (see [20, Sect. I.1] for details on the Frechet topology used).The full set of embeddings U " R2, denoted Emb(U,R2), is a open subset ofC!(U,R2). In particular, this implies that Emb(U,R2) is a Frechet manifold(see [20, Sect. I.4.1]).

Since U ! R2 it holds that Emb(U,R2) contains the identity mapping on U,which we denote by Id. The tangent space TIdEmb(U,R2) at the identity isgiven by the smooth vector fields on U, which we denote by X(U). Notice thatthe vector fields need not be tangential to the boundary "U . Also notice thatX(U) is a Frechet Lie algebra with bracket given by minus the Lie derivative onvector fields, i.e., if #, $ # X(U), then ad#($) = $£#$.

Let g = dx%dx+dy%dy be the standard Euclidean metric on R2. Considerthe subspace of C!(U,R2) consisting of maps that preserve the metric up tomultiplication with elements in the space F(U) of smooth real valued functionson U. That is, the subspace

C!c (U,R2) = {% # C!(U,R2);%"g = Fg, F # F(U)}.

3 This generalisation of EPDi! has not yet been worked out in detail in the literature.However, it is likely that the approach developed in [19] for free boundary flow canbe used with only minor modifications.

4 One can also look at the generalisation of EPDi! to embeddings from a Kleingeometry perspective. Indeed, let Di!U(R2) denote the di!eomorphisms that leavesthe domain U invariant. Then the embeddings Emb(U,R2) can be identified withthe space of co-sets Di!(R2)/Di!U(R2).

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This subspace is topologically closed in C!(U,R2). The set of conformal embed-dings Con(U,R2) = Emb(U,R2) & C!

c (U,R2) is an open subset of C!c (U,R2)

and a Frechet submanifold of Emb(U,R2). The tangent space TIdCon(U,R2) isgiven by

Xc(U) = {# # X(U);£#g = div(#)g},which follows by straightforward calculations. Notice that Xc(U) is a subalgebraof X(U), since

££!$g = £#£$g $£$£#g = £#(div($)g)$£$(div(#)g)

=!£# div($)$£$ div(#)

"g + div($)£#g $ div(#)£$g# $% &

0

= div(£#$)g.

In the forthcoming, we identify the plane R2 with the complex numbers Cthrough (x, y) '" z = x + iy. Hence, the vector fields X(U) are identified withsmooth complex valued functions on U, and Xc(U) with the space of holomorphicfunctions.

The complex L2 inner product on X(U) is given by

((#, $))L2(U) :=

'

U#(z)$(z) dA(z),

where dA = dx * dy is the canonical volume form on R2. Correspondingly, wealso have the real L2 inner product given by

(#, $)L2(U) :=

'

Ug(#, $)dA = Re((#, $))L2(U).

Also, we have the more general class of real and complex H1" inner products

given by

(#, $)H1"(U) := (#, $)L2(U) +

!

2(#x, $x)L2(U) +

!

2(#y, $y)L2(U),

((#, $))H1"(U) := ((#, $))L2(U) +

!

2((#x, $x))L2(U) +

!

2((#y, $y))L2(U),

where ! + 0 and #x, #y respectively denotes derivatives with respect to theCartesian coordinates (x, y). Notice that if #, $ # Xc(U), then

((#, $))H1"(U) = ((#, $))L2(U) + !((##, $#))L2(U)

where ## and $# denote complex derivatives.The class of inner products ((·, ·))H1

"(U) on TIdEmb(U,R2) = X(U) induces acorresponding class of Riemannian metrics on Emb(U,R2) by

T%Emb(U,R2), T%Emb(U,R2) - (U, V ) '" (U . &$1, V . &$1)H1"(%(U)). (1)

Note that &$1 is well-defined as a map &(U) " U since & is an embedding.Also note that the restriction of the metric (1) to the submanifold of di!eomor-phisms Di!(U) ! Emb(U,R2) yields to the “ordinary” H1

" metric on Di!(U)corresponding to the EPDi! equations.

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3 Derivation of the Geodesic Equation

In this section we derive the geodesic equation on Con(U,R2) for the class ofH1

"(U) metrics given by (1). These equation are given by the Euler-Lagrangeequations with respect to the quadratic Lagrangian on Con(U,R2) given by

L(&, &) =1

2(& . &$1, & . &$1)H1

"(%(U)) =1

2((& . &$1, & . &$1))H1

"(%(U)), (2)

where & # T%Emb(U,R2) corresponds to the time derivative.As a first step, we have the following result.

