Association Fields via Cuspless SubRiemannian Geodesics in SE(2)
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Abstract

show that \(\mathcal{R}\) is contained in half space x≥0 and (0,y _{fin})≠(0,0) is reached with angle π,

show that the boundary \(\partial\mathcal{R}\) consists of endpoints of minimizers either starting or ending in a cusp,

analyze and plot the cones of reachable angles θ _{fin} per spatial endpoint (x _{fin},y _{fin}),

relate the endings of association fields to \(\partial\mathcal {R}\) and compute the length towards a cusp,

analyze the exponential map both with the common arclength parametrization t in the subRiemannian manifold \((\mathrm{SE}(2),\mathrm{Ker}(\sin\theta{\rm d}x +\cos\theta {\rm d}y), \mathcal{G}_{\xi}:=\xi^{2}(\cos\theta{\rm d}x+ \sin\theta {\rm d}y) \otimes(\cos\theta{\rm d}x+ \sin\theta{\rm d}y) + {\rm d}\theta \otimes{\rm d}\theta)\) and with spatial arclength parametrization s in the plane \(\mathbb{R}^{2}\). Surprisingly, sparametrization simplifies the exponential map, the curvature formulas, the cuspsurface, and the boundary value problem,

present a novel efficient algorithm solving the boundary value problem,

show that subRiemannian geodesics solve Petitot’s circle bundle model (cf. Petitot in J. Physiol. Paris 97:265–309, [2003]),

show a clear similarity with association field lines and subRiemannian geodesics.
Keywords
SubRiemannian geometric control Association fields Pontryagin’s maximum principle Boundary value problem Geodesics in rototranslation space1 Introduction
Curve optimization plays a major role both in imaging and visual perception. In imaging there exist many works on snakes and active contour modeling, whereas in visual perception illusionary contours arise in various optical illusions [48, 52]. Mostly, these optimal curve models rely on Euler’s elastica curves [33] (minimizing \(\int(\kappa ^{2}+ \xi^{2}) {\rm d}s\)) to obtain extensions where typically external forces to the data are included, cf. [5, 18, 21, 60, 61].
On top of that elastica curves relate to modes of the direction process (for contourcompletion [24]) where the direction of an oriented random walker is deterministic and its orientation is random. Such deterministic propagation only makes sense when the initial orientation is sharply defined. Instead Brownian motion with random behavior both in spatial propagation direction and in orientation direction [1, 22, 25], relates to hypoelliptic diffusion on the planar rototranslation group. Such a Brownian motion models contour enhancement [25] rather than contour completion [24], see [28] for a short overview. The corresponding Brownian bridge measures [27, 67] (relating to socalled completion fields in imaging [4, 24, 63, 64]) tend to concentrate towards optimal subRiemannian geodesics [12, 15, 22, 26, 47, 56]. So both elastica curves and subRiemannian geodesics relate to two different fundamental leftinvariant stochastic processes [28] on subRiemannian manifolds on the 2DEuclidean motion group SE(2), (respectively to the direction process [24, 48] and to hypoelliptic Brownian motion [1, 22, 25]).

Every cuspless subRiemannian geodesic (stationary curve) is a global minimizer [15, 16].

The EulerLagrange ODE for normalized curvature \(z=\kappa/\sqrt {\kappa^{2}+\xi^{2}}\) can be reduced to a linear one.

The boundary value problem can be tackled via effective analytic techniques.

The locations where global optimality is lost can be derived explicitly.

