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  • Computational Science Center, University of Vienna, Vienna, Austria

Variational Methods in Shape Analysis


The concept of a shape space is linked both to concepts from geometry and from physics. On one hand, a path-based viscous flow approach leads to Riemannian distances between shapes, where shapes are boundaries of objects that mainly behave like fluids. On the other hand, a state-based elasticity approach induces a (by construction) non-Riemannian dissimilarity measure between shapes, which is given by the stored elastic energy of deformations matching the corresponding objects. The two approaches are both based on variational principles. They are analyzed with regard to different applications, and a detailed comparison is given.


The analysis of shapes as elements in a frequently infinite-dimensional space of shapes has attracted increasing attention over the last decade. There are pioneering contributions in the theoretical foundation of shape space as a Riemannian manifold as well as path-breaking applications to quantitative shape comparison, shape recognition, and shape statistics. The aim of this chapter is to adopt a primarily physical perspective on the space of shapes and to relate this to the prevailing geometric perspective. Indeed, we here consider shapes given as boundary contours of volumetric objects, which consist either of a viscous fluid or an elastic solid.

In the first case, shapes are transformed into each other via viscous transport of fluid material, and the flow naturally generates a connecting pathin the space of shapes. The viscous dissipation rate - the rate at which energy is converted into heat due to friction - can be defined as a metric on an associated Riemannian manifold. Hence, via the computation of shortest transport paths one defines a distance measure between shapes.

In the second case, shapes are transformed via elastic deformations, where the associated elastic energy only depends on the final stateof the deformation and not on the path along which the deformation is generated. The minimal elastic energy required to deform an object into another one can be considered as a dissimilarity measure between the corresponding shapes.

In what follows we discuss and extensively compare the path-based and the state-based approach. As applications of the elastic shape model, we consider shape averages and a principal component analysis of shapes. The viscous flow model is used to exemplarily cluster 2D and 3D shapes and to construct a flow type nonlinear interpolation scheme. Furthermore, we show how to approximate the viscous, path-based approach with a time-discrete sequence of state-based variational problems.

Background A review of different shape space concepts

The structure of shape spaces and statistical analyses of shapes have been examined in various settings, and applications range from the computation of priors for segmentation [16, 43, 17] and shape classification [25, 50, 48, 44] to the construction of standardized anatomical atlases [37, 66, 14]. Among all existing approaches, a number of different concepts of a shape are employed, including landmark vectors [16, 39], planar curves [41, 84, 52], surfaces in R3 [25, 40, 24], boundary contours of objects [44, 31, 67], multiphase objects [83] as well as the morphologies of images [22].

The analysis of a shape space is typically based on a notion of a distance or dissimilarity measure d( ⋅, ⋅) between shapes [50, 31, 67, 10, 54, 51], whose definition frequently takes a variational form. This distance can be used to define an average [27, 67] or a median [28, 4] of given shapes according to for p = 1 and p = 2, respectively (cf. Sect. 4.1.1). Likewise, shape variations can be obtained by a principal component analysis (PCA, cf. Sect. 4.1.2) or a more general covariance analysis in a way which is consistent with the dissimilarity measure between shapes [11, 16, 27, 68]. From the conceptional point of view, one can distinguish two types of these dissimilarity or distance measures which may be characterized as rather state based or path based, respectively. While the first approach is independent of the notion of paths of shapes, the latter distance definition requires the computation of an optimal, connecting path in shape space. In some cases, both concepts coincide: The Euclidean distance between two points, e.g., can equivalently be interpreted in a state-based manner as the norm of the difference vector or as the length of the shortest connecting path (we shall provide a physical interpretation for each case in Sect. 3.1).

The notion of a shape space was already introduced by Kendall in 1984 [39], who considers shapes as k-tuples of points in Rd , endowed with the quotient metric of Rkd with respect to similarity transforms. Often, however, a shape space is just modeled as a linear vector space which is not invariant with respect to shift or rotation a priori. In the simplest case, such a shape space is made up of vectors of landmark positions, and distances between shapes can be evaluated in a state-based manner as the Euclidean norm of their difference. Chen and Parent [12] investigated averages of 2D contours already in 1989. Cootes et al. perform a PCA on training shapes with consistently placed landmarks to obtain priors for edge-based image segmentation [16]. Hafner et al. use a PCA of position vectors covering the proximal tibia to reconstruct the tibia surface just from six dominant modes [35]. Perperidis et al. automatically assign consistent landmarks to training shapes by a non-rigid registration as a preprocessing step for a PCA of the cardiac anatomy [63]. Söhn et al. compute dominant eigenmodes of landmark displacement on human organs, also using registration for preprocessing [73].

As an infinite-dimensional vector space, the Lebesgue-space L 2has served as shape space, where again shape alignment is a necessary preprocessing step. Leventon et al. identify shapes with their signed distance functions and impose the Hilbert space structure of L 2on them to compute an average and dominant modes of variation [43]. Tsai et al. apply the same technique to 3D prostate images [79]. Dambreville et al. also compute shape priors, but using characteristic instead of signed distance functions [19].

A more sophisticated state-based shape space is obtained by considering shapes as subsets of an ambient space with a metric d( ⋅, ⋅) and endowing them with the Hausdorff distance

between any two shapes . Charpiat et al. employ smooth approximations of the Hausdorff distance based on a comparison of the signed distance functions of shapes [10]. For a given set of shapes, the gradient of the shape distance functional at the average shape is regarded as shape variation of the average and used to analyze its dominant modes of variation [11]. Frame indifference is mimicked by an inner product that weights rotations, shifts, scalings, and the orthogonal complement to these transformations differently. Charpiat et al. also consider gradient flow morphing from one shape onto another one which can be regarded as a means to obtain meaningful paths even in shape spaces with state-based distance measures.

An isometrically invariant distance measure between shapes (or more general metric spaces) that is also not based on connecting paths is provided by the Gromov-Hausdorff distance, which can be defined variationally as
where is a distance measure between points in . The Gromov-Hausdorff distance represents a global, supremum-type measure of the lack of isometry between two shapes. Memoli and Sapiro use this distance for clustering shapes described by point clouds, and they discuss efficient numerical algorithms to compute Gromov-Hausdorff distances based on a robust notion of intrinsic distances on the shapes [50]. Bronstein et al. incorporate the Gromov-Hausdorff distance concept in various classification and modeling approaches in geometry processing [6]. Memoli investigates the relation between the Gromov-Hausdorff distance and the Hausdorff distance under the action of Euclidean isometries as well as L p -type variants of the Gromov-Hausdorff distance [49].

In [46], Manay et al. define shape distances via integral invariants of shapes and demonstrate the robustness of this approach with respect to noise.

Another distance or dissimilarity measure which also measures the lack of isometry between shapes can be obtained by interpreting shapes as boundaries of physical objects and measuring the (possibly nonlinear) deformation energy of an elastic matching deformation ϕ between two objects [67, 36]. Since, by the axiom of elasticity, this energy solely depends on the original and the final configuration of the deformed object but not on the deformation path, the elastic dissimilarity measure can clearly be classified as state based (as will be detailed in Sect. 3.2.2). This physical approach comes along with a natural linearization of shapes via boundary stresses to perform a covariance analysis [68] and will be presented in Sect. 4.1. Pennec et al. define a nonlinear elastic energy as the integral over the ambient space of an energy density that depends on the logarithm of the Cauchy-Green strain tensor [62, 61], which induces a symmetric state-based distance.

Typical path-based shape spaces have the structure of a Riemannian manifold. Here, the strength of a shape variation is measured by a Riemannian metric, and the square root of the Riemannian metric evaluated on the temporal shape variation is integrated along a path of shapes to yield the path length. The length of the shortest path between two shapes represents their geodesic distance d( ⋅, ⋅). Averages are obtained via the Fréchet mean [30], which was further analyzed by Karcher [38]. There is also a natural linear representation of shapes in the tangent space at the Fréchet mean via the logarithmic map, which enables a PCA.

