Geodesics on Shape Spaces
Bounded Variation and Sobolev Metrics
This paper studies the space of planar curves endowed with the Finsler metric over its tangent space of displacement vector fields. Such a space is of interest for applications in image processing and computer vision because it enables piecewise regular curves that undergo piecewise regular deformations, such as articulations. The main contribution of this paper is the proof of the existence of the shortest path between any two -curves for this Finsler metric. Such a result is proved by applying the direct method of calculus of variations to minimize the geodesic energy. This method applies more generally to similar cases such as the space of curves with metrics for integer. This space has a strong Riemannian structure and is geodesically complete. Thus, our result shows that the exponential map is surjective, which is complementary to geodesic completeness in infinite dimensions. We propose a finite element discretization of the minimal geodesic problem, and use a gradient descent method to compute a stationary point of the energy. Numerical illustrations show the qualitative difference between and geodesics.
2010 Mathematics Subject Classification: Primary 49J45, 58B05; Secondary 49M25, 68U05.
Keywords: Geodesics ; Martingale ; -curves ; shape registration
This paper addresses the problem of the existence of minimal geodesics in spaces of planar curves endowed with several metrics over the tangent spaces. Given two initial curves, we prove the existence of a minimizing geodesic joining them. Such a result is proved by the direct method of calculus of variations.
We treat the case of -curves and -curves ( integer). Although the proofs’ strategies are the same, the and cases are slightly different and the proof in the case is simpler. This difference is essentially due to the inherent geometric structures (Riemannian or Finslerian) of each space.
We also propose a finite element discretization of the minimal geodesic problem. We further relax the problem to obtain a smooth non-convex minimization problem. This enables the use of a gradient descent algorithm to compute a stationary point of the corresponding functional. Although these stationary points are not in general global minimizers of the energy, they can be used to numerically explore the geometry of the corresponding spaces of curves, and to illustrate the differences between the Sobolev and metrics.
1.1 Previous Works
Shape spaces as Riemannian spaces.
The mathematical study of spaces of curves has been largely investigated in recent years; see, for instance, [50, 28]. The set of curves is naturally modeled over a Riemannian manifold . This consists in defining a Hilbertian metric on each tangent plane of the space of curves, i.e. the set of vector fields which deform infinitesimally a given curve. Several recent works [29, 15, 49, 48] point out that the choice of the metric notably affects the results of gradient descent algorithms for the numerical minimization of functionals. Carefully designing the metric is therefore crucial to reach better local minima of the energy and also to compute descent flows with specific behaviors. These issues are crucial for applications in image processing (e.g. image segmentation) and computer vision (e.g. shape registration). Typical examples of such Riemannian metrics are Sobolev-type metrics [39, 37, 41, 40], which lead to smooth curve evolutions.
Shape spaces as Finslerian spaces.
It is possible to extend this Riemannian framework by considering more general metrics on the tangent planes of the space of curves. Finsler spaces make use of Banach norms instead of Hilbertian norms . A few recent works [28, 49, 16] have studied the theoretical properties of Finslerian spaces of curves.
Finsler metrics are used in  to perform curve evolution in the space of -curves. The authors make use of a generalized gradient, which is the steepest descent direction according to the Finsler metric. The corresponding gradient flow enables piecewise regular evolutions (i.e. every intermediate curve is piecewise regular), which is useful for applications such as registration of articulated shapes. The present work naturally follows . Instead of considering gradient flows to minimize smooth functionals, we consider the minimal geodesic problem. However, we do not consider the Finsler metric favoring piecewise-rigid motion, but instead the standard -metric. In , the authors study a functional space similar to by considering functions with finite total generalized variation. However, such a framework is not adapted to our applications because functions with finite total generalized variation can be discontinuous.
Our main goal in this work is to study the existence of solutions, which is important to understand the underlying space of curves. This is the first step towards a numerical solution to the minimal path length problem for a metric that favors piecewise-rigid motion.
Geodesics in shape spaces.
The computation of geodesics over Riemannian spaces is now routinely used in many imaging applications. Typical examples of applications include shape registration [38, 45, 42], tracking , and shape deformation . In , the authors study discrete geodesics and their relationship with continuous geodesics in the Riemannian framework. Geodesic computations also serve as the basis to perform statistics on shape spaces (see, for instance, [45, 2]) and to generalize standard tools from Euclidean geometry such as averages , linear regression , and cubic splines , to name a few. However, due to the infinite dimensional nature of shape spaces, not all Riemannian structures lead to well-posed length-minimizing problems. For instance, a striking result [29, 48, 49] is that the natural -metric on the space of curves is degenerate, despite its widespread use in computer vision applications. Indeed, the geodesic distance between any pair of curves is equal to zero.
