1

Elastic Geodesic Paths in Shape Space ofParametrized Surfaces

Sebastian Kurtek†, Eric Klassen‡, John C. Gore∗, Zhaohua Ding∗, and Anuj Srivastava†

Abstract—This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it providesefficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces.The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preservingtransformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unitsphere in R

3) and impose a Riemannian metric on it in such a way that the re-parameterization group acts on this space by isometries.Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, weuse a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a techniquepresented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solutionprovides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideasusing examples from shape analysis of anatomical structures and other general surfaces.

Index Terms—shape analysis, Riemannian distance, parameterization invariance, path-straightening, geodesics

✦

1 INTRODUCTION

Shape is an important feature of objects and can be im-mensely useful in characterizing objects for the purposeof detection, tracking, classification, and recognition.As an example, it plays an important role in medicalimage analysis where advances in non-invasive imagingtechnology have enabled researchers to study biologicalvariations of anatomical structures. Studying shapes of3D anatomical structures in the brain is of particularinterest because many diseases can potentially be linkedto alterations of these shapes. Shape analysis of surfaceshas also become important in biometrics, graphics, 3DTV, computer vision, etc.

There has been a significant amount of research andactivity in the general area of shape analysis. By shapeanalysis we mean a set of tools for comparing, matching,deforming, and modeling shapes. The main differencesamongst different tools proposed so far lie in the mathe-matical representations and metrics used in the analysis.For example, in shape analysis of planar objects (objectsin 2D images), a variety of mathematical representations,including binary images, sampled points (active shapemodels [6]), ordered points (landmark-based shape anal-ysis [10]), medial axes [32], level sets [29], and others,have been used. These different representations, alongwith their corresponding choices of metrics, lead todifferent solutions with their respective strengths and

This paper was presented in part at the IEEE Conference on CVPR, SanFrancisco, June 2010.

• †Department of Statistics, Florida State University, Tallahassee, FL 32306.• ‡Department of Mathematics, Florida State University, Tallahassee, FL

32306.• ∗Vanderbilt Univ. Institute of Imaging Science, Nashville, TN, 37232.

limitations. A natural representation for shape analy-sis of boundaries of planar objects uses parameterizedcurves, although historically that representation has beenunder-utilized. One of the main reasons for its limiteduse has been the issue of parameterization. While a re-parameterization of a curve does not change its shape, itdoes change the coordinate, angle, or curvature functions(as functions of the parameterization) along the curvesand any comparison directly involving those functionswill be affected. How can we deal with this shape-preserving but unconventional transformation? The so-lution comes from choosing representations and metricsin such a way that the resulting geodesics betweencurves are invariant to their re-parameterizations, in ad-dition to the standard shape-preserving transformationssuch as rotation, translation and uniform scaling (seee.g. [37], [19], [38], [33]). For instance, Srivastava et al.[33], [18], [19] use a square-root velocity function (SRVF)

q(t) = β(t)√|β(t)|

, to analyze the shape of a parameterized

curve β : [0, 1] → Rn under the standard L

2 metric. Thereare several reasons for selecting such a representation;three important ones are:

1) Under the L2 metric, the action of the re-

parameterization group is by isometries, i.e. ifq1, q2 are two SRVFs and γ : [0, 1] → [0, 1] isany re-parameterization function, then ‖q1 − q2‖ =‖(q1, γ) − (q2, γ)‖, where (qi, γ) denotes the SRVFof the re-parameterized curve and ‖ · ‖ denotesthe L

2 norm. This helps define a proper metric onthe shape space (representation space modulo there-parameterization group), and ultimately makesshape analysis invariant to re-parameterization us-ing distances of the type: minγ ‖q1 − (q2, γ)‖.

2) Under the SRVF representation an elastic Rieman-

2

nian metric defined in [28] becomes the standardL2 metric enabling its use in the previous item.

3) This framework allows pair-wise matchingof points on curves using optimal re-parameterizations while simultaneouslycomputing distances.

In this paper we are focused on shape analysis ofboundaries formed by 3D objects. In particular, we focuson shape analysis of parametrized surfaces of genus zero(i.e. diffeomorphic to S

2), and we are interested in aRiemannian framework that allows comparison, match-ing, deformation, averaging, and modeling of observedshapes. Motivated by the Riemannian shape analysisof curves, we pose the following question: What is anatural representation of surfaces and a corresponding metricthat together allow for a parameterization-invariant shapeanalysis? The solution to this question is one of the maincontributions of this paper. Additionally, we present anefficient framework for solving a fundamental problemin 3D shape analysis: How to compute geodesic paths be-tween given parameterized surfaces under the chosen metric?

As a motivating example, consider the three toy heartsurfaces in Figure 2. These surfaces have the same shapebut different parameterizations (displayed next to thesurfaces). A framework not invariant to parameteriza-tion would result in a non-zero distance between thesesurfaces despite their shapes being identical.

Fig. 1. The same surface with different parameterizations.

1.1 Past and Current Methods

Similar to curves, there have been several analogousrepresentations of surfaces. Many groups have proposedmethods for studying the shapes of surfaces by embed-ding them in volumes and deforming these volumesunder the LDDMM framework [13], [17], [8], [7], [35].While these methods are both prominent and pioneeringin medical image analysis, they are typically compu-tationally expensive since they try to match not onlythe objects of interest but also some background spacecontaining them. An alternative approach is based onmanually-generated landmarks under the Kendall shapetheory [10] and active shape models [6]. Others study 3Dshape variabilities using level sets [26], curvature flows[16], or point cloud matching via the iterative closestpoint algorithm [1]. Also, there has been remarkablesuccess in the use of medial representations for shapeanalysis, especially in medical image analysis, see e.g.[3], [12].

However, the most natural representation for studyingshapes of 3D objects seems to be parameterized surfaces.In case of parameterized surfaces, there is an additionalissue of handling the parameterization variability. Somepapers, e.g. using SPHARM [20], [4] or SPHARM-PDM[34], [11], tackle this problem by choosing a fixed param-eterization that is analogous to the arc-length parameter-ization on curves. Kilian et al. [21] presented a techniquefor computing geodesics between triangulated meshes(discretized surfaces) but at their given parameteriza-tions. Similar to the elastic representations of curves,we would like to include the parameterization variablein the analysis. This inclusion results in an improvedregistration of features across surfaces. Of course, thequestion is: How can we include the parameterizationvariable in our shape analysis? A large set of papersin the literature treat parameterization (or registration)as a pre-processing step [36]. In other words, they takea set of surfaces and use some energy function, suchas the entropy [5] or the minimum description length[9], to register points across surfaces. Once the surfacesare registered, they are compared using standard pro-cedures. There are several fundamental problems withthis approach. Firstly, the energy used for registrationdoes not lead to a proper distance on the shape space ofsurfaces. Secondly, due to a registration procedure basedon ensembles, the distance between any two shapes endsup being dependent on the other shapes in the ensemble.The registration and the comparisons of surfaces here aretypically disjoint procedures and under different metrics.The contrast between these methods and our approachis presented in Figure 2.

To the best of our knowledge, there are very fewtechniques in the literature on a Riemannian shape anal-ysis of parameterized surfaces that can provide geodesicpaths and be invariant to re-parameterization.

3

Fig. 2. We seek a joint framework for registration andshape analysis.