A Riemannian shape space which might still be regarded as rather state- than path-oriented is given by the space of polygonal medial axis representations, where each shape is described by a polygonal lattice and spheres around each vertex [87]: Here, the Lie group structure of the medial representation space can be exploited to approximate the Fréchet mean as exponential map of the average of the logarithmic maps of the input. Fletcher et al. perform a PCA on these log-maps to obtain the dominant geometric variations of kidney shapes [27] and brain ventricles [26]. Fuchs and Scherzer use the PCA on log-maps to obtain the covariance of medial representations, and they use a covariance-based Mahalanobis distance to impose a new metric on the shape manifold. This metric is employed to obtain priors for edge-based image segmentation [32, 33].

Kilian et al. compute and extrapolate geodesics between triangulated surfaces of fixed mesh topology, using isometry invariant Riemannian metrics that measure the local distortion of the grid [40]. Eckstein et al. employ different metrics in combination with a smooth approximation to the Hausdorff distance to perform gradient flows for shape matching [24]. Liu et al. use a discrete exterior calculus approach on simplicial complexes to compute geodesics and geodesic distances in the space of triangulated shapes, in particular taking care of higher genus surfaces [45].

An infinite-dimensional Riemannian shape space has been developed for planar curves. Klassen et al. propose to use as a Riemannian metric, the L 2-metric on variations of the direction or curvature functions of arclength-parameterized curves. They implement a shooting method to find geodesics [41], while Schmidt and Cremers present an alternative variational approach [70]. Srivastava et al. assign different weights to the L 2-metric on stretching and on bending variations and obtain an elastic model of curves [75]. Michor and Mumford examine Riemannian metrics on the manifold of smooth regular curves [51]. They show the standard L 2-metric in tangent space, leading to arbitrarily short geodesics and hence employ a curvature-weighted L 2-metric instead. Yezzi and Mennucci resolved the problem taking into account the conformal factor in the metric [84]. Sundaramoorthi et al. use Sobolev metrics in the tangent space of planar curves to perform gradient flows for image segmentation via active contours [76]. Michor et al. discuss a specific metric on planar curves, for which geodesics can be described explicitly [52]. In particular, they demonstrate that the sectional curvature on the underlying shape space is bounded from below by zero, which points out a close relation to conjugate points in shape space and thus to only locally shortest geodesics. Finally, Younes considers a left-invariant Riemannian distance between planar curves by identifying shapes with elements of a Lie group acting on one reference shape [85].

When warping objects bounded by shapes in Rd , a shape tube in Rd + 1 is formed. Delfour and Zolésio [20] rigorously develop the notion of a Courant metric in this context. A further generalization to classes of non-smooth shapes and the derivation of the Euler-Lagrange equations for a geodesic in terms of a shortest shape tube is investigated by Zolésio in [88].

Dupuis et al. [23] and Miller et al. [54, 53] define the distance between shapes based on a flow formulation in the embedding space. They exploit the fact that in case of sufficient Sobelev regularity for the motion field von the whole surrounding domain Ω, the induced flow consists of a family of diffeomorphisms. This regularity is ensured by a functional ∫0 1Ω Lvv dx dt, where Lis a higher-order elliptic operator [76, 85]. Geometrically, ∫ Ω Lvv dxis the underlying Riemannian metric, and we will discuss related, path-based concepts in Sect. 3.2.1. Under sufficient smoothness assumptions, Beg et al. derive the Euler-Lagrange equations for the diffeomorphic flow field [3]. To compute geodesics between hypersurfaces in the flow of diffeomorphism framework, a penalty functional measures the distance between the transported initial shape and the given end shape. Vaillant and Glaunès [80] identify hypersurfaces with naturally associated two forms and used the Hilbert space structures on the space of these forms to define a mismatch functional. The case of planar curves is investigated under the same perspective by Glaunès et al. in [34]. To enable the statistical analysis of shape structures, parallel transport along geodesics is proposed by Younes et al. [86] as the suitable tool to transfer structural information from subject-dependent shape representations to a single template shape.

In most applications, shapes represent boundary contours of physical objects. Fletcher and Whitaker adopt this viewpoint to develop a model for geodesics in shape space which avoids overfolding [29]. Fuchs et al. [31] propose a Riemannian metric on a space of shape contours, motivated by linearized elasticity. This metric can be interpreted as the rate of physical dissipation during the deformation of a viscous liquid object [83, 82] and will be elaborated in Sect. 4.2.

Finally, a shape space is sometimes understood as a manifold, learnt from training shapes and embedded in a higher-dimensional (often linear) space. Many related approaches are based on kernel density estimation in feature space. Here, the manifold is described by a probability distribution in the embedding space, which is computed by mapping points of the embedding space into a higher-dimensional feature space and assuming a Gaussian distribution there. In general, points in feature space have no exact preimage in shape space, so that approximate preimages have to be obtained via a variational formulation [64]. Cremers et al. use this technique to obtain 2D silhouettes of 3D objects as priors for image segmentation [17]. Rathi et al. provide a comparison between kernel PCA, local linear embedding (LLE), and kernel LLE (kernel PCA only on the nearest neighbors) [65]. Thorstensen et al. approximate the shape manifold using weighted Karcher means of nearest neighbor shapes obtained by diffusion maps [77].

Mathematical Modeling and Analysis

Recalling the Finite-Dimensional Case

At first, let us investigate distances and their relation to concepts from physics in the simple case of Euclidian space. In Euclidean space, shortest paths are straight lines, and they are unique, so that the distance computation involves only the states of the two end points: The geodesic distance between any two points x1,x2Rd is given by the norm of the difference, ∥ x 2x 12, which implies the equivalence of the state-based and the path-based perspective. A corresponding physical view might be the following. Considering that - by Hooke's law - the stored elastic energy of an elastic spring extended from x 1to x 2is given by for the spring constant C, the distance can be interpreted in a state-based manner as the square root of the elastic spring energy (Fig. 1). Likewise, from a path-based point of view, the minimum dissipated energy of a dashpot which is extended from x 1to x 2at constant speed within the fixed time interval [0, 1] reads Diss = ∫0 1v 2 2 dt = 2μx 2x 1 2 2, where 2μ is the dashpot parameter and the velocity is given by v = x 2x 1. Using this physical interpretation, we can express for instance the arithmetic mean of a given set of points x1,...,xnRd either as the minimizer of the total elastic deformation energy in a system, where the average xis connected to each x i by elastic springs or as the minimizer of the total viscous dissipation when extending dashpots from x i to x.
Fig. 1 The force Fof an elastic spring between x 1and x 2is proportional to (x 2x 1), as well as the force Fof a dashpot which is extended from x 1to x 2within time 1 at constant velocity v. The spring energy reads and the dashpot dissipation Diss = ∫ \nolimits \nolimits Fv dt = 2μx 2x 1 2 2

Before we investigate the same concepts on more general Riemannian manifolds, let us briefly recall some basic notation. A Riemannian manifold is a set that is locally diffeomorphic to Euclidean space. Given a smooth path x(t) ∈ , t ∈ [0, 1], we can define its derivative at time tas a tangent vector to at x(t). The vector space of all such tangent vectors makes up the tangent space T x(t) , and it is equipped with the metric g x(t)( ⋅, ⋅) as the inner product. The length of a path x(t) ∈ , t ∈ [0, 1], is defined as , and locally shortest paths are denoted geodesics. They can be shown to minimize [21, Lemma 2.3]. Let us emphasize that a general geodesic is only locally the shortest curve. In particular, there might be multiple geodesics of different length connecting the same end points. The geodesic distance between two points is the length of the shortest connecting path. Finally, for a given xthere is a bijection exp x : T x of a neighborhood of 0 ∈ T x into a neighborhood of xthat assigns to each tangent vector vT x the end point of the geodesic emanating from xwith initial velocity vand running over the time interval [0, 1] [42, Theorem 1.6.12] or [74, Chap. 9, Theorem 14].