The study of the geodesic distance over shape spaces (modeled as curves, surfaces, or diffeomorphisms) has been widely developed in the past ten years [31, 9, 8]. We refer the reader to  for a review of this field of research. These authors typically address the questions of existence of the exponential map, geodesic completeness (the exponential map is defined for all time), and the computation of the curvature. In some situations of interest, the shape space has a strong Riemannian metric (i.e., the inner product on the tangent space induces an isomorphism between the tangent space and its corresponding cotangent space) so that the exponential map is a local diffeomorphism. In  the authors describe geodesic equations for Sobolev metrics. They show in Section 4.3 the local existence and uniqueness of a geodesic with prescribed initial conditions. This result is improved in , where the authors prove the existence for all time. Both previous results are proved by techniques from ordinary differential equations. In contrast, local existence (and uniqueness) of minimizing geodesics with prescribed boundary conditions (i.e. between a pair of curves) is typically obtained using the exponential map.
In finite dimensions, existence of minimizing geodesics between any two points (global existence) is obtained by the Hopf-Rinow theorem . Indeed, if the exponential map is defined for all time (i.e. the space is geodesically complete) then global existence holds. This is, however, not true in infinite dimensions, and a counterexample of non-existence of a geodesic between two points over a manifold is given in . An even more pathological case is described in , where an example is given where the exponential map is not surjective although the manifold is geodesically complete. Some positive results exist for infinite dimensional manifolds (see in particular Theorem B in  and Theorem 1.3.36 in ) but the surjectivity of the exponential map still needs to be checked directly on a case-by-case basis.
In the case of a Finsler structure on the shape space, the situation is more complicated, since the norm over the tangent plane is often non-differentiable . This non-differentiability is indeed crucial to deal with curves and evolutions that are not smooth (we mean evolutions of non-smooth curves). That implies that geodesic equations need to be understood in a weak sense. More precisely, the minimal geodesic problem can be seen as a Bolza problem on the trajectories . In  several necessary conditions for existence of solutions to Bolza problems in Banach spaces are proved within the framework of differential inclusions. Unfortunately, these results require some hypotheses on the Banach space (for instance the Radon-Nikodym property for the dual space) that are not satisfied by the Banach space that we consider in this paper.
We therefore tackle these issues in the present work and prove existence of minimal geodesics in the space of curves by a variational approach. We also show how similar techniques can be applied to the case of Sobolev metrics.
Section 2 deals with the Finsler space of -curves. Our main contribution is Theorem 2.25 proving the existence of a minimizing geodesic between two -curves. We also explain how this result can be generalized to the setting of geometric curves (i.e. up to reparameterizations).
Section 3 extends these results to -curves with integer, which gives rise to Theorems 3.4 and 3.8. Our results are complementary to those presented in  and  where the authors show the geodesic completeness of curves endowed with the -metrics with integer. We indeed show that the exponential map is surjective.
Section 4 proposes a discretized minimal geodesic problem for and Sobolev curves. We show numerical simulations for the computation of stationary points of the energy. In particular, minimization is made by a gradient descent scheme, which requires, in the -case, a preliminary regularization of the geodesic energy.
2 Geodesics in the Space of -Curves
In this section we define the set of parameterized -immersed curves and we prove several useful properties. In particular, in Section 2.2, we discuss the properties of reparameterizations of -curves.
The space of parameterized -immersed curves can be modeled as a Finsler manifold as presented in Section 2.3. Then, we can define a geodesic Finsler distance and prove the existence of a geodesic between two -curves (Sections 2.4). Finally, we define the space of geometric curves (i.e., up to reparameterization) and we prove similar results (Section 2.5). We point out that, in both the parametric and the geometric case, the geodesic is not unique in general. Through this paper we identify the circle with .
2.1 The Space of -Immersed Curves
Let us first recall some needed defintions.
Definition 2.1 (-functions).
We say that is a function of bounded variation if its first variation is finite:
Several times in the following, we use the fact that the space of functions of bounded variation is a Banach algebra and a chain rule holds. We refer to [1, Theorem 3.96, p. 189] for a proof of these results.
We say that if and its second variation is finite:
For a sake of clarity we point out that, as , for every -function, the first variation coincides with the -norm of the derivative. Moreover, by integration by parts, it holds
The -norm is defined as
The space can also be equipped with the following types of convergence, both weaker than the norm convergence:
Weak* topology. Let and . We say that weakly* converges in to if
where denotes the weak* convergence of measures.