SubRiemannian geodesics are parametrization independent in the rototranslation group SE(2), which is encoded via a pinwheel structure of cortical columns in the primary visual cortex [50, 51].
 P
 Fix ξ>0 and boundary conditions \(g_{in}=(x_{in},y_{in},\theta _{in}), g_{fin}=(x_{fin},y_{fin},\theta_{fin})\in\mathbb{R}^{2}\times S^{1}\). On the space of (regular enough) planar curves, parameterized by planar arclength s>0, we aim to find the solutions of:$$\begin{aligned} &\mathbf{x}(0)=(x_{in},y_{in}),\quad\quad\mathbf{x}(\ell )=(x_{fin},y_{fin}) , \\ &\dot{\mathbf{x}}(0)=(\cos(\theta_{in}), \sin(\theta_{in})) , \end{aligned}$$(1)$$\begin{aligned} &\dot{\mathbf{x}}(\ell)=(\cos(\theta_{fin}), \sin(\theta _{fin})), \\ & \int_0^\ell\sqrt{\xi^2+(\kappa (s))^2}~ds\to\min~~(\mbox{with $\ell$ free}). \end{aligned}$$(2)
This variational problem was studied as a possible model of the mechanism used by the visual cortex V1 to reconstruct curves which are partially hidden or corrupted. This model was initially due to Petitot (see [50, 51] and references therein). Subsequently, the subRiemannian structure was introduced in the problem by Petitot [52] for the contact geometry of the fiber bundle of the 1jets of curves in the plane (the polarized Heisenberg group), whereas Citti and Sarti [22] introduced the subRiemannian structure in SE(2) in problem P. The group of planar rotations and translations SE(2) is the true symmetry group underlying problem P. Therefore, we build on the SE(2) subRiemannian viewpoint first proposed by Citti and Sarti [22], and we solve their cortical model for all appropriate endconditions. The stationary curves of problem P were derived by the authors of this paper in [12, 26]. The problem was also studied by Hladky and Pauls in [40], and by BenYosef and BenShahar in [11].
In this article we will show that the model coincides^{1} with the circle bundle model by Petitot [52] and that its minimizers correspond to spatial projections of cuspless subRiemannian geodesics within \(\mathbb {R}^{2}\rtimes S^{1}\).
Remark 1.1
We will see in the following that this set \(\mathcal{R}\) is the set of all endpoints in \(\mathbb{R}^{2} \times S^{1}\) that can be connected with a cuspless stationary curve of problem P, starting from (0,0,0).
Remark 1.2
The physical dimension of parameter ξ is [Length]^{−1}. From a physical point of view it is crucial to make the energy integrand dimensionally consistent. However, the problem with (x(0),θ(0))=(0,0,0) and ξ>0 is equivalent up to a scaling to the problem with ξ=1: The minimizer x of P with ξ>0 and boundary conditions (0,0) and (x _{1},θ _{1}) relates to the minimizer \(\overline{\mathbf{x}}\) of P with ξ=1 and boundary conditions (0,0) and (ξ x _{1},θ _{1}), by spatial rescaling: \(\mathbf{x}(s)=\xi^{1} \overline{\mathbf{x}}(s)\). Therefore, in the remainder of this article we just consider the case ξ=1 for simplicity.
 1.