Lemma 1. For any (&, &) # TEmb(U,R2) it holds that

((& . &$1, & . &$1))H1"(%(U)) = ((&#&,&#&))L2(U) + !((&#, &#))L2(U).

Proof. Let f be any complex valued function on U. By a change of variablesw = &(z) we obtain

'

%(U)f . &$1(w)dA(w) =

'

Uf(z)|&#(z)|2dA(z).

For the first term we take f(z) = &(z)&(z) which yields

((& . &$1, & . &$1))L2(%(U)) =

'

U&(z)&(z)|&#(z)|2dA(z) = ((&#&,&#&))L2(U).

For the second term we take first notice that

(& . &$1)#(w) =&# . &$1(w)

&# . &$1(w)

and then we take f(z) = &#(z)/&#(z). The result now follows. /0

We are now ready to derive the Euler-Lagrange equation from the variationalprinciple. Indeed, we look for at curve & : [0, 1] " Con(U,R2) such that

d

d'

(((&=0

' 1

0

1

2((&&(t) . &&(t)

$1, &&(t) . &&(t)$1))H1

"(%(U)) dt = 0

for all variations && : [0, 1] " Con(U,R2) such that &&(0) = &(0), &&(1) = &(1)and &0 = &. To simplify notation we introduce

( =d

d'

(((&=0

&&.

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Notice that ((0) = ((1) = 0. Using Lemma 1 and the fact that di!erentiationcommutes with integration we obtain

0 =

' 1

0

)((&#&,

d

d'

(((&&#&&&))L2(U) + !((&#,

d

d'

(((&&#&))L2(U)

*dt

=

' 1

0

)((&#&,(#&+ &#())L2(U) + !((&#, (#))L2(U)

*dt

=

' 1

0

)((&#&,(#&$ &#( +

d

dt(&#()))L2(U) $ !((&#,(#))L2(U)

*dt

=

' 1

0

)$ (( d

dt(&#&),&#())L2(U) + ((&#&,(#&$ &#())L2(U) $ !((&#,(#))L2(U)

*dt,

where in the last two equalities we use integration by parts over the time variableand the fact that ( vanishes at the endpoints. Notice that there are now no timederivatives on (. Thus, by the fundamental lemma of calculus of variations wecan remove the time integration and thereby obtain a weak equation which mustbe fulfilled at each point in time. In order to obtain a strong formulation, weneed also to isolate ( from spatial derivatives. The standard approach of usingintegration by parts introduces a boundary integral term. In most examplesof calculus of variations, this boundary term either vanishes (in the case of aspace of tangential vector fields), or it can be treated separately giving rise tonatural boundary conditions (in the case of a space where vector fields can havearbitrary small compact support). However, in the case of conformal mappings,there is always a global dependence between interior points, and points on theboundary (since holomorphic functions cannot have local support). Hence, weneed an appropriate analogue of integration by parts which avoids boundaryintegrals. For this, consider the adjoint operator of complex di!erentiation, i.e.,an operator "%

z : Xc(U) " Xc(U) such that

((#, $#))L2(U) = (("%z #, $))L2(U), 1 #, $ # Xc(U).

Notice that "%z depends on the domain U. In the case of the unit disk U = D,

it holds that "%z #(z) = "z(z2#(z)) = 2z#(z) + z2##(z). In the general case, this

operator is more complicated, but can still be computed under the assumptionthat a conformal embedding U " D is known (see [21]).

Using the operator "%z we can now proceed as follows

0 = $(( ddt

(&#&),&#())L2(U) + ((&#&,(#&$ &#())L2(U) $ !((&#,(#))L2(U)

= $((|&#|2&+ &&#&#,())L2(U) $ ((&&#&#,())L2(U)

+ ((&#&, ((&)# $ (&#))L2(U) $ !(("%z &#,())L2(U)

= $((|&#|2&+ &&#&# + &&#&# + !"%z &#,())L2(U) + (("%

z (&#&),(&))L2(U)

= $(((|&#|2 + !"%z "z)&+ &(&#&# + &#&#)$ &"%

z (&#&),())L2(U).