1.2 Our Approach

To illustrate our approach, let f1 and f2 denote twosurfaces; f1 and f2 are elements of an appropriate spaceF , which is made precise later, and let 〈〈·, ·〉〉 be thechosen Riemannian metric on F . Then, under certainconditions, the geodesic distance between shapes of f1and f2 will be given by quantities of type:

(This assumes that translation and scaling variabil-ity in f1 and f2 has already been removed.) HereF (t) is a path in F indexed by t, and the quantity∫ 1

0〈〈Ft(t), Ft(t)〉〉(1/2)dt denotes the length of F , L(F ).

The minimization inside the brackets, thus, denotes theproblem of finding a geodesic path (locally the shortestpath) between the surfaces f1 and O(f2◦γ), where O andγ stand for an arbitrary rotation and re-parameterizationof f2, respectively. The minimization outside the bracketseeks the optimal rotation and re-parameterization ofthe second surface so as to best match it with the firstsurface. In simple words, the outside optimization solvesthe registration or matching problem while the insideoptimization solves for both an optimal deformation(geodesic) and a formal distance (geodesic distance)between shapes. An important strength of this approachis, thus, that the registration and distance-based com-parison are solved jointly rather than sequentially.Another strength is that this framework can be easilyextended to different types of surfaces.

The rest of this paper is organized as follows. Section2 describes a convenient mathematical representation ofembedded surfaces and introduces a parameterization-invariant Riemannian metric for shape analysis of suchsurfaces. It establishes the pre-shape space, the shape-preserving transformation groups, and the shape spaceas a quotient space of the pre-shape space. While Sec-

tion 3 presents a path-straightening method for findinggeodesics in the pre-shape space, Section 4 presentsa registration method and an algorithm for findinggeodesics in shape spaces. It also gives some examplesof geodesics using different surfaces. Section 5 presentsan application of this method to classification of mathe-matics deficiency.

2 MATHEMATICAL REPRESENTATION

Let S denote a 2D smooth surface with genus zero.We will represent the surface S with its embeddingf : S

2 → R3. The function f is also called a parame-

terization of S, parameterized by an element of S2. Let

the set of parameterized surfaces be F = {f : S2 7→

R3|∫

S2‖f(s)‖2ds < ∞ and f is smooth}, where ds

is the standard Lebesgue measure on S2. We choose

the natural Riemannian structure in the tangent space,Tf(F): for any two elements m1,m2 ∈ Tf(F), definean inner product: 〈m1,m2〉 =

∫

S2〈m1(s),m2(s)〉ds, where

the inner product inside the integral is the standardEuclidean product. The resulting L

2 distance between

any two points f1, f2 ∈ F is(∫

S2‖f1(s)− f2(s)‖2ds

)1/2,

and the geodesic path connecting them in F is a “straightline”: β(t) = tf2+(1−t)f1. One can represent surfaces aselements of F as stated here and use the L

2 distance tocompare shapes of surfaces. Although this frameworkis very common and seemingly convenient, it is notsuitable for analyzing shapes of surfaces as it is notinvariant to re-parameterizations.

We explain this point further. Let Γ be the set ofall diffeomorphisms of S

2. This set will act as the re-parametrization group for surfaces. Γ is a Lie groupwith composition as the group operation and the identitymapping as the identity element γid. The natural actionof Γ on F is on the right by composition: for a γ ∈ Γ,f ∈ F , the re-parameterized surface is given by f ◦ γ. Inorder to unify all representations of a surface, we definethe orbit of f , under the action of Γ as: [f ] = {f◦γ|γ ∈ Γ}.The quotient space F/Γ is the set of all such orbits, andwe would like to put a natural metric on it. If Γ acts onF by isometries, this would be feasible. The L

2 metricwould simply descend to a metric on the quotient space.So we check the isometry condition:

‖f1 ◦ γ − f2 ◦ γ‖ =

(∫

S2

‖f1(γ(s))− f2(γ(s))‖2ds

)1/2

=

(∫

S2

‖f1(s)− f2(s)‖2J−1γ (s)ds

)1/2

6= ‖f1 − f2‖ ,

where Jγ(s) is the Jacobian of γ at s. This inequalitycomes from the fact that γ, in general, is not areapreserving and hence the Jacobian is not one at allpoints. This lack of isometry means that the L

2 distancebetween any two surfaces will not be the same if theyare re-parameterized by the same element of Γ. Thisalso implies degeneracy, that is, it may be possible tocarefully choose γ such that the L

2 distance betweenany two surfaces in F is arbitrarily close to zero. Onesolution is to restrict to only those re-parameterizationsthat are area-preserving or some subset of these [15].

4

However, this is a severe restriction and may not beable to provide a good matching of surfaces and thusresult in an unreliable measure of their differences. Ourapproach is to define a new metric in the space F suchthat the isometry condition under the action of the re-parameterization group is satisfied.

Let {ux(s), uy(s)} form an orthonormal basis of thetangent space of S2 at s. Then the directional derivativeof a function f at a point s in the direction ux is fx(s) =ux(s)f(s) and in the direction uy is fy(s) = uy(s)f(s).To endow F with a Riemannian metric, we begin bydefining a new representation of surfaces [25], [24]:

Definition 1. Define the mapping Q : F → L2 as Q(f)(s) =

√

‖a(s)‖f(s), where ‖a(s)‖ = ‖fx(s) × fy(s)‖ is the areamultiplication factor of f at s.

Here ‖ · ‖ denotes the standard 2-norm of a vector inR

3. The factor ‖a(s)‖ is the ratio of infinitesimal areas ofthe surface at f(s) and the domain at s. For any f ∈ F ,we will refer to q(s) ≡ Q(f)(s) as the q-map of f . SinceF is a set of smooth surfaces, the set of all q-maps is asubset of L

2(S2,R3), henceforth denoted by L2. Figure

3(a) displays the mapping Q between F and L2. We can

(a) (b)

Fig. 3. (a) Mapping Q : F → L2. (b) Its differential at f ,

Q∗,f : Tf(F) → L2.

now define a new action of Γ on L2, the space of q-maps,

as follows:

Definition 2. If a surface f is re-parameterized by γ, thenits q-map is given by

√

Jγ(q ◦ γ). This defines a right actionof Γ on L

2 by L2 × Γ → L

2 as (q, γ) =√

Jγ(q ◦ γ).An important fact about the map Q is that if we re-

parameterize a surface by γ and then obtain its q-map(Definition 1), or if we obtain its q-map (Definition 1) andthen act by γ (Definition 2), the result will be the same.In other words, the diagram in Figure 4 is commutative.The proof of this statement follows.

Proof: First, we focus on computing aγ = (f ◦γ)x × (f ◦ γ)y. Let γ(s) = (γ1(s), γ2(s))T and f(s) =(f1(s), f2(s), f3(s))T . Then, (f ◦ γ)x = fxγ

1x + fyγ

2x and

similarly, (f ◦ γ)y = fxγ1y + fyγ

2y . We can write

∂(f ◦ γ)

∂x=

f1xγ

1x + f1

yγ2x

f2xγ

1x + f2

yγ2x

f3xγ

1x + f3

yγ2x

,∂(f ◦ γ)

∂y=

f1xγ

1y + f1

yγ2y

f2xγ

1y + f2

yγ2y

f3xγ

1y + f3

yγ2y

.

We can now compute aγ , which yields,

γ1xγ2yf

2xf

3y + γ2xγ

1yf

3xf

2y − γ1xγ

2yf

3xf

2y − γ2xγ

1yf

2xf

3y

γ1xγ2yf

3xf

1y + γ2xγ

1yf

1xf

3y − γ1xγ

2yf

1xf

3y − γ2xγ

1yf

3xf

1y

γ1xγ2yf

1xf

2y + γ2xγ

1yf

2xf

1y − γ1xγ

2yf

2xf

1y − γ2xγ

1yf

1xf

2y

.