Direct derivation of the EulerLagrange equation. E.g. the approach by Mumford [48], yielding a direct approach to the ODE for the curvature, see Appendix A.
 2.
The Pontryagin Maximum principle: A geometrical control theory approach based on Hamiltonians, cf. [3, 12, 47, 53] and Appendix D.
 3.
The Bryant and Griffith’s approach (based on the works by MarsenWeinstein on reduction in theoretical mechanics [44]) using a symplectic differential geometrical approach based on Lagrangians [26, App. A], cf. [19].
The first approach very efficiently produces only the EulerLagrange equation for the curvature of stationary curves, but lacks integration of a single curve and lacks a geometric study of the continuum of all stationary curves that arise by varying the possible boundary conditions.
The second approach includes profound geometrical understanding from a Hamiltonian point of view and deals with local optimality [3] of stationary curves.
The third approach^{3} takes a Lagrangian point of view and provides additional differential geometrical tools from theoretical mechanics that help integrating and structuring the canonical equations. These additional techniques will be of use in deriving semianalytic solutions to the boundary value problem and in the modeling of association fields.
Firstly, application of Mumford’s approach for deriving the ODE for curvature of elastica, to problem P is relatively straightforward, see Appendix A, but does not explicitly involve geometrical control and the Frenet formula still needs to be integrated.
Thirdly, application of the Bryant and Griffith’s (Lagrangian) approach to problem P will yield a canonical Pfaffian system on an extended manifold whose elements involve both position, orientation, control (curvature and length), spatial momentum and angular momentum. We will show that the essential part of this Pfaffian system is equivalent to \(\nabla_{\dot{\gamma}} p = 0\) where ∇ denotes a Cartan connection and p denotes momentum as a covector within \(T^{*}(\mathbb {R}^{2}\rtimes S^{1})\). This fundamental identity allows us to analytically solve the boundary value problem.
1.1 Lift problem \(\bf{P}\) to the rototranslation group
 Since in this problem we are taking v(⋅)∈L ^{1}([0,ℓ]), the curve \(\gamma=(x(\cdot),y(\cdot),\theta(\cdot)):[0,\ell]\to \mathbb{R} ^{2}\times S^{1}\) is absolutely continuous and curve \(\mathbf{x}=(x(\cdot ),y(\cdot)):[0,\ell]\to\mathbb{R}^{2}\) is in Sobolev space \(W^{2,1}([0,\ell],\mathbb{R}^{2})\).
 P _{curve}:
 Fix ξ>0 and boundary conditions \((x_{in},y_{in},\theta _{in}), (x_{fin},y_{fin},\theta_{fin})\in\mathbb{R}^{2}\times S^{1}\), with (x _{ in },y _{ in })≠(x _{ fin },y _{ fin }). In the space of integrable (possibly nonsmooth) controls \(v(\cdot ):[0,\ell]\to\mathbb{R}\), we aim to solve:$$\begin{aligned} &(x(0),y(0),\theta(0))=(x_{in},y_{in},\theta_{in}), \\ &(x(\ell ),y(\ell),\theta(\ell))=(x_{fin},y_{fin},\theta_{fin}), \\ & \left( \begin{array}{c} \frac{dx}{ds}(s)\\ \frac{dy}{ds}(s)\\ \frac{d\theta }{ds}(s) \end{array} \right)=\left( \begin{array}{c} \cos(\theta(s)) \\ \sin(\theta(s)) \\ 0 \end{array} \right)+v(s) \left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right), \\ & \int_0^\ell\sqrt{\xi^2 + \kappa(s)^2}~{\rm d}s= \int_0^\ell\sqrt{\xi^2 + v(s)^2}{\rm d}s\\ &\quad\to\min\quad (\mbox{here } \ell\geq0 \mbox{ is free}) \end{aligned}$$(11)