Thus, this relation must hold for all holomorphic functions (. However, theexpression in the first slot of the inner product is not holomorphic, so it needs

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to be orthogonally projected back to the set of holomorphic functions. UsingHodge theory for manifolds with boundary, one can show that the orthogonalcomplement of Xc(U) in X(U) with respect to the real inner product (·, ·)L2(U)

is given by

Xc(U)& = {# # X(U); #(z) = "xF $ i"yF + "yG+ i"xG, F,G # F0(U)},

where F0(U) = {F # F(U);F |'U = 0} are the smooth functions that vanish atthe boundary. This result is obtained in [21]. Since Xc(U)& is invariant undermultiplication with i, it is also the orthogonal complement with respect to thecomplex L2 inner product.

Now we finally arrive at the strong formulation of the Euler-Lagrange equa-tions

d

dt

!A(&)&

"$ &"%

z (&#&) = A(&)!"xF $ i"yF + "yG+ i"xG

",

"z& = 0,

F |'U = G|'U = 0,

(3)

where A(&) = |&#|2 +!"%z "z is the inertia operator (self adjoint with respect to

the L2 inner product) and where the second to last equation means that & is con-strained to be holomorphic. Indeed, one may think of equation (3) as a Lagrange-D’Alembert equation for a system with configuration space Emb(U,R2) which,by Lagrangian multipliers (F,G), is constrained to the submanifold Con(U,R2).

In the special case U = D we get

d

dt

!A(&)&

"$ &"z(z

2&#&) = A(&)!"xF $ i"yF + "yG+ i"xG

",

"z& = 0,

F |'U = G|'U = 0.

(4)

3.1 Weak Geodesic Equation in the Right Reduced Variable

It is also possible to derive the geodesic equation using the right reduced variable# = &.&$1, as is typically done for geodesic equations on di!eomorphism groupswith invariant metric (see e.g. [22, 23, 24]). However, there is a di!erence betweenthe setting of embeddings and that of di!eomorphism groups, since # is definedon &(U), which is not fixed in the embedding setting. Nevertheless, the “movingdomain” ) = &(U) simply moves along the flow, i.e., points on the boundaryfollows the flow of the vector field #. For details of this setting in the two casesof unconstrained embeddings and volume preserving embeddings, see [19].

Let *& be a variation of * as above, and let #& = *& . *$1& . Using the calculus

of Lie derivatives, direct calculations yield

d

d'

(((&=0

#& = $ +£$#

d

d'

(((&=0

1

2(#, #)H1

"((#(U)) = (#,£$#)L2(((U)) + !(##,£$##)L2(((U))+

(#, div($)#)L2(((U)) + !(##, div($)##)L2(((U))

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where $ = dd&

((&=0

&& . &$1. From the variational principle

d

d'

(((&=0

' 1

0L(&&, &&) dt =

d

d'

(((&=0

' 1

0

1

2(#&, #&)H1

"((#(U)) dt = 0

we now obtain the weak form of the equation in terms of the variables (&, #) as

' 1

0

)(#, $ + 2£$# + div($)#)L2(((U))+

!(##, $# + "z£$# +£$## + div($)##)L2(((U))

*dt = 0.

Passing now to the complex inner product, we use the formulas

((#, $))L2(U) = (#, $)L2(U) + i(#, i$)L2(U)

g(#, div($)#) + ig(#, div(i$)#) = 2#$##

£i$# = i£$#

which yields the weak equation

' 1

0

)((#, $ + 2£$# + 2$##))L2(((U))

+ !((##, $# + "z($##) + 2£$#

#))L2(((U))

*dt = 0. (5)

Together with the equation & = #.&, this is thus a weak form of the equation (3),but expressed in the variables (&, #).

4 Totally Geodesic Submanifolds

In this section we investigate special solutions to equation (4). The approachfor doing so is to find a finite dimensional submanifold of Con(D,R2) such thatsolutions curves starting and ending on this submanifold actually lie on thesubmanifold.

Recall that a submanifold N ! M of a Riemannian manifold (M, g) is totallygeodesic with respect to (M, g) if geodesics in N (with respect to g restrictedto N) are also geodesics in M . For a thorough treatment of totally geodesicsubgroups of Di!(M) with respect to various metrics, see [24].

Consider now the submanifold of linear conformal transformations

Lin(D,R2) =+& # Con(D,R2);&(z) = cz, c # C

,.

Proposition 1. Lin(D,C) is totally geodesic in Con(D,C) with respect to theH1

" metric given by (1).