This expression can be rewritten such that aγ = aJγ ,where Jγ is the Jacobian of γ. Given this, we can verifythe validity of Definition 2:

(q, γ) =√

‖aγ‖(f ◦ γ) =√

Jγ√

‖a‖(f ◦ γ) =√

Jγ(q ◦ γ).

Fig. 4. Representation of surface f using the q-map.

2.1 Riemannian Metric & Isometry Condition

We choose the natural L2 metric on the space of q-

maps. That is, for any two elements w1, w2 ∈ Tq(L2),

define an inner product: 〈w1, w2〉 =∫

S2〈w1(s), w2(s)〉ds.

The Riemannian metric that we will use on F is thepullback of the L

2 metric from the space of q-maps.For this purpose, we first derive the differential of Qat f , denoted by Q∗,f . This is a linear mapping betweentangent spaces Tf(F) and L

2 as shown in Figure 3(b).For a tangent vector v ∈ Tf (F) and r ∈ R, the mappingQ∗,f : Tf (F) → TQ(f)(L

2) is given by:

Q∗,f(v) =d

dr|r=0Q(f + rv)

=1

2√

‖a‖(∂‖af+rv‖

∂r|r=0)f +

√

‖a‖ v. (2)

Since∂‖af+rv‖

∂r |r=0 = (a·av)‖a‖ , where av = vx×fy+fx×vy ,

we obtain

Q∗,f(v) =1

2‖a‖ 32

(a · av)f +√

‖a‖ v. (3)

In this equation a depends only on f while av dependson both f and v. We use this differential of Q to definea Riemannian metric on F as follows.

Definition 3. For any f ∈ F and any v1, v2 ∈ Tf (F), definethe inner product:

〈〈v1, v2〉〉f ≡ 〈Q∗,f (v1), Q∗,f (v2)〉 , (4)

where the inner product on the right side is the standardinner product in L

2.

With this induced metric, F becomes a Riemannianmanifold and we want to compute geodesic distancesbetween two points, say f1 and f2, in F under this

5

metric. To write the metric in Definition 3 in full detail,we use the expression for Q∗,f(v) given in Eqn. 3:

〈〈v1, v2〉〉f = 〈Q∗,f (v1), Q∗,f(v2)〉

= 〈 1

2‖a‖ 32

(a · av1)f +√

‖a‖ v1,

1

2‖a‖ 32

(a · av2)f +√

‖a‖ v2〉

= 〈 1

4‖a‖3 (a · av1)f, (a · av2)f〉

+ 〈 1

2‖a‖ [(a · av2)v1 + (a · av1)v2], f〉

+ 〈‖a‖v1, v2〉.

An important property of this metric is that the actionof Γ on F is by isometries.

Proposition 1. For any surface f ∈ F , a γ ∈ Γ and twotangent vectors v1, v2 ∈ Tf (F), we have:

〈〈v1 ◦ γ, v2 ◦ γ〉〉f◦γ = 〈〈v1, v2〉〉f . (5)

Proof:

〈Q∗,(f◦γ)(v1 ◦ γ), Q∗,(f◦γ)(v2 ◦ γ)〉L2

=

∫

S2

〈1

2‖a(γ(s))‖32

(a(γ(s)) · av1◦γ(γ(s)))f(γ(s))

+√

‖a(γ(s))‖ v1(γ(s)) ,1

2‖a(γ(s))‖32

(a(γ(s)) · av2◦γ(γ(s)))f(γ(s))

+√

‖a(γ(s))‖ v2(γ(s))〉 ds

=

∫

S2

〈1

2‖a(s)‖32 Jγ(s)

32

(a(s) · av1(s))Jγ(s)2f(s)

+√

‖a(s)‖Jγ(s) v1(s) ,1

2‖a(s)‖32 Jγ(s)

32

(a(s) · av2(s))Jγ(s)2f(s)

+√

‖a(s)‖Jγ(s) v2(s)〉 Jγ(s)−1

ds

=

∫

S2

〈1

2‖a(s)‖32

(a(s) · av1(s))f(s) +√

‖a(s)‖ v1(s) ,

1

2‖a(s)‖32

(a(s) · av2(s))f(s) +√

‖a(s)‖ v2(s)〉 ds

= 〈Q∗,f (v1), Q∗,f (v2)〉L2 .

since s = γ(s) and ds = Jγ(s)ds.

2.2 Pre-Shape and Shape Space

Shape analysis of surfaces can be made invariant tocertain global transformations by normalizing. The trans-lation of surfaces is easily taken care off by center-

ing: fcentered(s) = f(s) −∫S2

f(s)‖a(s)‖ds∫S2

‖a(s)‖ds. Scaling can be

removed by re-scaling all surfaces to have unit area,

fscaled(s) =f(s)√∫

S2‖a(s)‖ds

.

With a slight abuse of notation, we define the spaceof normalized surfaces as F . F forms the pre-shapespace in our analysis. The remaining groups – rotationand re-parameterization – are dealt with differently, by

removing them algebraically from the representationspace.

1) Rotation Group, SO(3): The rotation group SO(3)acts on F , SO(3) × F → F according to (O, f) =Of , for O ∈ SO(3) and f ∈ F . It is easy to checkthat the action of SO(3) on F under the inducedmetric is by isometries.

2) Re-Parametrization Group, Γ: The re-parameterization group Γ acts on F according toF × Γ → F by (f, γ) = (f ◦ γ). As discussed inSection 2.1 the re-parametrization group Γ acts onF by isometries under the induced metric.

Proposition 2. The actions of Γ and SO(3) on F commute.

Since the actions of SO(3) and Γ commute we can definean action of the product of the groups on F . The orbitof a surface f is given by:

[f ] = closure{O(f ◦ γ)|O ∈ SO(3), γ ∈ Γ} (6)

and the set of all [f ] is defined to be S = {[f ]|f ∈ F}.Since the orbits under Γ are not closed, we use theirclosures to define equivalence classes.

The next step is to define geodesic paths in F andS. We start with the case of F ; the geodesic distancebetween any two points f1, f2 ∈ F , dF(f1, f2), is givenby:

minF : [0, 1] → F

F (0) = f1, F (1) = f2

(∫ 1

0

〈〈Ft(t), Ft(t)〉〉(1/2)

dt

)

. (7)

We will use a path-straightening approach for solvingthis problem in Section 3. Once we have an algorithmfor finding geodesics in F , we can obtain geodesicsand geodesic lengths in S by solving an additionalminimization problem over SO(3)× Γ as stated in Eqn.1. This problem searches over the orbit [f2] so that thegeodesic distance between f1 and an element of [f2] isminimized. We will use a gradient-based approach tosolve this problem in Section 4.

3 GEODESICS IN THE PRE-SHAPE SPACE

Consider the problem of finding geodesics between sur-faces f1 and f2 in F using a path-straightening approach.This method was first described in [22] and was laterused for finding geodesics between elastic curves in [33].The basic idea here is to connect f1 and f2 by anyinitial path, e.g. using a straight line under the L

2 metric,and then iteratively “straighten” it until it becomes ageodesic. This update is performed using the gradientof an appropriate energy function. Earlier works onpath-straightening involved non-linear manifolds insidea larger vector space such that the Riemannian metricon the manifold was a restriction of the standard metricon the larger space. As seen next, the current case isdifferent. The space of parameterized surfaces is a vectorspace, but the metric on this space is non-standard.