 Problem P _{MEC} has a solution by Chow’s and Fillipov’s theorems [3] regardless the choice of endcondition and has been completely solved in a series of papers by one of the authors (see [47, 55, 56]). It gives rise to a subRiemannian distance on the subRiemannian manifold within SE(2) as we will explain next.
 P _{MEC}:
 Fix ξ>0 and boundary conditions \((x_{in},y_{in},\theta _{in}), (x_{fin},y_{fin},\theta_{fin})\in\mathbb{R}^{2}\times S^{1}\). In the space of L ^{∞} controls \(\tilde{u}(\cdot),\tilde {v}(\cdot):[0,\ell]\to\mathbb{R}\), solve:$$\begin{aligned} &(x(0),y(0),\theta(0))=(x_{in},y_{in},\theta_{in}), \\ &(x(T),y(T),\theta(T))=(x_{fin},y_{fin},\theta_{fin}) , \\ & \left( \begin{array}{c} \frac{dx}{dt}(t)\\ \frac{dy}{dt}(t)\\ \frac{d\theta }{dt}(t) \end{array} \right)=\tilde{u}(t) \left( \begin{array}{c} \cos(\theta(t)) \\ \sin(\theta(t)) \\ 0 \end{array} \right)+\tilde{v}(t) \left( \begin{array}{c} 0\\ 0\\ 1 \end{array} \right) \\ & \int_0^T\sqrt{\xi^2\tilde {u}(t)^2+\tilde{v}(t)^2}~{\rm d}t \\ &\quad \to\min\quad (\mbox{here } T\geq0 \mbox{ is free}) \end{aligned}$$(12)
Remark 1.3
The subRiemannian structure is 3D contact and analytic and therefore we have nonexistence of abnormal extrema and all minimizers are analytic, where we note that distribution Δ is 2generating cf.[3, Chap. 20.5.1].
Problem P _{MEC} is to be considered as an auxiliary mechanical problem (of optimal path planning of a moving car carrying a steering wheel and the ability to drive both forwardly and backwardly) associated to P _{curve}. To this end we stress that P _{MEC} cannot be interpreted as a problem of reconstruction of planar curves, [14]. The problem is that the minimizing curve γ=(x,θ):[0,T]→SE(2) may have a vertical tangent vector (i.e. in θdirection) in between the ending conditions, which causes a cusp in the corresponding projected curve t↦x(t) in the plane, see Fig. 2. Such a cusp corresponds to a point on an optimal path where the car is suddenly set in reverse gear.
Problem P _{MEC} is invariant under monotonic reparameterizations and at a cusp spatial arclength parametrization breaks down. If \((x_{fin},y_{fin},\theta_{fin}) \in\mathcal{R}\) no such cusps arise and P _{MEC} and P _{curve} are equivalent [15, 16] and we can use arclength parametrization also in P _{MEC} (in which case the first controlvariable is set to 1, since \(\langle\omega^{2}\vert_{\gamma (s)},\dot{\gamma}(s)\rangle=1\)). In [16] we have proven the following Theorem.
Definition 1
Let \(\mathcal{R} \subset\mathrm{SE}(2)\) denote the set of endpoints in SE(2) that can be reached from e with a stationary curve of problem P _{curve}.
Theorem 1