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Proof. If t '" &(t) is a path in Lin(D,C), i.e., &(z) = cz with c # C, then# = & . &$1 is of the form #(z) = az with a # C. Now, let t '" (&, #) fulfill thevariational equation (5) for each variation of the form $(z) = bz with b # C. Weneed to show that t '" (&, #) then fulfills the equation for any variation of theform $(z) = zk (since the monomials span the space of holomorphic functions).Thus, for the first term in (5) we get

((#, $ + 2£$# + 2$##))L2(%(D)) = ((#, $ + 4$## $ 2##$))L2(%(D))= ((&# · # . &),&# · ($ + 4$## $ 2##$) . &))L2(D)

= |c|4((az, bzk + 4kbazk $ 2abzk))L2(D).

(6)

where in the first line we use the conformal change of variables formula forintegrals. Now, since the monomials are orthogonal with respect to ((·, ·))D theexpression vanish whenever k 2= 1. For the second term, notice that ## = a isconstant. Now, if $ is constant, then all the terms $#, $##,£$## vanish. If $ = zk

with k + 2, then the second term !((##, $# + "z(#$#) + 2£$##)) is of the form

!

'

cDa(b+ 3ab)kzk$1 dA(z) = !|c|2a(b+ 3ab)

'

Dkzk$1

which vanishes for every a, c1 # C. This concludes the proof. /0

We now derive a di!erential equation for the totally geodesic solutions inLin(D,C) in term of the variables (c, a) corresponding to &(z) = cz and #(z) =az. From & = # .& it follows that c = ac. Next, we plug the ansatz into the weakequation (5), and use that b vanish at the endpoints, which yields the equations

c = ac

a(2c+ !) = $4a2c$ !a2(7)

These equations thus gives special solutions to equation (4). We obtain thatd2

dt2 (c2 + !c) = 0, so we can analytically compute these special solutions. Notice

that if a and c are initially real, then both a and c stay real, so the even smallersubmanifold of pure scalings is also totally geodesic.

Fig. 1 gives a visualization of total geodesic solutions where the two endtransformations are given first by a pure scaling and second by a pure rotation.Notice that within the submanifold Lin(D,C), the smaller submanifold of scalingsis totally geodesic, as is shown in the left figure. However, the submanifold ofrotations is not, as is shown in the right figure.

Remark 1. By using again the weak form (5) of the governing equation onecan further show that the submanifold of translations is not totally geodesic inCon(D,C). Nor is the submanifold of a"ne conformal transformations.

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t

yx

c = 0.2 c = e1.6"i

Fig. 1. Geodesic curve from &0(z) = z to &1(z) = cz for di!erent values of cand ! = 0. The mesh lines show how the unit circle evolves. Notice that thescaling geodesic stays a scaling (left figure), whereas the rotation geodesic picksup some scaling during its time evolution (right figure).

5 Numerical Discretization

In this section we describe a method for numerical discretization of the equa-tions (4). The basic idea is to obtain a spatial discretization of the phase spacevariables (&, &) # TCon(D,R2) by truncation of the Taylor series. Thus, we usea Galerkin type approach for spatial discretization. For time discretization wetake a variational approach, using the framework of discrete mechanics (cf. [25]).

The discrete configuration space is given by

Qn = {& # Con(D,R2);&(z) =n$1-

i=0

cizi, ci # C}.

Notice thatQn is an n–dimensional submanifold of Con(D,R2). Since Con(D,R2)is a open subset of the vector space of all holomorphic maps on D (in the Frechettopology), it holds that the discrete configuration space Qn is an open subsetof spanC{1, z, . . . , zn$1} 3 Cn. Thus, each tangent space T%Qn is identifiedwith Cn by taking the coe"cients of the finite Taylor series. Together with therestricted H1

" metric, Qn is a Riemannian manifold.Let U, V # T%Con(D,R2), and let (ak)!k=0, (bk)

!k=0 respectively be their Tay-

lor coe"cients. Then it holds that

((U, V ))L2(D) =1

+

!-

k=0

(1 + i)aibi ,

which follows since ((zi, zj))L2(D) = ,ij(i+ 1)/+.

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The next step is to obtain a numerical method that approximates the geodesics.For this we use the variational method obtained by the discrete Lagrangian onQn ,Qn given by

Ld(&k,&k+1) = hL! &k + &k+1

2# $% &%k+1/2

,&k+1 $ &k

h

"

=1

2h((&#

k+1/2(&k+1 $ &k),&#k+1/2(&k+1 $ &k)))L2(D)+

!