We can now define the (possibly non-unique, cf. Sect. 5) mean xof a number of npoints x 1, , x nin analogy to the Euclidian case as , where d( ⋅, ⋅) is the Riemannian distance on . This average is uniquely defined as long as the geodesics involved in the distance computation are unique, and it has been investigated in differential geometry by Karcher [38]. Furthermore, on a Riemannian manifold , the inverse exponential map log x = exp x − 1provides a method to obtain representatives log x (x i ) ∈ T x of given input points x iin the (linear) vector space T x (Fig. 2). On these, we can perform a PCA, which is by definition a linear statistical tool.
Fig. 2 The logarithmic map assigns each point x i on the manifold a vector in the tangent space T x , which may be seen as a linear representative

In a Riemannian space , the path-based approach can immediately be applied by exploiting the Riemannian structure, and can be considered as the energy dissipation spent to move a point from x(0) to x(1) along a geodesic. The logarithms log x (x i ) in this model correspond to the initial velocities of the transport process leading from xto x i . When applying the state-based elastic model in , however, there is no mechanically motivated notion of paths and thus also no logarithmic map. Only if we suppose that the Riemannian structure of the space is not induced by changes in the inner structure of our objects, the physical model based on elastic springs still coincides with the viscous model: We consider elastic springs stretched on the surface and connecting the points xand x i with a stored energy . Then, as before in the Euclidian case, a state-based average xof input points x 1, , x n can be defined. Furthermore, interpreting spring forces acting on xand pointing toward x i as linear representatives of the input points x i , one can run a PCA on these forces as well. However, for any reasonable (even finite-dimensional) model of shape space, objects are not rigid, and the inner relation between points as subunits (such as the vertex points of polygonal shapes) essentially defines the Riemannian (and thus the path-based) structure of the space : The rate of dissipation along a path in shape space depends on the interaction of object points. Physically, the corresponding point interaction energy is converted into thermal energy via friction. This dissipation depends significantly on the path in shape space traversed from one shape to the other. In contrast, when applying the state-based approach to the same shape space, we directly compare the inner relations between the subunits, i.e., we have no history of these relations. This comparison can be quantified based on a stored (elastic) interaction energy which is then a quantitative measure of the dissimilarity of the two objects but in general no metric distance.

Path-Based Viscous Dissipation Versus State-Based Elastic Deformation for Non-rigid Objects

In the following, we will especially consider two different physically motivated perspectives on a shape space of non-rigid volumetric objects in more detail. In the first case, we will adopt a path-based view, motivated by the theory of viscous fluids, while the second, state-based approach will be motivated by elasticity.

We will regard shapes as boundaries of domains which will be interpreted as physical objects. The resulting shape space structure depends on the particular type of physical objects : An interpretation of as a blob of a viscous fluid will yield an actually Riemannian, path-based shape space, while the interpretation as an elastic solid results in a state-based perspective, which will turn out to be non-Riemannian by construction.

Path-Based, Viscous Riemannian Setup

Shapes will be modeled as the boundary contour of a physical object that is made of a viscous fluid. The object might be surrounded by a different fluid (e.g., with much lower viscosity and compression modulus), nevertheless, without any restriction we will assume void outside the object in the derivation of our model. Here, viscositydescribes the internal resistance in a fluid and is a macroscopic measure of the friction between fluid particles, e.g., the viscosity of honey is significantly larger than that of water. The friction is described in terms of the stress tensor σ = (σ ij ) ij = 1, …d , whose entries describe a force per area element. By definition, σ ij is the force component along the ith coordinate direction acting on the area element with a normal pointing in the jth coordinate direction. Hence, the diagonal entries of the stress tensor σ refer to normal stresses, e.g., due to compression, and the off-diagonal entries represent tangential (shear) stresses. The Cauchy stress law states that due to the preservation of angular momentum, the stress tensor σ is symmetric [13].

In a Newtonian fluid, the stress tensor is assumed to depend linearly on the gradient : = of the velocity v. In case of a rigid body motion the stress vanishes. A rotational component of the local motion is generated by the antisymmetric part of the velocity gradient, and it has the local rotation axis ∇ ×vand local angular velocity | ∇ ×v | [78]. Thus, as rotations are rigid body motions, the stress only depends on the symmetric part of the velocity gradient. For an isotropic Newtonian fluid we get σ ij = λδ ijk (ε[v]) kk + 2με[v] ij , or in matrix notation σ = λtrε[v]1 + 2με[v], where 1 is the identity matrix. The parameter λ is denoted Lamé's first coefficient. The local rate of viscous dissipation - the rate at which mechanical energy is locally converted into heat due to friction - can now be computed as
This is in direct correspondence to the mechanical definition of the stress tensor σ as the first variation of the local dissipation rate with respect to the velocity gradient, i.e., σ = δ Dv diss. Indeed, by a straightforward computation we obtain Here, trε[v]2measures the averaged local change of length and trε[v]2the local change of volume induced by the transport. Obviously divv = tr(ε[v]) = 0 characterizes an incompressible fluid.
Now, let us consider a path of objects connecting with and generated by a time-continuous deformation. If each point of the object at time t ∈ [0, 1] moves in an Eulerian framework at the velocity v(t, x) ( ), so that the total deformation of into can be obtained by integrating the velocity field vin time, then the accumulated global dissipation of the motion field vin the time interval [0, 1] takes the form
This is the same concept as employed by Dupuis et al. [23] and Miller et al. [53] in their pioneering diffeomorphism approach. They minimize a dissipation functional under the simplifying assumption that the material behaves equally viscous inside and outside the object. Also, is replaced by a higher-order quadratic form Lvvwhich plays the role of the local rate of dissipation in a multipolar fluid model [57]. Multipolar fluids are characterized by the fact that the stresses depend on higher spatial derivatives of the velocity. If the quadratic form associated with Lacts only on ε[v] and is symmetric, then rigid body motion invariance is incorporated in the multipolar fluid model (cf. Sect. 4.2). In contrast to this approach, we here measure the rate of dissipation differently inside and outside the object and rely on classical (monopolar) material laws from fluid mechanics.
On this physical background we will now derive a Riemannian structure on the space of shapes in an admissible class of shapes S. The associated metric on the (infinite-dimensional) manifold Sis in abstract terms a bilinear mapping that assigns each element an inner product on variations of (cf. Sect. 3.1above). The associated length of a tangent vector is given by . Furthermore, as we have already seen above the length of a differentiable curve is then defined by , where is the temporal variation of at time t. The Riemannian distance between two shapes and on Sis given as the minimal length taken over all curves with and or equivalently (cf. Sect. 3.1above) as the length of a minimizer of the functional . For shapes an infinitesimal variation of a shape is associated with a transport field . This transport field is obviously not unique. Indeed, given any vector field won with for all (where denotes the (d − 1)-dimensional tangent space to at x), the transport field v + wis another possible representation of the shape variation . Let us denote by the affine space of all these representations. As a geometric condition for we obtain for all , where denotes the outer normal to in . Given all possible representations we are interested in the optimal transport, i.e., the transport leading to the least dissipation. Thus, using definition (1) of the local dissipation rate we finally define the metric as the minimal dissipation rate on motion fields vwhich are consistent with the variation of the shape δS,
Let us remark that we distinguish explicitly between the metric on motion fields and the metric on shape variations. Finally, integration in time leads to the total dissipation (2) to be invested in the transport along a path in the shape space S. This implies the following definition of a time-continuous geodesic path in shape shape:
Definition 1 (Geodesic path). Given two shapes and in a shape space S, a geodesic path between and is a curve with and which is a local solution of
among all differentiable paths in S.
The Riemannian distance between two shapes and induced by this definition is given by the length of the shortest (geodesic) path between the two shapes, i.e.,
Figure 3shows two different paths between the same pair of shapes, one of them being a (numerically approximated) geodesic. Note that the chosen dissipation model combines the control of infinitesimal length changes via trε[v]2, and the control of compression via trε[v]2. Figure 4 evaluates the impact of these two terms on the shapes along a geodesic path.
Fig. 4 Two geodesic paths between dumbbell shapes varying in the size of the ends. In the top examplethe ratio λ ∕ μ between the dissipation parameters is 0. 01 (leading to rather independent compression and expansion of the ends since the associated change of volume implies relatively low dissipation), and 100 in the bottom row(now mass is actually transported from one end to the other). The underlying texture on the objects is aligned to the transport direction, and the absolute value of the velocity vis color coded as
Fig. 3 A geodesic (top, path length L = 0. 2225 and total dissipation Diss = 0. 0497) and a non-geodesic path (bottom, L = 0. 2886, Diss = 0. 0880) between an A and a B. The intermediate shapes of the bottom row are obtained via linear interpolation between the signed distance functions of the end shapes. The local dissipation rate is color coded as

State-Based, Path-Independent Elastic Setup

Now, objects bounded by a shape contour are no longer composed of a viscous fluid but are considered to be elastic solids. To describe object deformations, we aim for an elastic energy which is not restricted to small displacements and which is consistent with first principles. Alongside the shape space modeling, we will recall some background from elasticity. For details we refer to the comprehensive introductions in the books by Ciarlet [15] and Marsden and Hughes [47].