\((x_{fin},y_{fin},\theta_{fin}) \in\mathcal{R}\) if and only if P _{curve} has a unique minimizing geodesic which exactly coincides with the unique minimizer of P _{MEC}.

\((x_{fin},y_{fin},\theta_{fin}) \notin\mathcal{R}\) if and only if problem P _{curve} is illdefined (i.e. P _{curve} does not have a minimizer).^{7}
2 Structure of the Article
Firstly, in Sect. 3 we consider the origin of the problem of finding cuspless subRiemannian geodesics in \((\mathrm {SE}(2),\Delta , \mathcal{G}_{\beta})\), which includes cortical modeling of the primary visual cortex and association fields.
In Sect. 4 we provide a short road map on how to connect two natural parameterizations. The cuspless subRiemanian geodesics in the subRiemannian manifold \((\mathrm{SE}(2),\Delta,\mathcal{G}_{\beta})\) can be properly parameterized by the subRiemannian arclength parametrization (via t) or by spatial arclength parametrization (via s). Parametrization via t yields the central part of the mathematical pendulum phase portrait (recall Eq. (10)), whereas parametrization via s yields a central part of a hyperbolic phase portrait (recall Eq. (4)). The hyperbolic phase portrait does not coincide with a local linearization approximation (as in HartmanGrobman’s theorem [38]). In fact, it is globally equivalent to the relevant part of the pendulum phase portrait (i.e. the part associated to cuspless subRiemannian geodesics). The involved coordinate transforms are global diffeomorphisms.
In Sect. 5 we define the exponential map [2, 47] for P _{curve} and P _{MEC}. Then we show that the set \(\mathcal{R} \subset\mathrm{SE}(2)\) (consisting of admissible endconditions) equals the range of the exponential map of P _{curve}. We will provide novel explicit formulas for the exponential map for P _{curve} using spatial arc length parametrization s and moreover, for completeness and comparison, in Appendix B we will also provide explicit formulas for the exponential map of P _{MEC} that were previously derived in previous work [47] by one of the authors.
We show that the exponential map of P _{curve} follows by restriction of P _{MEC} to the strip \((\nu,c) \in[0,2\pi] \times\mathbb{R}\), see Fig. 9. A quick comparison in Appendix B learns us that spatial arclength parametrization (also suggested in [22]) simplifies the formulas of the (globally minimizing, cuspless) geodesics of P _{curve} considerably.
 1.
show that \(\mathcal{R}\) is contained in half space x≥0 and (0,y _{fin})≠(0,0) is reached with angle π,
 2.
show in Theorem 6 that the boundary \(\partial \mathcal{R}\) consists of the union of endpoints of minimizers either starting or ending in a cusp and a vertical line \(\mathfrak{l}\) above (0,0,0), and we compute the total spatial arclength towards a cusp,
 3.
analyze and plot the cones of reachable angles θ _{fin} per spatial endpoint (x _{fin},y _{fin}),
 4.
prove homeomorphic and diffeomorphic properties of the exponential map in Theorem 6,
 5.
show in Lemma 8 that geodesics that end with a cusp at \(\theta_{fin}=\frac{\pi}{2}\) are precisely those with stationary curvature (\(\dot{\kappa}(0)=0\)) at the origin.
In Sect. 7 we solve the boundary value problem, where we derive a (semi)analytic description of the inverse of the exponential map and present a novel efficient algorithm to solve the boundary value problem. This algorithm requires numerical shooting only in a small subinterval of [−1,1], rather than a numerical shooting algorithm in \(\mathbb{R}^{2}\times S^{1}\).
In Sect. 8 we show a clear similarity of cuspless subRiemannian geodesics and the association field lines from psychophysics [34] and neurophysiology [52]. This is not surprising as we will show that subRiemannian geodesics allowing xparametrization, exactly solve the circle bundle model for association fields by Petitot, cf. [52]. It is remarkable that the endings of association fields are close to the cuspsurface \(\partial\mathcal{R}\), which we underpin with Lemma 8 and Remark 8.1.
For a concise overview of previous mathematical models for association fields and their direct relation to the cuspless subRiemannian geodesic model proposed in this article we refer to the final subsection in Appendix G.
3 Origin of Problem \(\bf{P}\): Cortical Modeling
In a simplified model (see [51, p. 79]), neurons of V1 are grouped into orientation columns, each of them being sensitive to visual stimuli at a given point of the retina and for a given direction on it. The retina is modeled by the real plane.
The Euclidean motion group acts transitively and free on the space of positions and orientations, allowing us to identify the coupled space of positions and orientations \(\mathbb{R}^{2}\rtimes S^{1}\) with the rototranslation group \(\mathrm{SE}(2)=\mathbb{R}^{2} \rtimes SO(2)\). This imposes a natural Cartan connection [26, 52] on the tangent bundle \(T(\mathbb{R}^{2}\rtimes S^{1})\) induced by the pushforward of the leftmultiplication of SE(2) onto itself.
In this article we will show that subRiemannian geodesics closely model the association fields from psychophysics and that the location of cusps seems to provide a reasonable grouping criterium to connect two local orientations (consistent with endings of the association field), see Fig. 8. Next we will show that it does not matter whether one lifts problem P (given by Eqs. (1) and (2)) to the projective line bundle or to the group of rotations and translations in the plane.
3.1 No Need for Projective Line Bundles in P _{curve}
The P _{MEC} problem on \((\mathrm{SE}(2)=\mathbb {R}^{2}\rtimes S^{1}, \Delta=\mathrm {Ker}(\omega^{3}), \mathcal{G}_{\xi})\) can as well be formulated on the projective line bundle P ^{1} [14, 52] where antipodal points on the sphere S ^{1} are identified. See also [13].