2h((&#

k+1 $ &#k,&

#k+1 $ &#

k))L2(D),

where h > 0 is the step size. The discrete action is thus given by

Ad(&0, . . . ,&N ) =N$1-

k=0

Ld(&k,&k+1). (8)

Now, a method for numerical computation of geodesics originating from theidentity and ending at a known configuration is obtained as follows.

Algorithm 1 Given & # Con(D,R2), an approximation to the geodesic curvefrom the identity element &0(z) = z to & is given by the following algorithm:

1. Set &N = Pn&, where Pn : Con(D,R2) " Qn is projection by truncation ofTaylor series.

2. Set initial guess &k = (1$ k/N)&0 + k/N&N for k = 1, . . . , N $ 1.3. Solve the minimization problem

min%1,...,%N!1'Qn

Ad(&0, . . . ,&N )

with a numerical non-linear numerical minimization algorithm.

Remark 2. In practical computations we use Cn instead of Qn. Thus, as a laststep one have to check that the solution obtained fulfils that &k # Qn, i.e.,&#k(z) 2= 0 for z # D. For short enough geodesics, this is guaranteed by the fact

that &#0(z) = 1.

Remark 3. Notice that we solve the problem as a two point boundary valueproblem. Thus, we assume that the final state &N is known. In future work we willconsider the more general optimal control problem, where the final configurationis determined by minimimizing a functional, such as sum-of-squares of the pixeldi!erence between the destination and target image.

5.1 E!cient Evaluation of the Discrete Action

Evaluation of the discrete action functional (8) requires computation of the Tay-lor coe"cients of the product &#

k+1/2(&k+1 $ &k). The “brute force” algorithm

for doing this requires O(n2) operations. However, it is well known that FFTtechniques can be used to accelerate such computations. Using this, we now givean O(Nn log n) algorithm for evaluation of the discrete action.

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Algorithm 2 Given &0, . . . ,&N , an e"cient algorithm for computing the dis-crete action Ad(&0, . . . ,&N ) is given by:

1. For each k = 0, . . . , N , compute the Taylor coe"cients of &#k. This requires

O(Nn) operations.2. Compute

A1 =N$1-

k=0

!

2h((&#

k+1 $ &#k,&

#k+1 $ &#

k))L2(D).

This requires O(Nn) operations.3. Set ak # C2n to contain the Taylor coe"cients of &#

k+1/2 as its first n $ 1

elements, and then zero padded. This requires O(Nn) operations.4. Set bk # C2n to contain the Taylor coe"cients of (&k+1 $ &k)/h as its first

n elements, and then zero padded. This requires O(Nn) operations.5. Compute

ak = FFT(ak), bk = FFT(bk).

This requires O(Nn log n) operations.6. Compute component-wise multiplication

ck = akbk.

This requires O(Nn) operations.7. Compute

ck = IFFT(ck).

This requires O(Nn log n) operations.8. Let %k be the polynomial with Taylor coe"cients given by ck. Then compute

A2 =h

2

N$1-

k=0

((%k,%k))L2(D).

This requires O(Nn) operations.

Finally, the discrete action is now given by Ad(&0, . . . ,&N ) = A1 +A2.

6 Experimental Results

In this section we use the numerical method developed in the previous sectionto confirm that the discrete Lagrangian method is able to calculate geodesics(solutions to (4)) with moderately large deformations. The purpose of theseexperiments is to demonstrate our numerical method: we leave further investi-gations of conformal image registration for another paper. In all the examples wecompute geodesics starting from the identity &0(z) = z and ending at some poly-nomial &1 (of relatively low order). We consider both simple linear and heavilynon-linear warpings. The simulations are carried out with two di!erent values of

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the parameter ! to illustrate how the geodesics depend on the metric. The datais given in Table 2.

Fig. 2 shows the geodesics corresponding to scaling and rotation. Notice thatthe geodesics stay in the submanifold Lin(D,R2), as predicted by Proposition 1.Also notice the di!erence between large and small !. For small !, the scalingcoe"cient behaves (almost) like d2

dt2 c21 = 0, which is the solution of equation (7)

with ! = 0, while the scaling coe"cient behaves (almost) like d2

dt2 c1 = 0, whichis the asymptotic solution of equation (7) as ! " 4.