6

Let F : [0, 1] → F denote a path in F . The energy ofthe path F under the induced metric is defined to be:

E[F ] =

∫ 1

0

〈〈Ft, Ft〉〉F dt

Using Defn. 3=

∫ 1

0

〈Q∗,F (Ft), Q∗,F (Ft)〉 dt

=

∫ 1

0

(〈 1

4‖A‖3 (A ·At)2F, F 〉

+ 〈 1

‖A‖ (A ·At)Ft, F 〉

+ 〈‖A‖Ft, Ft〉)dt

=

∫ 1

0

∫

S2

[1

4‖A‖3 (A · At)2(F · F )

+1

‖A‖(A · At)(Ft · F )

+ ‖A‖(Ft · Ft)]dsdt .

In this derivation we have suppressed the argument t forall the quantities. Also, we use A(t) to imply a(F (t)). Itis well known that a critical point of E is a geodesicpath in F . To find a critical point, we are going touse the gradient ∇EF which, in turn, is approximatedusing directional derivatives, ∇EF (G), where G ∈ G is aperturbation of the path F . Here G denotes the set of allpossible perturbations of F . Figure 5 is a depiction of thisapproach. We start with an initial path F and iterativelyupdate it in the direction of ∇E until we arrive at thecritical point F ∗, which is the desired geodesic.

Fig. 5. Computation of geodesic paths using the path-straightening algorithm.

3.1 Directional Derivative of EIn this section we present the derivation of ∇EF (G). Thisdirectional derivative will be used to approximate thegradient of E. The derivative of E in the direction G isgiven by ∇EF (G) =

ddǫE(F + ǫG)|ǫ=0. The energy of the

perturbed path is:

E[F + ǫG] =

∫ 1

0

〈Q∗,F+ǫG(Ft + ǫGt), Q∗,F+ǫG(Ft + ǫGt)〉L2dt.

(8)

In order to write down an analytical formula for theenergy gradient, we first derive some terms that areneeded later. All of these terms involve F and are,therefore, functions of t, x, and y. These arguments aresuppressed for the sake of brevity. Since A = Fx × Fy

and At = Ft,x × Fy + Fx × Ft,y , we have:

• δGA ≡ Gx × Fy + Fx ×Gy .• δGAt ≡ Gt,x×Fy +Ft,x×Gy +Gx×Ft,y +Fx×Gt,y.

Here we use the notation δGA to represent the directionalderivative of the function A in the direction G. Similarly,we can obtain the directional derivatives of other terms:

• δG(F · F ) = 2(F ·G)• δG(Ft · Ft) = 2(Ft ·Gt)• δG(F · Ft) = (Ft ·G)(F ·Gt)

• δG‖A‖ = (A·δGA)‖A‖

• δG(A · At) = (A · δGAt)(At · δGA).We are now ready to write down the analytic formula forthe path-straightening energy gradient, which is givenby:

∇EF (G) =d

dǫE(F + ǫG)|ǫ=0

=

∫ 1

0

∫

S2

(−3(A · δGA)4‖A‖5 (A ·At)

2(F · F )

+1

2‖A‖3 (A ·At)(A · δGAt +At · δGA)(F · F )

+1

2‖A‖3 (A ·At)2(F ·G)

+−(A · δGA)

‖A‖3 (A · At)(Ft · F )

+1

‖A‖(A · δGAt +At · δGA)(Ft · F )

+1

‖A‖(A · At)(Ft ·G+ F ·Gt)

+(A · δGA)

‖A‖ (Ft · Ft)

+ 2‖A‖(Ft ·Gt)) dsdt

The terms inside the integrals may be re-written as H1 ·δGA+H2 · δGAt +H3 ·G+H4 ·Gt, where,

H1 = [A

‖A‖(

−3

4‖A‖4(A ·At)

2(F · F )−1

‖A‖2(A ·At)(Ft · F )

+(Ft · Ft)) +At

‖A‖((A · At)

2‖A‖2(F · F ) + (F · Ft))],

H2 = [A

‖A‖((A · At)

2‖A‖2(F · F ) + (F · Ft))],

H3 = [1

2‖A‖3(A · At)

2F +

1

‖A‖(A · At)Ft], and

H4 =1

‖A‖(A ·At)F + 2‖A‖Ft] .

Furthermore, we can re-write terms H1 · δGA and H2 ·δGAt and combine them as follows:

H1 · δGA = H1 · (Fx ×Gy) + (Gx × Fy) ·H1

= (H1 × Fx) ·Gy + (Fy ×H1) ·Gx,

H2 · δGAt = (Gt,x × Fy) ·H2 +H2 · (Ft,x ×Gy)

+ (Gx × Ft,y) ·H2 +H2 · (Fx ×Gt,y)

= (Fy ×H2) ·Gt,x + (H2 × Ft,x) ·Gy

+ (Ft,y ×H2) ·Gx + (H2 × Fx) ·Gt,y, and

H1 · δGA+H2 · δGAt = (Fy ×H2) ·Gt,x

+ (H2 × Ft,x +H1 × Fx) ·Gy

+ (Ft,y ×H2 + Fy ×H1) ·Gx

+ (H2 × Fx) ·Gt,y.

7

We now have:

(Fy ×H2) ·Gt,x + (H2 × Ft,x +H1 × Fx) ·Gy

+(Ft,y ×H2 + Fy ×H1) ·Gx + (H2 × Fx) ·Gt,y

+H3 ·G+H4 ·Gt

= H3 ·G+H4 ·Gt +M1 ·Gx +M2 ·Gy

+M3 ·Gt,x +M4 ·Gt,y .

Thus, the final expression for the directional derivativeof the energy function is:

∇EF (G) =

∫ 1

0

∫

S2

(H3 ·G+H4 ·Gt +M1 ·Gx

+ M2 ·Gy +M3 ·Gt,x +M4 ·Gt,y)dsdt.

3.2 Orthonormal Basis of GIn order to approximate ∇EF , we utilize the directionalderivative of E. To compute this derivative, we usean orthonormal basis of G, P = {pi|i = 1, 2, ...}, andset ∇EF =

∑∞i=1(∇EF (pi))pi. The next question is:

How can we form a basis for G? Each perturbationG : S2 × [0, 1] → R

3 has three arguments, x, y, t, wherex and y are the coordinates on S

2, and t is the timeindex along the path. We begin by defining two basesP s : S2 → R and P t : [0, 1] → R. There is an additionalrestriction on P t, that is, P t(0) = 0, P t(1) = 0 becausewe do not want to perturb the starting and the endpoints of the path F . In order to define P s we utilize thespherical harmonics functions. It is well known that anysquare integrable function on S

2 can be expressed as alinear combination of spherical harmonics. The basis P t

is defined as follows: P t = {sin(2πit), cos(2πit)− 1|1 ≤i ≤ c3, i ∈ Z}. We use all possible products of P s, P t

to form a basis P : S2 × [0, 1] → R. The final step is todefine the full basis P : S2× [0, 1] → R

3 by utilizing threecopies of P :

P (x, y, t) =

P (x, y, t)

P (x, y, t)

P (x, y, t)

.