Strict topology. Let and . We say that strictly converges to in if
Note that the following distance
is a distance in inducing the strict convergence.
Proposition 2.2 (weak* convergence).
Let . Then weakly* converges to in if and only if is bounded in and strongly converges to in .
Proposition 2.3 (embedding).
The following continuous embeddings hold:
In particular (see Claim 3 p.218 in ) we have
We refer to  for a deeper analysis of -functions.
We can now define the set of -immersed curves and prove that it is a manifold modeled on . In the following we denote by a generic -curve and by its derivative. Recall also that, as is a -function of one variable, it admits a left and right limit at every point of and it is continuous everywhere except on a (at most) countable set of points of . The space of smooth immersion of is defined by
The natural extension of this definition to -curves is
where denotes the segment connecting the two points. This definition implies that is locally the graph of a function. However, in the rest of the paper, we relax this assumption and work on a larger space under the following definition.
Definition 2.4 (-immersed curves).
A -immersed curve is any closed curve satisfying
We denote by the set of -immersed curves.
Although a bit confusing, we preferred to work with this definition of immersed curves, since it is a stable subset of under reparameterizations. Note that this definition allows for cusp points and thus curves in cannot be in general viewed as the graph of a function. Condition (2.4) allows one to define a Frénet frame for a.e.- by setting
Finally we denote by the length of defined as
The next proposition proves a useful equivalent property of (2.4):
Every satisfies (2.4) if and only if
As , it admits a left and right limit at every point of so that we can define the following functions:
where and are continuous from the left and the right, respectively and satisfy
Let us suppose that verifies (2.4) and . Then we can define a sequence such that , and (up to a subsequence) we have for some . Now, up to a subsequence, the sequence is a left-convergent sequence (or a right-convergent sequence), which implies that . This is of course in contradiction with (2.4). The right-convergence case is similar.
Now, let us suppose that satisfies (2.7) so that and also satisfy (2.7). Then if for some , for every there exists such that , which is in contradiction with . This proves that the left limit is positive at every point. By using we can similarly show that the right limit is also positive, which proves (2.4). ∎
We can now show that is a manifold modeled on since it is open in .
is an open set of .
Let . We prove that
In fact, by (2.1), we have , so that every curve such that
Remark 2.7 (immersions, embeddings, and orientation).
Moreover, as -immersed curves can have some self-intersections, the standard notion of orientation (clockwise or counterclockwise) defined for Jordan’s curves cannot be used in our case. The interior of a -immersed curve can be disconnected and the different branches of the curve can be parameterized with incompatible orientations. For example, there is no standard counterclockwise parameterization of the curve in Fig.1.
In order to define a suitable notion of orientation, we introduce the notion of orientation with respect to an extremal point. For every and we say that is an extremal point for if lies entirely in a closed half-plane bounded by a line through .
We also suppose that the Frénet frame denoted is well defined at , where denotes here the unit outward normal vector. Then, we say that is positively oriented with respect to if the ordered pair gives the counterclockwise orientation of . For example the curve in Fig.1 is positively oriented with respect to the point but negatively oriented with respect to .
2.2 Reparameterization of -Immersed Curves
In this section we introduce the set of reparameterizations adapted to our setting. We prove in particular that it is always possible to define a constant speed reparameterization.
Moreover, we point out several properties describing the relationship between the convergence of parameterizations and the convergence of the reparameterized curves. On one hand, in Remark 2.9 we underline that the reparameterization operation is not continuous with respect to the -norm. On the other hand, Lemma 2.12 proves that the convergence of the curves implies the convergence of the respective constant speed parameterizations.
Definition 2.8 (reparameterizations).
We denote by the set of homeomorphisms such that . The elements of are called reparameterizations. Note that any can be considered as an element of by the lift operation (see ). Moreover the usual topologies (strong, weak, weak*) on subsets of will be induced by the standard topologies on the corresponding subsets of .
The behavior of curves under reparameterizations is discontinuous due to the strong topology as described below.
Remark 2.9 (discontinuity of the reparameterization operation).
In this remark we give a counterexample to the following conjecture: for every and for every sequence of parameterizations strongly converging to we obtain that strongly converges to in .
This actually proves that the composition with a reparameterization is not a continuous function from the set of reparameterizations to .
We consider the curve drawn in Fig. 2 and we suppose that it is counterclockwise oriented and that the corner point corresponds to the parameter . Note also that the second variation of is represented by a Dirac delta measure in a neighborhood of , where is a vector such that .