Fig. 3 shows the geodesics corresponding to various non-linear transforma-tions. Although the di!erences in the geodesic paths for di!erent values of ! aresmall, we notice that for higher values of ! the geodesic are “more regular” atthe cost of occupying more volume. This is especially clear in Example 3, where,halfway thorough the geodesic, the “bump” on the right of the shape behavesdi!erently for the two values of !.

We anticipate that the method will allow us to study the metric geometry ofthe conformal embeddings as was done by Michor and Mumford [17] for metricson planar curves and to determine, for example, which conformal embeddingsare markedly closer in one metric than another, and how the geodesic pathsdi!er between di!erent metrics and between di!erent groups, for example, bypassing to a smaller group (e.g. the Mobius group) or to a larger one (e.g. thefull di!eomorphism pseudogroup for embeddings).

7 Conclusions

Motivated by the preference of Thompson [3] for ‘simple’ warps in his examplesof how images of one species can be deformed into those of a related species, wehave derived the geodesic equation for planar conformal di!eomorphisms usingtheH1

" metric. We have chosen conformal warps as they were used by Thompson,and are very simple. Of course, the animals that Thompson was interested inare actually three-dimensional, and for any number of dimensions bigger than 2the set of conformal warps is rather restricted. However, our intention is to startwith conformal warps and to continue to build progressively more complex setsof deformations, rather than working always in the full di!eomorphism group asis conventional.

The conformal warps admit the rigid transformations of rotation and trans-lation as special cases, and we have shown that these linear conformal trans-formations are totally geodesic in the conformal warps with respect to the H1

"

metric that we have considered.

We have also provided a numerical discretization of the geodesic equation,and used it to demonstrate the e!ects of the parameters of the conformal warps.In future work we will use the algorithm that we have developed here to performimage registration based on conformal warps using the LDDMM framework andapply it to images such as real examples of those drawn by Thompson.

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Annotation Coe!cients Choice of ! Type

Example 1 (a,b) c0 = 0c1 = 0.5

! = 0.1 in (a)! = 100 in (b)

Scaling

Example 2 (a,b) c0 = 0c1 = exp(0.4"i)

! = 0.1 in (a)! = 100 in (b)

Rotation

Example 4 (a,b) c0 = 0.0185475c1 = 0.8034225c2 = #0.13933275c3 = #0.23849625c4 = #0.18597975c5 = #0.0125472c6 = 0.18020775c7 = 0.27937125

! = 0.1 in (a)! = 10 in (b)

Non-linear

Example 5 (a,b) c0 = 0.00674 + 0.053125ic1 = 0.77654 + 0.103125ic2 = 0.109424 + 0.103125ic3 = #0.052777 + 0.103125ic4 = #0.115049 + 0.103125ic5 = #0.0409141 + 0.103125ic6 = 0.126201 + 0.103125ic7 = 0.288402 + 0.103125i

! = 0.1 in (a)! = 10 in (b)

Non-linear

Table 2. Data used in the various examples. The polynomial for the final con-formal mapping is of the form &1(z) =

.n$1k=0 ckz

k with n = 16. Coe"cients notlisted are zero. In all examples, we used N = 20 discretisation points in time.

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Example 1(a)

(b)

Example 2(a)

(b)

Fig. 2. In Example 1, geodesics in the H1" metric connecting the identity z '" z

to z '" 0.5z are calculated using the discrete Lagrangian method. In (a), ! =0.1 and in (b), ! = 100. Both geodesics coincide with the analytic solution toequation (7). In Example 2, the geodesic connecting z '" z to z '" e0.4)iz iscalculated, again matching the analytic solution perfectly, illustrating the e!ectof !.

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Example 3(a)

(b)

Example 4(a)

(b)

Fig. 3. Examples 3 and 4 illustrate two geodesics in the H1" metric on conformal

embeddings calculated using the discrete Lagrangian method. In (a), ! = 0.1 andin (b), ! = 100. In both examples, the target di!eomorphism &1 has been chosento be highly non-linear (see Table 2 for the exact data used). Little di!erence isvisible to the eye between the two values of !: in Example 3, a small bump onthe right side of the boundary behaves di!erently.

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Acknowledgements

This work was funded by the Royal Society of New Zealand Marsden Fund andthe Massey University Postdoctoral Fellowship Fund. The authors would like tothank the reviewers for helpful comments and suggestions.