We orthonormalize this basis using the Gram-Schmidtprocedure under the following metric (given two basiselements G1, G2 ∈ G):

(G1, G2) =

∫ 1

0

∫

S2

(G1 ·G2 +G1t ·G2

t +G1x ·G2

x

+ G1y ·G2

y +G1t,x ·G2

t,x +G1t,y ·G2

t,y)dsdt.

In practice, we use a subset containing 1400 basis ele-ments of P to approximate ∇EF (G).

The accuracy of this path-straightening algorithm de-pends on the number of basis elements used to ap-proximate the gradient. As we increase the number ofbasis elements, the approximation of the geodesic pathimproves. In Figure 6, we plot the distance in F betweenthe two displayed surfaces as a function of the degreeof spherical harmonics used in the basis construction.In this example, we fix the value of c3 to 2. We note

that as the degree increases, the distance between thesurfaces decreases and eventually stabilizes. The sameholds when we increase the value of c3, while holdingthe degree of spherical harmonics constant. This factorseems to play a smaller role in the accuracy of thegeodesic computation.

(a) (b)

Fig. 6. (a) Decrease in the distance as the degree ofspherical harmonics increases from 1 to 12 (the factor c3is kept constant at 2). (b) Decrease in the distance as thefactor c3 increases from 1 to 5 (the degree of sphericalharmonics is kept constant at 3).

Now we show some examples of geodesics in Fobtained using path-straightening. To demonstrate theeffectiveness of path-straightening, we consider a specialcase where f1 = f2 = f and initialize a path whereF (t) 6= f for t in the interior of the path. Of course,we expect the geodesic path to be F ∗(t) = f , a con-stant path. The results are displayed in Figure 7. Usingpath-straightening, we obtain a 91.4% decrease in theenergy function, and the resulting path is visibly thesame surface. When we increased the number of basiselements used to approximate ∇E, we found the energydecrease to be even greater. These paths are computedand displayed by discretizing at times ti = (i− 1)/6, i =1, 2, . . . , 7.

Some additional examples of geodesics in F are pre-sented later in Section 4.4. Once we have a geodesic pathF ∗ between any two points, the distance in the pre-shapespace between f1 and f2, dF (f1, f2), is the length of F ∗,as specified in Eqn. 7.

4 GEODESICS IN SHAPE SPACE SNow, we consider the problem of finding geodesics be-tween surfaces in S. This requires solving an additionaloptimization over the product SO(3)× Γ.

4.1 Optimization Over Rotation

We can use a gradient approach for this optimization,but instead we will use an approximate albeit efficienttechnique based on Procrustes analysis. For a fixed γ ∈ Γ,

8

Initial Path Geodesic in F E

View 1

View 2

Fig. 7. An example of geodesic computation in F .

the minimization over SO(3) is performed as follows.Compute the 3 × 3 matrix C =

∫

S2f1(s)f2(s)

Tds. Then,using the singular value decomposition C = UΣV T , wecan define the optimal rotation as O∗ = UV T (if thedeterminant of C is negative, the last column of V T

changes sign).

4.2 Optimization Over Re-Parameterization

In order to solve the optimization problem over Γ inEqn. 1, we will use a gradient approach. Although thisapproach has an obvious limitation of converging to alocal solution, it is still general enough to be applicable togeneral cost functions. Additionally, we have tried to cir-cumvent the issue of a local solution by taking multipleinitializations. This is similar to the gradient approachtaken in Kurtek et al. [25], [24]; the difference lies in thecost function used for optimization. In earlier papers,we address a problem of the type minγ∈Γ ‖q1− (q2, γ)‖2,where q1 and q2 are q-maps of f1 and f2 and here weminimize a cost function of the type dF (f1, f2 ◦ γ)2.

(a) (b)

Fig. 8. (a) Map between Γ and F . (b) Map betweentangent spaces of F and Γ.

We begin by defining a map ψ : Γ → [f2] by ψ(γ) =f2 ◦ γ, for a fixed f2 ∈ [f2]. Figure 8(a) shows this mappictorially. The cost function for optimization relates tothe quantity inside the parenthesis in Eqn. 1. Rather thantaking that quantity itself, we minimize its square forbetter numerical stability. Using Eqn. 7, we can definethe cost function to be: H : Γ → R,

H [γ] = dF (f1, f2 ◦ γ)2 = dF (f1, ψ(γ))2 , (9)

Fig. 9. Four examples of basis elements of Tγid(Γ).

where f2 = f2 ◦ γ0, and γ0, and γ denote the currentand the incremental re-parameterizations, respectively.If we have an orthonormal basis for Tγid

(Γ) and thedifferential of ψ, we can compute the full gradient of Hat γid and use it to update γ0. This leaves two remainingissues: (1) the specification of an orthonormal basis ofTγid

(Γ) and (2) the derivation of the differential ψ∗,γid.

The tangent space of Γ at identity γid is:

Tγid(Γ) = {b : S2 → T (S2)|b is a smooth vector field

and is tangential to S2}. (10)

We are going to construct an orthonormal basis forTγid

(Γ). For this purpose, we use gradients of sphericalharmonics. We denote the full orthonormal basis setsas B. We show four examples of this basis in Figure 9.(Further details can be found in Kurtek et al. [24].)

Now we determine the differential of the group actionψ at the identity, ψ∗,γid

: Tγid(Γ) → Tf2([f2]).

Proposition 3. Let b be a tangent vector field on S2, i.e.,

an element of Tγid(Γ) and let f2 = (f1

2 , f22 , f

32 ), where each

f j2 is a real-valued function on S

2. Then, ψ∗,γid(b) will also

have three components, (ψ1∗,γid

, ψ2∗,γid

, ψ3∗,γid

), given by theformula:.

ψj∗,γid

(b) = ∇f j2 · b, j = 1, 2, 3, (11)

where ∇f j2 is the gradient of f j

2 .

Figure 8(b) is a pictorial depiction of the map ψ∗,γid,

between tangent spaces of Γ and F .Now the full gradient of H with respect to γ is an

element of Tγid(Γ) given by:

dγ =∑

bi∈B

〈〈Ft(1), ψ∗,γid(bi)〉〉F (1)bi (12)

9

where F is the current geodesic in F between f1 andf2. This linear combination of the orthonormal basiselements of Tγid

(Γ) provides an incremental update off2 in the orbit [f2]. The optimal re-parameterization γ∗

is defined as the concatenation of all incremental updatesdγ. Figure 10 gives a pictorial depiction of this algorithm.It shows the fixed surface f1 and the search over the orbitof f2 to reach f2 ◦ γ∗.

Fig. 10. Matching of surfaces through re-parameterization.

4.3 Geodesics in SThe joint optimization results from alternating betweenoptimizations over Γ and SO(3) until convergence. Animportant question in this gradient approach is the ini-tialization over Γ. We use the following strategy. First, wenote that SO(3) forms a subset of the re-parameterizationgroup, Γ. We initialize the gradient search by utilizingthe largest irreducible finite subgroup of SO(3), call itK . Denote by k1, k2, . . . , k60 the elements of K ; each ofthem acts on F according to f ◦ ki. The initialization foroptimization over Γ is a search over K by solving for:

i = argmini=1,2,...,60

dF (f1, Oi(f2 ◦ ki))2 . (13)

Here, Oi is the optimal rotation of f2 ◦ hi to best matchf1, obtained through Procrustes analysis.