Flipping only one of the boundary conditions is not possible as in this article we shall show that if \((x_{fin},y_{fin}, \theta_{fin}) \in \mathcal{R} \Rightarrow(x_{fin},y_{fin}, \theta_{fin}+\pi) \in (\mathbb{R} ^{2}\times S^{1}) \setminus\mathcal{R}\), i.e. when (x _{ fin },y _{ fin },θ _{ fin }) is an admissible ending condition then (x _{ fin },y _{ fin },θ _{ fin }+π) is not admissible.

If we both flip (i.e. θ↦θ+π) and switch both the initial and ending condition we get the same curve (in opposite direction).
Therefore, in this article we will not identify antipodal points and we focus on problem P _{curve} and its corresponding admissible boundary conditions (i.e. an explicit description of the set \(\mathcal {R}\subset\mathrm{SE}(2)\)).
4 Parametrization of Curves in P _{curve}
The natural parametrization for subRiemannian geodesics in P _{MEC} is the subRiemannian arclength parametrization. However, when considering only those subRiemannian geodesics in \((\mathrm{SE}(2),\Delta,\mathcal{G}_{\xi})\) without cusps (as in P _{curve}), i.e. the cuspless subRiemannian geodesics, the problem is actually a planar curve problem (as in P) and there it is more natural^{8} to use spatial arclength parametrization.
Lemma 1
Proof
5 Cusps and the Exponential Map Associated to P _{curve} and P _{MEC}
In order to express the exponential map associated to P _{curve}(for ξ=1) in spatial arclength parametrization we apply Bryant & Griffith’s approach [20], which was previously successfully applied to the elastica problem [19]. Here we will also include an additional viewpoint on this technical approach via the Cartan connection. In case the reader is not so much interested in the geometrical details and underpinnings, it is also possible to skip the following derivations and to continue reading starting from the formulas for the subRiemannian geodesics γ(s) in Theorem 3.
Remark 5.1
The Christoffel symbols \(c^{j}_{ki}\) of the Cartan connection ∇ on the tangent bundle T(SE(2)) expressed in reference frame \(\{\mathcal{A}_{i}\}_{i=1}^{3}\) equal minus the structure constants on the Lie algebra. The Christoffel symbols of the corresponding Cartan connection on the cotangent bundle T ^{∗}(SE(2)) w.r.t. reference frame \(\{\omega^{i}\}_{i=1}^{3}\) have opposite sign and are thereby equal to the structure constants \(c^{j}_{ik}=c^{j}_{ki}\).
Now that the preliminaries are done let us apply Bryant and Griffith’s method to P _{curve} in 4 steps.
Theorem 2
Proof
Remark 5.2
The first part ensures γ=(x,θ) is the horizontal lift from the planar curve x(s)=(x(s),y(s)), i.e. \(\theta(s)=\arg(\dot{x}(s)+i \dot{y}(s))\). The second part allows us to interpretate \(p=\sum_{i=1}^{3} \lambda_{i} \omega^{i}\) as a momentum covector.
Remark 5.3
In contrast to LeviCivita connections on Riemannian manifolds, the Cartan connection ∇ has torsion and thereby autoparallel curves do not coincide with geodesics. In fact, Theorem 12 in Appendix C shows that autoparallel curves are (horizontal) exponential curves.
Lemma 2
Proof
Corollary 1
Proof
Follows directly from the hyperbolic phase portrait induced by \(\ddot{z}=z\) and Theorem 2, and solving for respectively z(s)=1 and z(s)=0. □