References

[1] Younes, L.: Shapes and Di!eomorphisms. Applied Mathematical Sciences.Springer-Verlag (2010)

[2] Holm, D.D., Marsden, J.E.: Momentum maps and measure-valued solutions(peakons, filaments, and sheets) for the EPDi! equation. In: The breadth of sym-plectic and Poisson geometry. Volume 232 of Progr. Math. Birkhauser Boston,Boston, MA (2005) 203–235

[3] Thompson, D.: On Growth and Form. Cambridge University Press (1942)[4] Wallace, A.: D’Arcy Thompson and the theory of transformations. Nature Re-

views Genetics 7 (2006) 401–406[5] Trouve, A.: An infinite dimensional group approach for physics based models in

patterns recognition. Technical report, Ecole Normale Superieure (1995)[6] Dupuis, P., Grenander, U.: Variational problems on flows of di!eomorphisms for

image matching. Q. Appl. Math. (1998) 587–600[7] Trouve, A.: Di!eomorphisms groups and pattern matching in image analysis. Int.

J. Comput. Vision 28 (1998) 213–221[8] Joshi, S., Miller, M.: Landmark matching via large deformation di!eomorphisms.

Image Processing, IEEE Transactions on 9 (2000) 1357 –1370[9] Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: A gen-

eral framework. International Journal of Computer Vision 41 (2001) 61–84[10] Beg, M.: Variational and computational methods for flows of di!eomorphisms in

image matching and growth in computational anatomy. PhD thesis, John HopkinsUniversity (2003)

[11] Beg, M.F., Miller, M.I., Trouve, A., Younes, L.: Computing large deformationmetric mappings via geodesic flows of di!eomorphisms. International Journal ofComputer Vision 61 (2005) 139–157

[12] Bruveris, M., Gay-Balmaz, F., Holm, D.D., Ratiu, T.S.: The momentum maprepresentation of images. J. Nonlinear Sci. 21 (2011) 115–150

[13] Marsland, S., McLachlan, R.I., Modin, K., Perlmutter, M.: On a geodesic equationfor planar conformal template matching. In: Proceedings of the 3rd MICCAIworkshop on Mathematical Foundations of Computational Anatomy (MFCA’11),Toronto (2011)

[14] Lang, S.: Fundamentals of di!erential geometry. Volume 191 of Graduate Textsin Mathematics. Springer-Verlag, New York (1999)

[15] Ebin, D.G., Marsden, J.E.: Groups of di!eomorphisms and the notion of anincompressible fluid. Ann. of Math. 92 (1970) 102–163

[16] Shkoller, S.: Geometry and curvature of di!eomorphism groups with H1 metricand mean hydrodynamics. J. Funct. Anal. 160 (1998) 337–365

[17] Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves.J. Eur. Math. Soc. (JEMS) 8 (2006) 1–48

[18] Sharon, E., Mumford, D.: 2d-shape analysis using conformal mapping. Int. J.Comput. Vis. 70 (2006) 55–75

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65


[19] Gay-Balmaz, F., Marsden, J., Ratiu, T.: Reduced variational formulations in freeboundary continuum mechanics. J. Nonlinear Sci. (2012) online first.

[20] Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Amer.Math. Soc. (N.S.) 7 (1982) 65–222

[21] Marsland, S., McLachlan, R.I., Modin, K., Perlmutter, M.: Application ofthe hodge decomposition to conformal variational problems. arXiv:1203.4464v1[math.DG] (2011)

[22] Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Volume 125of Applied Mathematical Sciences. Springer-Verlag, New York (1998)

[23] Khesin, B., Wendt, R.: The Geometry of Infinite-dimensional Groups. Volume 51of A Series of Modern Surveys in Mathematics. Springer-Verlag, Berlin (2009)

[24] Modin, K., Perlmutter, M., Marsland, S., McLachlan, R.I.: On Euler-Arnoldequations and totally geodesic subgroups. J. Geom. Phys. 61 (2011) 1446–1461

[25] Marsden, J.E., West, M.: Discrete mechanics and variational integrators. ActaNumer. 10 (2001) 357–514

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Geodesic Warps by Conformal Mappingsrmclachl/VISI2316.pdf · 2012. 7. 26. · metric geometry of the conformal embeddings, similar to what was done for metrics on planar curves by