Now we present an example of optimization overSO(3)×Γ. Figure 11 displays the result of this minimiza-tion in matching two closed surfaces with dual bumpswith different placements. We display the matching be-tween surfaces by transferring the colormap from f1 ontothe corresponding points on f2. Thus, a good matchingimplies that similar features are shaded by similar colors.In the initial matching between the surfaces the bumpson the two surfaces do not match each other. But, afteroptimization, the grid on O∗(f2 ◦ γ∗) is stretched andcompressed such that the two bumps on the two surfacesare matched. The stretching and compression of the gridis clearly seen in the display of γ∗.

γ∗

Fig. 11. Comparison of initial matching between f1 andf2 and matching after gradient minimization.

Implementation: To compute geodesics in S, we needto solve two optimization problems in Eqn. 1. Althoughwe can use the methodology described in this paper tosolve for the optimal re-parameterization, γ∗, we use anapproximate but more efficient solution. It is importantto note that the two procedures provide very similarresults. We perform the joint optimization in two steps.Given two surfaces f1 and f2 as defined in previous sec-tions, we first obtain the optimal parameterization γ∗ off2 using the gradient technique described in [25]. Giventhe optimal parameterization of the second surface, weproceed to compute the geodesic path F ∗ in S. The resultof this procedure is an approximation of the geodesicdistance and path between f1 and O∗(f2 ◦ γ∗).Computational Cost: For these experiments, we used theMatlab environment on an Intel Xeon CPU (2.50GHz,8GB RAM, Windows XP). When we sample the surfaceswith 2500 points and use 1400 basis elements for thepath-straightening optimization, the average computa-tional cost for computing a geodesic in F (10 iterations)is approximately 90s, and in S it is approximately 100sdue to the extra computation of γ∗.

4.4 Comparisons of Geodesics in F and SIn this section we highlight the improvements in match-ing of surfaces during the optimization over SO(3) × Γby comparing geodesic paths between the same pairsof surfaces in F and S. In all of these experiments, wenotice that E[F ∗] for geodesics in S is typically an orderof magnitude smaller than the energies for geodesics inF .

In Figure 12, we display a simple example of geodesiccomputations for two surfaces with dual bumps. We alsodisplay the deformation vector field associated with theshooting vector, F ∗

t (0), for the geodesic, F ∗(t), computedin F . In the case of S, we display the tangent vectorfield resulting from the optimization over Γ and thedeformation vector field from the subsequent geodesiccomputation. We can clearly see from the geodesic pathand corresponding vector fields, that in the pre-shapespace the two bumps on the first surface are contracted

10

Geodesic in F ; E(F ∗) = 0.0502 Geodesic in S; E(F ∗) = 0.0096

(a) (b) (c)

Fig. 12. Top: Comparison of geodesics in the pre-shape and shape spaces. Bottom: (a) Shooting vector for thegeodesic in the pre-shape space. (b) Re-parameterization-based deformation vector field. (c) Shooting vector for thegeodesic in the shape space.

while new bumps are created. This results in the mid-point of the geodesic path having four smaller bumps. Inthe shape space, the effect is as if the bumps on the firstsurface move to the location of the bumps on the secondsurface. This geodesic is the result of improved matchingdue to the optimization over the re-parameterizationgroup.

Figure 13 (top row) displays an example of geodesicsbetween surfaces with different number and placementof peaks. The geodesic in F distorts the large peak onf1 as it is transformed into one of the peaks on f2.On the other hand, the geodesic in S preserves thesurface features much better. The second row displaysthe geodesic paths between identical heart surfaces butwith two different parameterizations, that is, f2=f1 ◦ γ.Since our metric is parameterization invariant, we expectthe distance between them to be zero. As shown in thefigure, the resulting geodesic in S is a constant and theassociated energy is almost zero. The third row displaysthe geodesics between a horse and a cow. Again, we see asignificant difference in geodesic energies. Furthermore,we can see nice feature preservation along the geodesicin the shape space. The legs of the horse match thelegs of the cow and the tail grows nicely. The lastrow displays the geodesic paths between surfaces oftwo chess pieces. The preservation of features alongthe geodesic in the shape space is clearly seen in thisexample. Figure 14 shows more examples of geodesicsbetween closed surfaces with various shapes. The lastrow in Figure 14 displays the geodesic paths betweensurfaces of a left pallidus and a right thalamus extractedfrom MRI scans of a human brain. Computing shapedifferences using geodesic paths between surfaces ofanatomical structures is a very important applicationof shape analysis. More natural geodesics provide uswith a more accurate measure of differences betweenanatomical surfaces, which can be critical in diseasediagnostics. Once again, the geodesic path in S has muchlower energy than that in F and represents a morenatural transformation from f1 to f2.

5 EXPERIMENTAL RESULTS

As mentioned earlier, the geodesic paths provide uswith tools for comparing, matching, and deformingparameterized surfaces. We suggest a comparison ofshapes of 3D objects using geodesic distances betweentheir boundary surfaces in the shape space. This sectionpresents a specific application to illustrate that idea.Classification of Anatomical Shapes

Shape analysis of anatomical structures can play animportant role in detection, classification, and moni-toring of different diseases, especially those that affectthe human brain. In this section, we study shapes ofseveral subcortical brain structures (hippocampus, puta-men, thalamus, caudate, etc.) of subjects and analyzetheir relationship to certain mathematical deficienciesexhibited by those subjects. We begin by computingthe pairwise geodesic distances between correspondingsubstructures for 106 subjects and we consider 20 suchstructures for each subject. Some examples of thesesubstructures are shown in Figure 15. We utilize a leave-

Caudate Putamen Thalamus Pallidus

Fig. 15. Examples of subcortical structures.

one-out nearest neighbor classifier to assign a subject tothe case or control group. Table 1 provides the singlestructure classification results.

The best single structure classifier is the left inferiorparietal lobe, which achieves a 63.2% classification rate.This result is supported by some previous studies, whichhave shown that the inferior and superior parietal lob-ules and the intraparietal sulcus (all part of the parietallobes) activate during a number deviants study [14]. Theother single classifiers, which yield best performance arethe right pallidus (59.4%), the left posterior cingulate

11

Pre-Shape Space Shape Space

E(F ∗) = 0.0379 E(F ∗) = 0.0118

E(F ∗) = 0.0896 E(F ∗) = 0.0028

E(F ∗) = 0.0409 E(F ∗) = 0.0211

E(F ∗) = 0.0757 E(F ∗) = 0.0618

Fig. 13. Comparison of geodesics in F and S, and their energies. Top: two simple surfaces with different number andplacements of peaks. Second row: two heart surfaces with different parameterizations. Third row: a horse and a cow.Last row: two chess pieces

gyrus (59.4%) and the left superior parietal lobe (58.5%).The pallidus is a major component of the basal ganglia,which along with the thalamus form a very importantconnection in the human brain. Previous studies haveshown that the basal ganglia play an important role inlearning and cognitive functioning [31], [27]. The pal-lidus has also been identified as a very important deter-minant in working memory, which is very closely linkedto mathematics performance [2]. Finally, the posteriorcingulate gyrus is known to be involved in memoryformation and retrieval, which can also play a role inmathematics performance.

We can improve classification rates by combiningdistances for individual structures. The objective hereis to maximize the classification rate of disease andcontrols using some combination of distances based onindividual structures. We exhaustively search over adiscrete set of weights by taking a few structures at atime. Let us define dtot =

∑ni=1 widi, where wi ≥ 0

such that∑n

i=1 wi = 1 and dis are the geodesic dis-tances for individual structures. We start with the singlestructure that provides the highest classification rate andproceed by adding new structures one at a time, suchthat the classification rate improves maximally at eachstep. We stop when no more improvement is noticed.