After these results on subRiemannian geodesics, we continue with solving for ∇p=0, Eq. (37). Problem P _{curve} is leftinvariant and in the next lemma we select a suitable point on each coadoint orbit to simplify the computations considerably.
Lemma 3
Applying the above Lemma and Eq. (29) provides the next theorem, Theorem 3, where we provide explicit analytical formulae for the geodesics by integration of the Pfaffian system. To this end we first need a formal definition of the operator that integrates the Pfaffian system Eq. (28) and produces the corresponding geodesic of P _{curve} in SE(2).
Definition 2
Remark 5.4
For sober notation we omit index e and write \(\widetilde{\mathrm{Exp}}=\widetilde{\mathrm{Exp}}_{e}\) and H(p)=H(e,p) for exponential map and Hamiltonian. Furthermore, we include a tilde in this exponential map associated to the geometrical control problem of P _{curve} to avoid confusion with the exponential map Exp:T _{ e }(SE(2))→SE(2) from Liealgebra to Lie group.
Remark 5.5
The dual vectors \(p_{0}= \pm{\rm d}\theta\) are not part of the domain of the exponential map as in these cases one would have \((z_{0},\dot{z}_{0})=(\pm1,0)=(z(s),\dot{z}(s))\) for all s≥0 and the subRiemannian geodesics in SE(2) propagate only in vertical direction, not allowing spatial arclength parameterization. See also [16, Remark 31].
Theorem 3
Corollary 2
Proof
Corollary 3
Geodesics with \(\mathfrak{c}=1\) admit simple formulas:
Corollary 4
Corollary 5
5.1 Relation Between the Exponential Mappings of P _{curve} and P _{MEC}
In Theorem 3 we have derived the exponential map of P _{curve} in terms of spatial arclength parametrization s, whereas in previous work [15] the exponential map of P _{MEC} is expressed in subRiemannian arclength t. For comparison see Appendix B.
On the one hand one observes that the exponential map of P _{curve} is much simpler when expressed in s and it is easier to integrate in current active shape models in imaging where the same kind of parametrization is used. On the other hand for P _{MEC} it is more natural to choose tparametrization as this parametrization does not beak down at cusps. The following theorem relates the exponential mappings for P _{curve} and P _{MEC}.
Theorem 4
Proof
We note that ℓ≤s _{ max } implies that the orbits do not hit the cusp lines in the pase portraits (i.e. z=1 and ν=0,2π) so that (ν(t),c(t)) stays within the central strip (i.e. ν(t)∈[0,2π]) indicated in Fig. 9. The rest follows by Lemma 1. □
6 The Set \(\mathcal{R}\) and the CuspSurface \(\partial \mathcal {R}\)
According to Theorem 1 the set of points in SE(2) that can be reached with a global minimizer from unity element g _{ in }=e=(0,0,0) is equal to \(\mathcal {R}\) given in Definition 1. Therefore, we first need to investigate this set in order to apply cuspless subRiemannian geodesics in vision applications. First of all we have the following characterization.
Theorem 5
Proof
Apply Theorems 1 and 3, where the analytic stationary solution curves of P _{curve} break down iff ℓ=s _{ max }(p _{0}) in which case tangents to geodesics are vertical due to \(z(\ell)=\frac{d\theta}{dt}(T)=1\). □
Theorem 6