For two objects and with shapes and , we assume a deformation ϕ to be defined on and constrained by the assumption . For practical reasons one might consider to be embedded in a very soft elastic material occupying for some computational domain Ω. There is an elastic energy associated with the deformation ϕ:Ω→Rd . By definition, elastic means that this energy solely depends on the state and not on the path along which the deformation proceeds in time. More precisely, for so-called hyper-elastic materials, is the integral of an energy density Wdepending solely on the Jacobian of the deformation ϕ, i.e.,
This elastic energy is considered as a dissimilarity measure between the shapes and . As a fundamental requirement one postulates the invariance of the deformation energy with respect to rigid body motions, for any orthogonal matrix QSO(d) and translation vector bRd (the axiom of frame indifference in continuum mechanics). From this, one deduces that the energy density only depends on the right Cauchy-Green deformation tensor . Hence, there is a function such that the energy density Wsatisfies for all FRd,d . The Cauchy-Green deformation tensor geometrically represents the metric measuring the deformed length in the undeformed reference configuration. For an isotropic material and for d = 3, the energy density Wcan be further rewritten as a function solely depending on the principal invariants of the Cauchy-Green tensor, namely , controlling the local average change of length, (cofF: = detF F T ), reflecting the local average change of area, and , which controls the local change of volume. For a detailed discussion we refer to [78, 15]. We shall furthermore assume that the energy density is polyconvex [18], i.e., a convex function of , , and , and that isometries, i.e., deformations with , are local minimizers with [15]. Typical energy densities in this class are of the form
for a 1, a 2 > 0 and a convex function Γ:[0,∞)→R with Γ(I 3) → for I 3 → 0 and I 3. In nonlinear elasticity such material laws have been proposed by Ogden [58], and for p = q = 2 (the case considered in our computations) we obtain the Mooney-Rivlin model [15]. The built-in penalization of volume shrinkage, i.e., , enables us to control local injectivity (cf. [2]).

Incorporation of such a nonlinear elastic energy allows to describe large deformations with strong material and geometric nonlinearities, which cannot be treated by a linear elastic approach (cf. Hong et al. [36]). Furthermore, it balances in an intrinsic way expansion and collapse of the elastic objects and hence frees us from imposing artificial boundary conditions or constraints.

As in the previous section, the local force per area, induced by the deformation, is described at a point by the Cauchy stress tensor σ. It is related to the first Piola-Kirchhoff stress tensor , which measures the force density in the undeformed reference configuration, by .

Based on these concepts from nonlinear elasticity, we can now define a dissimilarity measure on shapes
Figure 5shows some applications of this measure. Obviously, the elastic energy is in general not symmetric so that . Indeed, by construction d elast( ⋅, ⋅) does not impose a metric structure on the space of shapes (we refer to Sect. 3.2.3for a detailed discussion). Nevertheless, it can be applied to develop physically sound statistical tools for shapes such as shape averaging and a PCA on shapes, as outlined below in Sect. 4.1.
Fig. 5 Example of elastic dissimilarities between different shapes. The arrows indicate the direction of the deformation, the color coding represents the local deformation energy density (in the reference as well as the deformed state)
Let us make a brief remark on the mathematical relation between the two different concepts of elasticity and viscous fluids. If we assume the Hessian of the energy density Wat the identity to be given by (which can be realized in (5) for a particular choice of a 1, a 2, and Γ, depending on the exponents pand q), then by the ansatz ϕ(x) = x + τv(x) and a second-order Taylor expansion we obtain
In effect, the Hessian of the nonlinear elastic energy leads to the energy density in linearized, isotropic elasticity
for displacements uwith ϕ(x) = x + u(x). This energy density, acting on displacements u, formally coincides with the local dissipation rate diss[v], acting on velocity fields v, in the viscous flow approach.
Finally, let us deal with the hard constraint , which is often inadequate in applications. Due to local shape fluctuations or noise in the shape acquisition, the shape frequently contains details that are not present in and vice versa. These defects would imply high energies in a strict 1-1 matching approach. Hence, we have to relax the constraint and introduce some penalty functional. Here, we either measure the symmetric difference of the input shapes and the pullback of the shape given by
where AB = ABBA, or alternatively the volume mismatch

Conceptual Differences Between the Path- and State-Based Dissimilarity Measures

The concept of the state-based, elastic approach to dissimilarity measurement between shapes differs significantly from the path-based viscous flow approach. In the elastic setup, the axiom of elasticity implies that the energy at the deformed configuration is independent of the path from shape to shape along which the deformation is generated in time. Hence, there is no notion of shortest paths if we consider a purely elastic shape model, and different from a path-based approach there might not even exist an intermediate shape with .

Unlike in the elasticity model, in the Newtonian model of viscous fluids the rate of dissipation and the induced stresses solely depend on the gradient of the motion field v. Even though the dissipation functional (2) looks like the deformation energy from linearized elasticity as outlined above, the underlying physics is only related in the sense that an infinetisimal displacement in the fluid leads to stresses caused by viscous friction, and these stresses are immediately absorbed via dissipation.

Surely, every (path-based) Riemannian space is metrizable (and in that sense state-based), and for many sufficiently regular (state-based) metric spaces we can devise a corresponding (path-based) Riemannian metric. However, from our mechanical perspective, the conceptual difference between the path-based, viscous and the state-based elastic approach is striking. In the path-basedapproach, the structure of the space is too complicated for a closed formula of the geodesic distance, so that the actual computation of a path is required. In the state-basedapproach, there is either no underlying path (i.e., no such that for any 0 ≤ t 1t 2t 3 ≤ 1 we have ), or the shape space structure is simple enough to allow for a closed formula of the geodesic distance as in Euclidean space.

Mathematically, the path-based nature of the viscous flow approach and the fact that an inversion of the motion field v → − vleads to a path from shape to in shape space with the same dissipation and length, i.e.,
ensures that the associated distance d viscousis actually a metric. In particular, the symmetry condition and the triangle inequality hold. As we have already seen, the symmetry condition does not hold for the elastic dissimilarity measure. Also, the triangle inequality cannot be expected to hold. Indeed, if a deformation ϕ A, B maps onto and a deformation ϕ B, C maps onto , then ϕ A, C : = ϕ B, C ∘ ϕ A, B deforms onto . However, based on our elastic model, is considered to be stress free when applying the deformation ϕ B, C (although it is actually obtained as the image of object under the deformation ϕ A, B ). Hence, the "history" of the deformation ϕ A, B is lost when measuring the energy of ϕ B, C . In addition, the energy density is highly nonlinear. As a consequence, in general we cannot expect . Indeed, Fig. 6 gives an example where the triangle inequality holds in the viscous, path-based and fails in the elastic, state-based approach. Furthermore, Fig. 7 depicts another example for the lack of symmetry already apparent in Fig. 5 with a particularly pronounced mechanical difference of the two dissimilarity measures.
Fig. 7 The state-based elastic dissimilarity measure d elastis not symmetric (as opposed to the path-based, viscous distance d viscous): In this example, it costs much more energy to drag out the protrusion than to push it in. The color coding represents the local deformation energy density in the reference and the deformed configuration
Fig. 6 Left: viscosity-based (time-discrete) geodesics between the shapes at the corners (the shapes are taken from [31]). The triangle inequality holds. Right: elastic dissimilarities between the same shapes, where the arrows point from the reference to the deformed configuration. The triangle inequality does not hold

Numerical Methods and Case Examples

Elasticity-Based Shape Space

In this section we will perform a statistical analysis on shapes up to the second moment, i.e., we will consider shape averaging and a principal component analysis on shapes as two exemplary applications of the state-based elastic shape space.