One combination of three structures resulted in a 71.7%classification rate. This combination consisted of: 0.5984L Inferior Parietal Lobe + 0.1014 R Anterior CingulateGyrus + 0.3002 R Putamen.

6 CONCLUSION AND DISCUSSION

Shape analysis of 3D objects is very important in manyscientific fields. We have proposed a novel Rieman-nian framework for computing geodesic paths betweenshapes of parameterized surfaces. These geodesics areinvariant to rigid motion, scaling and most importantlyre-parametrization of individual surfaces. The geodesiccomputation is based on a path-straightening techniquethat iteratively corrects paths between surfaces untilgeodesics are achieved. The iterative update is based onthe gradient of a path energy; this gradient is approx-imated using a large number of basis elements in theperturbation space. We have presented some examples ofgeodesics between surfaces in the pre-shape and shapespaces and utilized the distances between surfaces forclassification of anatomical shapes. An important appli-cation of this framework is in computations of means,covariances, and probability models to capture shapevariability within shape classes [23]. In this framework,

12

Pre-Shape Space Shape Space

E(F ∗) = 0.0535 E(F ∗) = 0.0293

E(F ∗) = 0.0323 E(F ∗) = 0.0153

E(F ∗) = 0.0410 E(F ∗) = 0.0107

E(F ∗) = 0.1012 E(F ∗) = 0.0353

E(F ∗) = 0.0239 E(F ∗) = 0.0046

Fig. 14. Top: a turtle and a hydrant. Second row: a bird and a duck. Third row: two shoes. Fourth row: a heart and aduck. Last row: left pallidus and right thalamus.

TABLE 1Classification performance for individual substructures.

Structure Caudate Hippocampus Pallidus Inferior Parietal Lobe Posterior Cingulate Gyrus InsulaSide L R L R L R L R L R

Our Method (%) 49.1 51.9 52.8 53.8 53.8 59.4 63.2 51.9 59.4 55.7

Structure Thalamus Precentral Putamen Superior Parietal Lobe Anterior Cingulate Gyrus PrecuneusSide L R L R L R L R R R

Class. Rate (%) 53.8 50.9 56.6 49.1 52.8 45.3 58.5 46.2 45.3 46.2

we assume spherical parameterizations of surfaces. Ob-taining such parameterizations for arbitrary surfaces isnot straightforward, and we are exploring approachesfrom graphics such as the one presented in [30]. Thereare two main limitations of this work. First, the metricused here is not invariant to translations. Second, thereis no physical interpretation of the metric induced onthe space of parameterized surfaces. One would like ametric, which can be interpreted as a combination ofbending and stretching. In the future, we would liketo explore alternative choices for the metric, which canavoid these problems.

As stated earlier, this framework is easily extended toother types of parameterized surfaces, such as quadrilat-

eral surfaces. Due to limited space, we have not provideddetails for this case. Instead, we provide some examplesfor illustration purposes. First, we show the effect ofoptimizing over the re-parameterization group in Figure16. In this example, f1 has one high peak, while f2 hasone high peak and one low peak. In the initial matchingbetween the surfaces the two high peaks do not matcheach other. But, after applying the optimal rotation O∗

and re-parameterizaton γ∗ to f2 the high peak on f1matches the high peak on O∗(f2 ◦ γ∗) very well. Inaddition, we provide two examples (in Figure 17) ofgeodesic path computations in the shape and pre-shapespace for toy surfaces. We can draw the same conclusions

13

Pre-Shape Space Shape Space

E(F ∗) = 0.0345 E(F ∗) = 0.0156

E(F ∗) = 0.1274 E(F ∗) = 0.0193

Fig. 17. Comparison of geodesics in F and S and their energies. Top: surfaces formed by images with two peakseach at different locations. Second row: surfaces of revolution.

γ∗

Fig. 16. Comparison of initial matching between f1 andf2 and matching after optimization over Γ.

from these examples as was done in Section 4.4. That is,the geodesic path energy is much smaller in the shapespace due to improved feature matching across surfaces.Acknowledgements: We would like to thank Mr. NathanLay from the Florida State University Department ofScientific Computing for his help in implementing themethods described in this paper. This research wassupported in part by AFOSR FA9550-06-1-0324, ONRN00014-09-1-0664, and NSF DMS-0915003 (AS).

REFERENCES

[1] A. Almhdie, C. Leger, M. Deriche, and R. Ledee. 3D registrationusing a new implementation of the ICP algorithm based on acomprehensive lookup matrix: Application to medical imaging.Pattern Recognition Letters, 28(12):1523–1533, 2007.

[2] M.H. Ashcraft and J.A. Krause. Working memory, math per-formance, and math anxiety. Psychonomic Bulletin and Review,14(2):243–248, 2007.

[3] S. Bouix, J.C. Pruessner, D.L. Collins, and K. Siddiqi. Hip-pocampal shape analysis using medial surfaces. NEUROIMAGE,25:1077–1089, 2001.

[4] C. Brechbuhler, G. Gerig, and O. Kubler. Parameterization ofclosed surfaces for 3D shape description. Computer Vision ImageUnderstanding, 61(2):154–170, 1995.

[5] J. Cates, M. Meyer, P.T. Fletcher, and R. Whitaker. Entropy-based particle systems for shape correspondence. In proc. of MIC-CAI Workshop Mathematical Foundations of Computational Anatomy,pages 90–99, 2006.

[6] T.F. Cootes, C.J. Taylor, D.H. Cooper, and J. Graham. Active shapemodels, – their training and application. Computer Vision and ImageUnderstanding, 61(1):38–59, 1995.

[7] J.G. Csernansky, L. Wang, D.J. Jones, D. Rastogi-Cru, G. Posener,J.A. Heydebrand, J.P. Miller, U. Grenander, and M.I. Miller.Hippocampal deformities in schizophrenia characterized by highdimensional brain mapping. American Journal of Psychiatry, 159:1–7, 2002.

[8] C. Davatzikos, M. Vaillant, S. Resnick, J.L. Prince, S. Letovsky, andR.N. Bryan. A computerized method for morphological analysisof the corpus callosum. Journal of Computer Assisted Tomography,20:88–97, 1996.

[9] R.H. Davies, C.J. Twining, T.F. Cootes, and C.J. Taylor. Building3-d statistical shape models by direct optimization. IEEE Trans.Medical Imaging, 29(4):961–981, 2010.

[10] I. L. Dryden and K. V. Mardia. Statistical Shape Analysis. JohnWiley & Son, 1998.

[11] G. Gerig, M. Styner, M.E. Shenton, and J.A. Lieberman. Shapeversus size: Improved understanding of the morphology of brainstructures. In Proc. of MICCAI, pages 24–32, 2001.

[12] K. Gorczowski, M. Styner, J.Y. Jeong, J.S. Marron, J. Piven, H.C.Hazlett, S.M. Pizer, and G. Gerig. Multi-object analysis of volume,pose, and shape using statistical discrimination. IEEE Trans.Pattern Analysis and Machine Intelligence, 32(4):652–666, 2010.

[13] U. Grenander and M. I. Miller. Computational anatomy: Anemerging discipline. Quarterly of Applied Mathematics, LVI(4):617–694, 1998.

[14] L. Gross. A neural seat for math? PLoS Biol, 4(5), 2006.

[15] X. Gu and B. C. Vemuri. Matching 3D shapes using 2D conformalrepresentations. In Proc. of MICCAI, pages 771–780, 2004.