Then we consider the family of parameterizations defined by
where the addition is considered modulo . This sequence of reparameterizations shifts the corner point on and converges -strongly to the identity reparameterization .
Moreover, we have that for every , which implies that converges to strongly in . However, similarly to , the second variation of is represented by a Dirac delta measure in a neighborhood of the parameter corresponding to the corner. Then
which implies that the reparameterized curves do not converge to the initial one with respect the -strong topology.
Remark 2.10 (constant speed parameterization).
Property (2.4) allows us to define the constant speed parameterization for every . We start by setting
where denotes the length of defined in (2.6) and where is a chosen basepoint belonging to . Now, because of (2.4), we can define and the constant speed parameterization of is given by . In order to prove that is invertible we apply the result proved in . In this paper the author gives a condition on the generalized derivative of a Lipschitz-continuous function in order to prove that it is invertible. We detail how to apply this result to our case.
Because of Rademacher’s theorem, as is Lipschitz-continuous, it is a.e. differentiable. Then we consider the generalized derivative at , which is defined as the convex hull of the elements of the form
where as and is differentiable at every . Such a set is denoted by and it is a non-empty compact convex set of . Now, in  it is proved that if then is locally invertible at . We remark that, in our setting, such a condition is satisfied because of (2.4) so that the constant speed parameterization is well defined for every .
Finally we remark that . For a rigorous proof of this fact we refer to Lemma 2.11.
The next two lemmas prove some useful properties of the constant speed parameterization.
If is such that and , then there exists a positive constant such that .
Recall that the reparameterization is the inverse of , where is a chosen basepoint belonging to . Then in particular we have
Moreover, because of (2.1), is bounded by . In particular we have and , and, by the chain rule for -functions, we also get with . We finally have
Then, by a straightforward calculation and the chain rule, we get that
which proves the lemma. ∎
Let be a sequence satisfying
and converging to in . Then in .
2.3 The Norm on the Tangent Space
We can now define the norm on the tangent space to at , which is used to define the length of a path. We first recall the main definitions and properties of functional spaces equipped with the measure .
Definition 2.13 (functional spaces w.r.t. ).
Let and . We consider the following measure defined as
Note that, as for every open set of the circle, we get
Moreover, the derivative and the -norm with respect to such a measure are given by
Note that, as , the above derivative is well defined almost everywhere. Similarly, the -norm is defined by
Moreover, the first and second variations of with respect to the measure are defined respectively by
Finally, is the space of functions belonging to with finite first variation . Analogously is the set of functions with finite second variation .
The next lemma points out some useful relationships between the quantities previously introduced.
For very the following identities hold:
follows by integrating by parts. follows from the definition of the derivative with respect to and (2.12). follows from and the definition of the derivative . ∎
For every it holds
We can now define the norm on the tangent space to .
Definition 2.16 (norm on the tangent space).
For every , the tangent space at to , which is equal to is endowed with the (equivalent) norm of the space
introduced in Definition 2.13. More precisely, the -norm is defined by
Finally, we recall that
Remark 2.17 (weighted norms).
Similarly to , we could consider some weighted -norms, defined as
where for . We can define the norm on the tangent space by the same constants.
One can easily satisfy that our results can be generalized to such a framework. In fact, this weighted norm is equivalent to the classical one and the positive constants do not affect the bounds and the convergence properties that we prove in this work.
The following proposition proves that and represent the same space of functions with equivalent norms.
Let . The sets and coincide and their norms are equivalent. More precisely, there exist two positive constants such that, for all
We suppose that is not equal to zero. For the -norms of , the result follows from (2.7) and the constants are given respectively by
Moreover, by Lemma 2.14 , the and -norms of the respective first derivative coincide. So it is sufficient to obtain the result for the second variation of .
By integration by parts, we have
where we used the fact that . This implies in particular that
Since and is a Banach algebra, we get
Now, as , applying the chain rule for -functions to , we can set
On the other hand, we have
and, because of Lemma 2.14, we get
Therefore, by the chain rule for -functions, the result is proved by taking the constant
The lemma ensues setting
2.4 Paths Between -Immersed Curves and Existence of Geodesics
In this section, we define the set of admissible paths between two -immersed curves and a Finsler metric on . In particular we prove that a minimizing geodesics for the defined Finsler metric exists for any given couple of curves.
Definition 2.19 (paths in ).
For every , we define a path in joining and as a function
For every , we denote the class of all paths joining and , belonging to