Elastic Shape Averaging

As usual, we consider objects as open sets in Rd with the object shape given as . Given nsufficiently regular shapes , i = 1, , n, we are interested in an average shape which reflects the geometric characteristics of the input shapes in a physically intuitive manner. Suppose denotes a candidate for this unknown shape. As it is characteristic for the elastic approach, the similarity of the input shapes to is measured by taking into account optimal elastic deformations with . The elastic energy of these deformations has the interpretation of a dissimilarity measure (cf. Sect. 3.2.2), so that we obtain a natural definition of an average shape as the minimizer of the sum of these terms (cf. Sect. 2).

Definition 2 (Elastic shape average). Given shapes in some shape space S, the elastic shape average is the minimizer of

If the input objects have Lipschitz boundary and the integrand of the deformation energy is polyconvex and bounded below by for p > d, C 1, C 2 > 0, the existence of a Hölder-continuous elastic shape average and deformations which realize the above infimum is guaranteed [81].

An example of a shape average is provided in Fig. 8 . Obviously, the process of shape averaging is a constrained variational problem in which we simultaneously have to minimize over ndeformations ϕ i and the unknown shape under the nconstraints .
Fig. 8 Elastic shape average (bottom right) of five human silhouettes. For the computation, all shapes have actually been described as phase fields, and the elastic deformations are extended outside the input objects (cf. Sect. 4.3.2). The objects are depicted along with their deformations ϕ i (acting on a checkerboard) and the distribution of local length change and volume change (range [0. 97, 1. 03] color coded as )
The necessary conditions for a set of minimizing deformations are the corresponding Euler-Lagrange equations. As usual, inner variations of one of the deformations lead to the classical system of PDEs for every deformation ϕ i on , meaning a divergence-free, equilibrized stress field (cf. Sect. 3.2.2). Furthermore, the coupling between the deformations via the constraints allows to derive a stress balance relation on : Consistent variation of all deformations ϕ i and the average by some displacement via (1 + δu) ∘ ϕ i and results in the optimality condition , which after integration by parts leads to for the outer normal to . We have here exploited on . Now, we consider displacements uwith local support and let this support collapse at some point xon . This yields the pointwise condition
for , where we have used the relation
between first Piola-Kirchhoff stress and Cauchy stress . Hence, the shape average can be interpreted as that stable shape at which the boundary stresses of all deformed input shapes balance each other (Fig. 9). Obviously, there is a straightforward generalization involving jumps of normal stresses on interior interfaces in case of multi-component objects.
Fig. 9 Sketch of the pointwise stress balance relation on the averaged shape
In order to ensure a certain regularity of the average shape , in addition to the sum of deformation energies in Definition 2one can consider a further energy contribution which acts as a prior on in the variational approach. In the exemplary computations shown (Figs. 10-12), the (d − 1)-dimensional Hausdorff measure has been employed as regularization.
Fig. 10 Average of 18 hand silhouettes (Taken from [16])
Fig. 12 Twenty-four given foot shapes (Courtesy of adidas), textured with the distance to the surface of the average foot (bottom-right). Values range from 6 mm inside the average foot to 6 mm outside, color coded as

Elasticity-Based PCA

As already explained in Sect. 3.1, a principal component analysis (PCA) is a linear statistical tool which decomposes a vector space into the direct sum of orthogonal subspaces. These subspaces are ordered according to the strength of variation which occurs along each subspace within a random set of sample vectors. We would like to interpret a given set of input shapes as such a random sample and perform a corresponding PCA, however, due to the linearity of a PCA we first have to identify linear representatives for each shape on which a PCA can then be performed. For a Riemannian shape space, we have outlined in Sect. 3.1that such linear representatives are given by the logarithmic map of the input shapes, but we have also learnt in Sect. 3.2.3that a state-based elastic shape space is incompatible with a Riemannian structure.
Fig. 11 Five segmented kidneys and their average (right). For the first two input kidneys the distribution of , , and is shown on sagittal cross-sections (the range [0. 85, 1. 15] is color coded as). While the first kidney is dilated toward the average, the second is compressed

To prepare the definition of appropriate linear representatives of shapes in an elastic shape space, let us briefly review the physical concept of boundary stresses. By the Cauchy stress principle, each deformation is characterized by pointwise boundary stresses on in the deformed configuration. The stress at some point xon is given by the application of the Cauchy stress tensor σ k to the outer normal ν on . The resulting stress σ k ν is a force density acting on a local surface element of . The shape is in an equilibrium configuration if the opposite force is applied as an external surface load (cf. Fig. 9). Otherwise, by the axiom of elasticity, releasing the object , the elastic body will snap back to the original reference configuration . Let us assume the relation between the energetically favorable deformation and its induced stresses to be one-to-one, so that the average shape can be described in terms of the input shape and the boundary stress σ k ν, and we write . Upon scaling the stress with a weight t ∈ [0, 1], we obtain a one-parameter family of shapes , connecting with . Thus, we can regard σ k ν as a representative of shape in the linear space of vector fields on .

Physically, it is more intuitive to identify a displacement u k instead of the normal stress σ k ν as the representative of an input shape . Hence, let us study how the average shape varies if we increase the impact of a particular input shape for some k ∈ { 1, , n}. For this purpose, we apply the Cauchy stress σ k ν to the average shape , scaled with a small constant δ. This additional boundary stress δσ k ν may be seen as a first Piola-Kirchhoff stress acting on the (reference) configuration . The elastic response is given by a correspondingly scaled displacement . Here, to properly incorporate the nonlinear nature of the second moment analysis, should be interpreted as the compound object which is composed of all deformed and thus prestressed input objects . This interpretation is reflected by the elastic material law employed to compute the displacements u k . In detail, u k is obtained as the minimizer of the free mechanical energy
under the constraints and of zero average translation and rotation. These displacements u k are considered as representatives of the variation of the average shape with respect to the input shape , on which a PCA will be performed.
As long as FW(F) is not quadratic in F, u k still solves a nonlinear elastic problem. The advantage of this nonlinear variational formulation is that it is of the same type as the one for shape averaging, and it encodes in a natural way the compound elasticity configuration of the averaged shape domain . However, for the linearization of shape variations we are actually only interested in the displacements δu k for small δ. Therefore, we consider the limit of the Euler-Lagrange equations for δ → 0 and after a little algebra obtain u k as the solution of the linearized elasticity problem
for the symmetrized displacement gradient under the constraints and , where the in general inhomogeneous and anisotropic elasticity tensor Creads

Next, for a PCA on the linearized shape variations u k we select a suitable inner product (metric) on displacements . Note that ginduces a metric on the associated boundary stresses so that instead of analyzing the u k the covariance analysis can equivalently be performed directly on the boundary stresses σ1ν, , σ n ν, which we originally derived as linear shape representatives. Indeed, the solvability condition is fulfilled, and thus the solution u k for given boundary stress σ k ν = Cuν is uniquely determined up to a linearized rigid body motion (i.e., an affine displacement with skew-symmetric matrix representation), which is fixed by the conditions of zero mean displacement and angular momentum for u. Then, due to the linearity of the operator σν↦u, the metric is bilinear and symmetric as well, and its positive definiteness follows from the positive definiteness of gand the injectivity of the map σν↦u.