[16] X. Gu, S. Wang, J. Kim, Y. Zeng, Y. Wang, H. Qin, and D. Samaras.Ricci flow for 3D shape analysis. In Proc. of IEEE InternationalConference on Computer Vision, 2007.

[17] S.C. Joshi, M.I. Miller, and U. Grenander. On the geometry and

14

shape of brain sub-manifolds. Pattern Recognition and ArtificialIntelligence, 11:1317–1343, 1997.

[18] S.H. Joshi, E. Klassen, A. Srivastava, and I. Jermyn. Remov-ing shape-preserving transformations in square-root elastic (SRE)framework for shape analysis of curves. In Proc. of 6th Intl. Conf.on Energy Minimization Methods in Computer Vision and PatternRecognition (EMMCVPR), Hubei, China, pages 387–398, 2007.

[19] S.H. Joshi, E. Klassen, A. Srivastava, and I.H. Jermyn. A novelrepresentation for Riemannian analysis of elastic curves in Rn. InProc. of IEEE Conference on Computer Vision and Pattern Recognition,pages 1–7, 2007.

[20] A. Kelemen, G. Szekely, and G. Gerig. Elastic model-basedsegmentation of 3d neurological data sets. IEEE Trans. MedicalImaging, 18(10):828–839, 1999.

[21] M. Kilian, N. J. Mitra, and H. Pottmann. Geometric modeling inshape space. In Proc. of SIGGRAPH, 2006.

[22] E. Klassen and A. Srivastava. Geodesics between 3D closedcurves using path-straightening. In Proc. of European Conferenceon Computer Vision, Lecture Notes in Computer Science, pages I: 95–106, 2006.

[23] S. Kurtek, E. Klassen, Z. Ding, M.J. Avison, and A. Srivastava.Parameterization-invariant shape statistics and probabilistic clas-sification of anatomical surfaces. In Gbor Szkely and Horst K.Hahn, editors, Proc. of Information Processing in Medical Imaging,volume 6801 of Lecture Notes in Computer Science, pages 147–158.Springer, 2011.

[24] S. Kurtek, E. Klassen, Z. Ding, S. Jacobson, J. Jacobson, M. Avison,and A. Srivastava. Parameterization-invariant shape comparisonsof anatomical surfaces. IEEE Trans. Medical Imaging, 30(3):849–858,2011.

[25] S. Kurtek, E. Klassen, Z. Ding, and A. Srivastava. A novelRiemannian framework for shape analysis of 3D objects. In Proc.of IEEE Computer Vision Pattern Recognition, 2010.

[26] R. Malladi, J.A. Sethian, and B.C. Vemuri. A fast level set basedalgorithm for topology-independent shape modeling. Journal ofMathematical Imaging and Vision, 6:269–290, 1996.

[27] F. McNab and T. Klingberg. Prefrontal cortex and basal gangliacontrol access to working memory. Nature Neuroscience, 11:103–107, 2008.

[28] W. Mio, A. Srivastava, and S. H. Joshi. On shape of plane elasticcurves. Intl. Journal of Computer Vision, 73(3):307–324, 2007.

[29] S. Osher and R. Fedkiw. Level Set Methods and Dynamic ImplicitSurfaces. Springer Verlag, 2003.

[30] E. Praun and H. Hoppe. Spherical parametrization and remesh-ing. In Proc. of ACM SIGGRAPH, volume 22, pages 340–349, 2003.

[31] C.A. Seger. The basal ganglia in human learning. Neuroscientist,12(4):285–290, 2006.

[32] K. Siddiqi and S. Pizer, editors. Medial Representations: Mathe-matics, Algorithms and Applications. Kluwer Academic Publishers,2006.

[33] A. Srivastava, E. Klassen, S.H. Joshi, and I.H. Jermyn. Shapeanalysis of elastic curves in euclidean spaces. IEEE Trans. PatternAnalysis and Machine Intelligence, 33(7):1415–1428, 2011.

[34] M. Styner, I. Oguz, S. Xu, C. Brechbuhler, D. Pantazis, J. Levitt,M.E. Shenton, and G. Gerig. Framework for the statistical shapeanalysis of brain structures using SPHARM-PDM. In Proc. ofMICCAI Open Science Workshop, 2006.

[35] M. Vaillant and J. Glaunes. Surface matching via currents. In Proc.of Information Processing in Medical Imaging, pages 381–392, 2005.

[36] O. van Kaick, H. Zhang, G. Hamarneh, and D. Cohen-Or. Asurvey on shape correspondence. In Proc. of Eurographics State-of-the-art Report, 2010.

[37] L. Younes. Computable elastic distance between shapes. SIAMJournal of Applied Mathematics, 58(2):565–586, 1998.

[38] L. Younes, P. W. Michor, J. Shah, D. Mumford, and R. Lincei.A metric on shape space with explicit geodesics. Matematica EApplicazioni, 19(1):25–57, 2008.

Sebastian Kurtek Sebastian Kurtek received his BS degree in Mathe-matics from Tulane University in 2007, and MS degree in Biostatisticsfrom the Florida State University in 2009, where he is currently a PhDcandidate.

Eric Klassen Eric Klassen is a professor of mathematics at FloridaState University. He earned his PhD in Mathematics at Cornell Universityin 1987, and his research areas include 3-dimensional topology, gaugetheory, Riemann surfaces, and computer image analysis.

John C. Gore John C. Gore received the Ph.D. degree in physics fromthe University of London, London, U.K., in 1976. He is the Director of theInstitute of Imaging Science and Hertha Ramsey Cress University Pro-fessor of Radiology and Radiological Sciences, Biomedical Engineering,Physics, and Molecular Physiology and Biophysics at Vanderbilt Univer-sity, Nashville, TN. His research interests include the development andapplication of imaging methods for understanding tissue physiology andstructure, molecular imaging, and functional brain imaging. Dr. Gore isa Member of National Academy of Engineering and an elected Fellowof the American Institute of Medical and Biological Engineering, theInternational Society for Magnetic Resonance in Medicine (ISMRM), andthe Institute of Physics (U.K.). In 2004, he was awarded the Gold Medalfrom the ISMRM for his contributions to the field of magnetic resonanceimaging. He is Editor-in-Chief of the journal Magnetic Resonance Imag-ing.

Zhaohua Ding Zhaohua Ding received the B.E. degree in biomedicalengineering from the University of Electronic Science and Technologyof China, Sichuan, in 1990, the M.S. degree in computer scienceand the Ph.D. degree in biomedical engineering, both from The OhioState University, Columbus, in 1997 and 1999, respectively. He wasa Research Fellow at the Department of Diagnostic Radiology, YaleUniversity, New Haven, CT, from 1999 to 2002. From July 2004, he wasan Assistant Professor at the Vanderbilt University Institute of ImagingScience and Department of Radiology and Radiological Sciences. Hisresearch focuses on processing and analysis of magnetic resonanceimages and clinical applications

Anuj Srivastava Anuj Srivastava is a Professor of Statistics at theFlorida State University in Tallahassee, FL. He obtained his MS andPhD degrees in Electrical Engineering from the Washington Universityin St. Louis in 1993 and 1996, respectively. After spending the year1996-97 at the Brown University as a visiting researcher, he joined FSUas an Assistant Professor in 1997. His research is focused on patterntheoretic approaches to problems in image analysis, computer vision,and signal processing. In particular, he has developed computationaltools for performing statistical inferences on certain nonlinear manifoldsand has published over 140 journal and conference articles in theseareas.