We consider two different inner products on displacements :
  • The L 2 -product.Given two square integrable displacements we define
    This product weights local displacements equally on the whole object .
  • The Hessian of the energy as inner product.Different from the L 2-metric, we now measure displacement gradients in a non-homogeneous way. We define
    for displacements with square integrable gradients. Hence, the contribution to the inner product is larger in areas of the compound object which are in a significantly stressed configuration.
Given an inner product, we can define the covariance operator Cov by
(note that the stresses σ k ν and thus also the displacements u k have zero mean due to (11)). Obviously, Cov is symmetric positive definite on span(u 1, , u n ). Hence, we can diagonalize Cov on this finite-dimensional space and obtain a set of g-orthonormal eigenfunctions and eigenvalues λ k > 0 with Cov w k = λ k w k . These eigenfunctions can be considered as the principal modes of variation of the average object and hence of the average shape , given the nsample shapes . Their eigenvalues encode the variation strength. The diagonalization of Cov can be performed by diagonalizing the symmetric matrix , where Λ = diag(λ1, λ2, ) and Ois orthogonal. The eigenfunctions are then obtained as .
Being displacements on , the modes of variation w k can easily be visualized via a scalar modulation δw k for varying δ (cf. the vizualization in Figs. 16 -18or the red lines in Figs. 13 and 15). If an amplified visualization of the modes is required, it is preferable to depict displacements w δ k which are defined as minimizers of the nonlinear variational energy (cf. (12)).
Fig. 13 First three dominant modes of variation for six input shapes (left), based on different metrics. Left: L 2-metric on displacements of a non-prestressed object (modes w k with ratios of 1, 0.23, 0.07). Middle: L 2-metric on displacements of the compound object \big{(} 1, 0.28, 0.03\big{)}. Right: energy Hessian-based metric on displacements of the compound object \big{(} 1, 0.61, 0.24\big{)}.
Fig. 15 First four modes of variation with ratios of 1, 0.88, 0.42, and 0.25 for the 18 hand silhouettes from Fig. 10
Fig. 16 Forty-eight input kidneys (Courtesy of Werner Bautz, radiology department at the University Hospital Erlangen, Germany) and their first four modes of variation with ratios of 1, 0.72, 0.37, and 0.31
Fig. 18 8 thorax CT scans from different patients (courtesy of Bruno Wirth, urology department at the Hospital zum hl. Geist, Kempen, Germany) and their first three modes of variation with ratios of 1, 0.12, and 0.07. Note that the thin lines which can be seen left of the heart correspond to contours of the liver, which are only visible in the first and last input image

Let us underline that this covariance analysis properly takes into account the usually strong geometric nonlinearity in shape analysis via the transfer of geometric shape variation to elastic stresses on the average shape, based on paradigms from nonlinear elasticity. Displacements or stresses are interpreted as the proper linearization of shapes. In abstract terms, either the space of displacements or stresses can be considered as the tangent space of shape space at the average shape, where the identification of displacements and stresses via (13) provides a suitable physical interpretation of stresses as shape variations.

The impact of the chosen metric.Naturally, the modes of variation depend on the chosen inner product. We have already mentioned that in order to be physically meaningful, the inner product should act on displacements u k of the compound object (which is composed of all deformed input shapes). If instead the u k were obtained by applying the boundary stresses σ k ν to an object which just looks like the average shape but does not contain the information how strongly the input shapes had to be deformed to arrive at the average, we obtain a different result (Fig. 13, left): If the prestressed state of some object regions is neglected, it becomes easier to deform them which causes the prediction of stronger variations. Figure 13 also hints at the differences between the employed metrics: The L 2-metric pronounces shape variations with large displacements even though they are energetically cheap (e.g., a rotation of some structure around a joint), while the Hessian of the elastic energy measures distances between displacements solely based on the associated change of elastic energy. Thus, displacements are weighted strongly in regions and directions which are significantly loaded.

The impact of the nonlinear elasticity model.Likewise, the particular choice of the nonlinear elastic energy density has a considerable effect on the average shape and its modes of variation. Figure 14 has been obtained using , where μ and λ are the coefficients of length and volume change penalization, respectively. A low penalization of volume changes apparently leads to independent compression and inflation at the dumbbell ends (left), while for deformations with a strong volume change penalization (right), material is squeezed from one end to the other. Here, the underlying metric is the based on the Hessian of the energy.
Fig. 14 First three modes of variation for eight dumbbell shapes, leftfor a 100 times stronger penalization of length than of volume changes (with ratios of 1, 0.22, 0.05), rightfor the reverse \big{(} 1, 0.41, 0.07\big{)}. Each row represents the variation of the average (middle shape) by δw k for the mode w k and varying δ
Figures 15-17show the dominant modes of variation for the examples from the previous section. A statistical analysis of the hand shapes in Fig. 15 has also been performed in [16] and [28], where the shapes are represented as vectors of landmark positions. The average and the modes of variation are quite similar, representing different kinds of spreading the fingers. The dominant modes of variation for a set of 48 three-dimensional kidney shapes is depicted in Fig. 16 , where for all modes w k we show the average (middle) and its variation according to δw k for varying δ. Local structures seem to be quite well represented and preserved during the averaging process and the subsequent covariance analysis compared to, e.g., the PCA on kidney shapes in [27] where a medial representation is used.
Fig. 17 The first six dominant modes of variation for the feet from Fig. 12

The PCA of the 24 foot shapes from Fig. 12 is shown in Fig. 17 and is much more intuitive than the color coding in Fig. 12 . The first mode apparently represents changing foot lengths, the second and third mode belong to different variants of combined width and length variation, and the fourth to sixth mode correspond to variations in relative heel position, ankle thickness, and instep height. Finally, Fig. 18 shows that the approach also works for image morphologies instead of shapes, using thorax CT scans as input. Here, the image edge set is considered as the corresponding shape, which is typically quite complex and characterized by nested contours. The first mode of variation represents a variation in chest size, the next mode corresponds to a change of heart and scapula shape, while the third mode mostly concerns the rib position.

Viscous Fluid-Based Shape Space

As explained in Sect. 3.2.1, the viscous fluid shape space is by construction a (infinite-dimensional) Riemannian manifold and as such is based on the computation of shape paths as opposed to state-based approaches like the elastic shape space from the previous section. In the elastic, state-based approach, we have to find for each pair of shapes and one single optimal matching deformation via which the similarity between and is determined. In contrast, here we require more information to measure the distance between the two shapes, namely an optimal velocity field at each time twithin the given time interval [0, 1]. In effect, this implies an increase of the dimension of the variational problem by the time component.

The two qualitatively different types of coordinates, the space coordinates (that span the space in which the shapes lie) and the time coordinate, are intuitively treated in different ways. One possibility is to regard the variational problem of computing a geodesic as a classical elliptic boundary value problem in time, in which each shape on a path seeks to be in equilibrium with its local neighborhood on the path. The equilibrizing force can be interpreted as an acceleration acting on the velocity field v. In this setting, it seems most natural to discretize first the time variable and approximate geodesics in shape space as discrete sequences of shapes, where each shape is connected to and equilibrates with its neighbors and the path length along the discrete path is approximated as a sum of approximations of the geodesic distance between neighboring shapes. The distance can be based on a matching deformation energy which will be elaborated on further down.

An alternative view starts from the underlying velocity field which generates the geodesic. Dupuis et al. [23] and Beg et al. [3] consider shapes (or rather images) embedded in a domain Ω⊂Rd . These shapes deform according to smooth, compactly supported velocity fields with . The regularity of the velocity fields is ensured by defining the path dissipation as ∫0 1Ω Lvv dx dtand the path length as for a differential operator Lof sufficiently high order (cf. Sect. 3.2.1). The corresponding shape deformation ϕ which is induced by the velocity field is obtained as the solution ϕ = ϕ1of the pointwise, Lagrangian ordinary differential equation .

In the first approach, the computation of a geodesic was seen as the concatenation of a number of local subproblems each of which represents the approximation of a geodesic segment between two intermediate shapes and each of which thus inherits the constraint that one shape is transferred exactly into the other. In contrast, in the second approach we have one single constraint, acting at the end of the geodesic and expressing that the accumulated flow ϕ deforms the starting shape into the final shape , .

Let us now focus on the first approach in which a geodesic path will be approximated via a finite sequence of shapes , connected by deformations which are optimal in a variational sense and fulfil the constraint .

Given two shapes , in some given space of shapes S, we define a discrete path of shapes as a sequence of shapes with and . For the time step the shape is supposed to be an approximation of with t k = kτ, where is a continuous path connecting and . For each pair of consecutive shapes and we now consider a matching deformation which satisfies . With each deformation ϕ k we associate a deformation energy of the same type as described in Sect. 3.2.2. If appropriately chosen, this energy will ensure sufficient regularity and a 1-1 matching property for deformations ϕ k with finite energy. As in elasticity, the energy is assumed to depend only on the local deformation, reflected by the Jacobian . Yet, different from elasticity, we suppose the material to relax instantaneously so that object is again in a stress-free configuration when applying ϕ k + 1at the next time step. Let us also emphasize that the stored energy does not depend on the deformation history as in most plasticity models in engineering. This energy is now employed to define time-discrete counterparts to the dissipation and length of continuous paths from Sect. 3.2.1.

Definition 3 (Discrete dissipation and discrete path length). Given a discrete path , its dissipation is defined as
where is a minimizer of the deformation energy under the constraint . Furthermore, the discrete path length is defined as

Let us make a brief remark on the proper scaling factors. The deformation energy is expected to scale like τ2(cf. (7)). Hence, the factor ensures the discrete dissipation measure to be conceptually independent of the time step size. The same holds for the discrete length measure .

To ensure that the above-defined dissipation and length of discrete paths in shape space are well defined, a minimizing deformation ϕ k of the elastic energy with has to exist. In fact, this holds for objects and with Lipschitz boundaries and for which there exists at least one bi-Lipschitz deformation of into for k = 1, , K[83].

With the notion of dissipation at hand, we can define a discrete geodesic path following the standard paradigms in differential geometry.

Definition 4 (Discrete geodesic path). A discrete path in a set of admissible shapes Sconnecting two shapes and is a discrete geodesic if there exists an associated family of deformations (ϕ k ) k = 1, …,K such that minimize the total energy over all intermediate shapes and all possible matching deformations with for k = 1, , K.
Examples of discrete geodesics are provided in Figs. 19 and 20. Apparently, the frame indifference and the (local) injectivity property of the matching deformations, which are ensured by the nonlinear deformation energy , allow the computation of reasonable discrete geodesics with only few intermediate shapes. Under sufficient growth conditions on the integrand of the deformation energy , the existence of discrete geodesics is guaranteed at least for certain compact sets Sof admissible shapes, e.g., shapes which can be described by spline curves with a finite set of control points from some compact domain and which satisfy a uniform cone condition in the sense that each is the tip of two cones with fixed height and opening angle which lie completely on either side of [83]. Such requirements on Sare necessary since the known regularity theory for deformation energies of the employed type does not allow to prove Lipschitz-regularity of optimal deformations so that the intermediate shapes might degenerate.
Fig. 19 Discrete geodesics between a straight and a rolled up bar, from first row to fourth row based on one, two, four, and eight time steps. The light gray shapes in the first, second, and third row show a linear interpolation of the deformations connecting the dark gray shapes. The shapes from the finest time discretization are overlayed over the others as thin black lines. In the last row the rate of viscous dissipation is rendered on the shape domains from the previous row, color coded as
Fig. 20 Discrete geodesic between a cat and a lion and between the hand shapes m336 and m324 from the Princeton Shape Benchmark [72]. For both examples, the local dissipation is color coded on slices through the shapes as

The discrete dissipation as the sum of matching deformation energies indeed represents an approximation to the time-continuous dissipation of a velocity field from Sect. 3.2.1. If a smooth path in shape space is considered which is interpolated at discrete times t k = kτ, k = 0, , K, and if for t ∈ [t k − 1, t k ), denotes the velocity field which generates the associated matching deformations ϕ k , then as the time step size decreases and v τconverges against a smooth velocity field v, the discrete dissipation converges against the time-continuous dissipation (2) induced by v(cf. [83] for details).

Within this framework of geodesics in shape space, the strict constraints that one shape is deformed exactly into another one are often inadequate in applications as has already been discussed in Sect. 3.2.2for the state-based, elastic setup. For the computation of an elastic dissimilarity measure, the single matching constraint could be relaxed as a mismatch penalty. In the Riemannian, viscous setting we pursue the same concept, however, the particular form of the employed constraints depends on the chosen view on shape geodesics. In the framework of geodesics as paths of diffeomorphisms, which we introduced at the beginning of this section, there is the single constraint , meaning that the induced diffeomorphism ϕ maps the initial shape onto the final shape . This constraint can be relaxed in the same manner as in Sect. 3.2.2via a penalty measuring the mismatch of the shapes or of the corresponding objects. For the time-discrete geodesic setting we have a sequence of matching constraints , k = 1, …, K, each of which can again be relaxed by the same means. In fact, we add to the discrete dissipation of a set (ϕ k ) k = 1, …, K of deformations a sum of mismatch penalties . In the limit for vanishing time step size and under the same conditions as above, this sum can be shown to converge against the optical flow type functional for the unit outward normal to the space time shape tube . Furthermore, with has been employed as regularization, which in the limit for τ → 0 converges against the integral .

Real-world objects are most often not only characterized by their outer contour but also contain internal structures that have to be matched properly when computing the similarity between two objects. As an example, consider the straight and the folded rod in Fig. 21 . The rods consist of three distinct components, which imposes a constraint on reasonable connecting paths: Each component is to be mapped onto its correct counterpart. A shortest path under this constraint obviously differs significantly from the geodesic which just matches the outer contours (cf. Fig. 19). This observation calls for a generalization of shapes, an example of which we have already seen in the context of an elastic shape space in Fig. 18 , where the edge set of an image was considered as a shape. Here, let us adopt a slightly different approach and regard shapes as being composed of a number of subcomponents. In detail, instead of a geodesic between just two shapes and , we now seek a geodesic path with for t ∈ [0, 1], between two collections of mseparate shapes, with and with . The geodesic path is supposed to be generated by a joint motion field . The single objects can then be regarded as the subcomponents of an overall object . The total dissipation along the path is measured exactly as before by
This naturally translates to the discrete dissipation of a path with K + 1 intermediate shape collections , k = 0, , K,
where the deformations ϕ k satisfy the constraints for k = 1, …, K, i = 1, …, m, and , , i = 1, …, m.
Fig. 21 Discrete geodesic between the straight and the folded bar from Fig. 31-19 , where the black region of the initial shape is constrained to be matched to the black region of the final shape. The bottom rowshows a color coding of the corresponding viscous dissipation. Due to the strong change in relative position of the black region, the intermediate shapes exhibit a strong asymmetry and high dissipation near the bar ends
The different object components can of course be assigned different material properties. Figure 22 shows frames from a real video sequence of moving white and red blood cells (top) as well as a discrete geodesic between the first and last frame (middle) for which the material parameters of the white blood cell were chosen twenty times weaker than for the red blood cells. The result is a nonlinear interpolation between distant frames which is in good agreement with the actually observed motion. Once geodesic distances between shapes are defined, one can statistically analyze ensembles of shapes and cluster them in groups based on the geodesic distance as a reliable measure for the similarity of shapes. Two exemplary examples are provided by the evaluation of geodesic distances between different 2D letters (Fig. 23, left) and between six different 3D foot shapes (Fig. 23, right). In the 2D example, we clearly identify three distinct clusters (Bs, Xs, and M).
Fig. 22 Top: frames from a real video sequence of a white blood cell among a number of red ones (courtesy Robert A. Freitas, Institute for Molecular Manufacturing, California, USA). Middle: computed discrete geodesic between the segmented shapes in the first and the last frame. Bottom: pushforward of the initial (first four shapes) and pullback of the final frame (last five shapes) according to the geodesic flow
Fig. 23 Left: pairwise geodesic distances between (also topologically) different letter shapes. Right: pairwise geodesic distances between different scanned 3D feet. The feet have volumes 499.5, 500.6, 497.6, 434.7, 432, and 381 cm3, respectively

A Collection of Computational Tools

So far, we have investigated some of the many aspects on mathematical models in shape space without any discussion of the corresponding computational tools and numerical algorithms. Hence, let us at least briefly mention some fundamental computational aspects to effectively deal with general classes of shapes as boundary contours of volumetric objects.

At first, we replace the strict separation between material inside the object and void outside by substituting the void with a material which is several orders of magnitude softer than inside the object. This relaxation is important with respect to the existence analysis and the stabilization of the computational method. In fact, we replace the deformation energy by the energy for a small constant η. In the implementation which underlies the above applications, for η = 10 − 4 one observes no significant qualitative impact of this regularization on the solution. Furthermore, as mentioned above, to ensure regularity of the shape contour , we take into account the area functional as a prior, weighted with a small factor.

Compared to a parametric description of shapes, e.g., as a polygonal line or a triangulated surface, an implicit description has several advantages. In particular, it does not require a remeshing even in case of large deformations, it allows for topological transitions without any extra handling of the associated singularities, and it can be combined with multi-scale relaxation schemes for an efficient minimization of the involved functionals.

In what follows, we consider a level set and a phase field description of shapes and outline the general framework of a multi-scale method based on finite element calculus. In fact, the phase field model has been used in the examples for the elastic shape averaging and the PCA, whereas the level set method has served as a numerical building block for the computation of time-discrete shape geodesics.

Shapes Described by Level Set Functions

The level set method first presented by Osher and Sethian [60] has been used for a wide range of applications [59, 71]. Burger and Osher gave an overview in the context of shape optimization [7]. To numerically solve variational problems in shape space, we assume a shape to be represented by the zero level set xΩ : u(x) = 0 of a scalar function u:Ω→R on a computational domain Ω⊂Rd . Furthermore, the zero super level set xΩ : u(x) > 0 defines the corresponding object domain . This shape description can be incorporated in a variational approach following the approximation proposed by Chan and Vese [9]. In fact, the partition of the domain Ωinto object and background is encoded via a regularized Heaviside function H εu. As in [9] we consider the function , where ε is a scale parameter representing the width of the smeared-out shape contour. Then, a deformation energy is approximated by
Furthermore, the energy measuring the volumetric mismatch between an object and the pullback of an object under a deformation ϕ can be approximated by
where u A , u B are level set representations of the shapes and , respectively. Finally, the surface area of a shape , which appears as a prior, is replaced by the total variation of H εu, and we obtain
Let us emphasize that in the actual energy minimization algorithm, the guidance of an initial zero level set towards the final shape relies on the nonlocal support of the derivative of the regularized Heaviside function (cf. [8]).

Shapes Described via Phase Fields

An alternative to a level set description of shapes is a phase field representation. Physically, the phase field approach is inspired by the observation that interfaces are usually not sharp but characterized by a diffusive transition. Mathematically, there are two basic types of such phase field representations, a single phase approach as the one presented by Ambrosio and Tortorelli [1] for the approximation of the Mumford-Shah model [56] and the double phase approach by Modica and Mortola [55] used to approximate surface integrals. In the shape context studied here, let us focus on the single phase model. Thus, a shape is encoded by a continuous, piecewise-smooth phase field function u:Ω→R which is zero on , but close to one everywhere else. The specific profile of the phase field function ufor a shape is determined via the phase field approximation
of the involved surface area . As in the above level set model the phase field parameter ε determines the width of the diffusive interface. Different from the level set model by Chan and Vese, the interface profile is not explicitly prescribed but implicitly encoded in the variational approach as the profile attained by minimizers of the functional. Based on this phase field model the penalty functional measuring the area mismatch between a shape and the pullback of a shape under a deformation ϕ can be approximated by

where u A , u B are phase fields representing the shapes and , respectively. In this type of models the deformation energy cannot be realized based on a phase field function udue to the fact that a single phase model allows to identify the shape itself but does not distinguish its inside and outside. Therefore, in the presented applications of elastic shape averaging and the elastic PCA the input objects and thus their characteristic functions were given a priori.

Multi-Scale Finite Element Approximation

For the spatial discretization of the functionals in the above variational approaches the finite element method can be applied. Hence, the level set function or the phase field u, representing a (unknown) shape , and the different components of the deformations ϕ are represented by continuous, piecewise multilinear (trilinear in 3D and bilinear in 2D) finite element functions Uand Φon a regular grid superimposed on the domain Ω = [0, 1] d . For the ease of implementation a dyadic grid resolution with 2 L + 1 vertices in each direction and a grid size h = 2L is chosen. Descent algorithm.The functionals depend nonlinearly both on the discrete deformations Φ(due to the concatenation UΦand the nonlinear integrand W( ⋅) of the deformation energy) as well as on the discrete level set or phase field functions U(e.g., due to the concatenation of the level set function with the regularized Heaviside function H ε( ⋅)). In our energy relaxation algorithm for fixed grid size, we employ a gradient descent approach. We constantly alternate between performing a single gradient descent step for all deformations and the level set or phase field functions. Numerical quadrature.Integral evaluations in the descent algorithm are performed by Gaussian quadrature of third order on each grid cell. For various terms we have to evaluate pullbacks UΦof a discretized level set function Uor a test function under a discretized deformation Φ. Let us emphasize that quadrature based on nodal interpolation of UΦwould lead to artificial displacements near the shape edges accompanied by strong artificial tension. Hence, in our algorithm, if Φ(x) lies inside Ωfor a quadrature point x, then the pullback is evaluated exactly at x. Otherwise, we project Φ(x) back onto the boundary of Ωand evaluate Uat that projection point. Cascadic multi-scale algorithm.The variational problem considered here is highly nonlinear, and for fixed time step size the proposed scheme is expected to have very slow convergence; also it might end up in some nearby local minimum. Here, a multilevel approach (initial optimization on a coarse scale and successive refinement) turns out to be indispensable in order to accelerate convergence and not to be trapped in undesirable local minima. Due to our assumption of a dyadic resolution 2 L + 1 in each grid direction, we are able to build a hierarchy of grids with 2 l + 1 nodes in each direction for l = L, , 0. Via a simple restriction operation we project every finite element function to any of these coarse grid spaces. Starting the optimization on a coarse grid, the results from coarse scales are successively prolongated onto the next grid level for a refinement of the solution [5]. Hence, the construction of a grid hierarchy allows to solve coarse scale problems in our multi-scale approach on coarse grids. Since the width ε of the diffusive shape representation should naturally scale with the grid width h, we choose ε = h.


Let us close with a comparison of path- and state-based shape space. Already in Sect. 3.2.3we have studied the difference between the state-based dissimilarity measure d elastand the path-based distance d viscous. Based on the applications considered in the previous sections let us compare the underlying concepts now more on a conceptual level of the geometry of shape space:
  • Non-uniqueness of shape averages. Due to the nonlinearity of the elastic variational problem, local minimizers of the elastic energy might be non-unique. There might even exist different minimizing deformations with the same elastic energy. Mechanically, this non-uniqueness is frequently associated with different buckling modes, which occur in case of large, geometrically nonlinear deformations. Hence, the shape average need not be uniquely defined, except in the small displacement case, where a linear elastic model (8) applies. In case of the path-based approach, (shortest) geodesics do not have to be unique either. Indeed, a geodesic is the unique shortest path only until the first conjugate point. Hence, the shape average is in a strict sense not well-defined if the distances are sufficiently large.
  • Different physical interpretation of the PCA. In the Riemannian setup with the metric being the rate of viscous dissipation, the corresponds to the initial velocity in the (optimal transport) flow of associated with shape into associated with the kth input shape . In the elastic model, the boundary stress results from the deformation ϕ k of onto the average object and effectively is the restoring force acting on the average shape . Via the linearized elasticity problem in the prestressed compound configuration of the average object , these restoring forces are identified with displacements u k . Depending on the model, either the flow velocities v k or the linear elastic displacements u k form the basis of a covariance analysis in the linear vector space of mappings . The outcome are principal shape variations of the average shape, either generated by motion fields or displacements, respectively.
  • Quantitative shape analysis. The Riemannian metric given by the rate of viscous dissipation in the path-based viscous fluid approach allows direct comparison of multiple ensembles of shapes via pairwise distance computations. Due to the lack of a triangle inequality this is possible only in a restricted sense in the state-based elastic approach, where dissimilarity measures for one fixed shape and a set of varying shapes can be computed.
  • The method of choice depends on the specific application. If shapes are considered as boundaries of objects with a viscous fluid inside then the path-based approach would be more appropriate. The state-based elastic approach is favorable for objects which behave more like deformable solids.


Level Set Methods Including Fast Marching MethodsMumford Shah, Phase Field ModelsNumerical Methods for Variational Approach in Image AnalysisShape SpacesVariational Approach in Image Analysis


The model proposed in Sect. 4.2has been developed in cooperation with Leah Bar and Guillermo Sapiro from the University of Minnesota. Benedikt Wirth has been funded by the Bonn International Graduate School in Mathematics. Furthermore, the work was supported by the Deutsche Forschungsgemeinschaft, SPP 1253 "Optimization with Partial Differential Equations." Part of Figs. 3 -4, and 19-23have been taken from [83], the results from Figs. 6 , 8, and 10-18stem from [67, 69].


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