\(\widetilde{\mathrm{Exp}}: \mathcal{D} \to\mathcal{R}\) is a homeomorphism if we equip \(\mathcal{D}\) and \(\mathcal{R}\) with the subspace topology.^{10}

\(\widetilde{\mathrm{Exp}}: \mathring{\mathcal{D}} \to \mathring {\mathcal{R}}\) is a diffeomorphism.
6.1 The Elliptic Integral in the Exponential Map
In this section we will first express the single elliptic integral arising in the exponential map in Theorem 3 in a standard elliptic integral and then we provide bounds for this integral from which one can deduce bounds on the set \(\mathcal{R}\).
Lemma 4
Proof
Using Eq. (77) and Eq. (38) we find \(1z(\tau)^{2}= \frac{1+\mathfrak{c}^{2}}{2} (1 c_{1}\cosh(2\tau)  c_{2} \sinh(2\tau))\) from which the result follows via v=iτ. □
For explicit bounds for the elliptic integral for the cases \(\mathfrak {c}<1\), where the subRiemannian geodesics are Ushaped, see Appendix H.
6.2 Observations and Theorems on \(\mathcal{R}\)
 1.
The range \(\mathcal{R}\) of the exponential map is a connected, noncompact set and its piecewise smooth boundary coincides with the cuspsurface, Eq. (55).
 2.
The range of the exponential map produces a reasonable criterium (namely condition (3)) to connect two local orientations. Consider the set of reachable cones depicted in Fig. 14.
 3.
The range of the exponential map of P _{curve} is contained in the halfspace x _{ fin }≥0 and θ _{ fin }=π can only be attained at x=0 and y≠0 where geodesics arrive at a cusp.
 4.The cone of reachable angles θ _{ fin } per position \((x_{fin}, y_{fin}) \in\mathbb{R}^{+} \times\mathbb{R}^{+}\), with \((x_{fin},y_{fin},\theta_{fin}) \in\mathcal{R}\) is either given bywith x _{ fin }=(x _{ fin },y _{ fin }) where θ _{endcusp}(x _{ fin }) denotes the final angle of the geodesic ending in (x _{ fin },⋅) with a cusp, and where θ _{begincusp}(x _{ fin }) denotes the final angle of a geodesic ending in (x _{ fin },⋅) starting with a cusp. In the second case there exist two geodesics ending in x _{ fin } with a cusp and we index these such that \(\theta_{\mathrm {endcusp}}^{1}<\theta_{\mathrm{endcusp}}^{2}\). Which of the two options applies depends on \(\mathbf{x}_{fin} \in \mathbb{R} ^{2}\). See Fig. 12.$$ \begin{aligned} &[\theta_{\mathrm{begincusp}}(\mathbf{x}_{fin}), \theta _{\mathrm{endcusp}}(\mathbf{x}_{fin})]\quad \textrm{or by }\\ &[\theta_{\mathrm{endcusp}}^{1}(\mathbf{x}_{fin}), \theta _{\mathrm{endcusp}}^{2}(\mathbf{x}_{fin})], \end{aligned} $$(57)
 5.The boundary of the range of the exponential map (given by Eq. (55)) is smooth except for 3 intersections between the surface induced by endpoints of geodesics starting from a cusp and the surface induced by endpoints of geodesics ending at a cusp. These intersections are given bywhere E(z,m) is given by Eq. (56).$$\begin{aligned}& \theta_{fin}=\pi\quad\textrm{and}\quad x_{fin}=0 \quad\textrm{and}\quad y_{fin} \leq 0, \\& \theta_{fin}=0 \quad\textrm{and }\\& y_{fin}= x_{fin} i E \biggl(i \,\mathrm{arcsinh}\, \frac{x_{fin}}{\sqrt{4x_{fin}^2}}, 1\frac{4}{x_{fin}^2} \biggr), \\& \textrm{and}\quad 0\leq x_{fin}<2, \\& \theta_{fin}=\pi\quad\textrm{and}\quad x_{fin}=0\quad \textrm{and}\quad y_{fin} \geq0, \end{aligned}$$
 6.
The critical surface splits the range of the exponential map into four disjoint parts, cf. Fig. 11. These parts \(\mathcal{C}_{1}^{1}\), \(\mathcal{C}^{0}_{1}\), \(\mathcal{C}_{2}^{+}\) and \(\mathcal{C}_{2}^{}\) directly relate to the splitting of the phase space, into the four parts \(C_{1}^{1}\), \(C^{0}_{1}\), \(C_{2}^{+}\) and \(C_{2}^{}\).
 7.
If \(g_{fin}=(x_{fin},y_{fin},\theta_{fin}) \in\mathcal{R}\) then \(g_{fin}=(x_{fin},y_{fin}, \theta_{fin}+\pi) \notin\mathcal{R}\).
Let’s underpin these observations with theorems.
Lemma 5
Let 0<a<b<1. Then \(\varPsi(a,b):=\frac{a}{\sqrt{1+b}} \frac{1}{2} \log( \frac{b+a}{ba} )<0\).
Proof
Ψ does not contain stationary points in the open region in \(\mathbb{R} ^{2}\) given by 0<a<b<1. At the boundary we have Ψ(0,b)=0 and lim_{ b↓a } Ψ(a,b)=−∞ and \(\varPsi(a,1)= \frac{a}{\sqrt{2}}\frac{1}{2} \log( \frac {1+a}{1a} )\) and \(\frac{\partial\varPsi(a,1)}{\partial a}<0\) so Ψ(a,b)<Ψ(0,1)=0 for 0<a<b<1. □
Theorem 7
The range \(\mathcal{R}\) of the Exponential map of P _{curve} is contained within the half space x≥0. In particular, its boundary \(\partial\mathcal{R}\) (i.e. the cuspsurface) is contained within x≥0.
Proof
For analysis of \(\mathcal{R}\) and \(\partial\mathcal{R}\) and for (semi)analytically solving of the boundary value problem the following identities (due to Theorem 3) come at hand.
Lemma 6
Theorem 8
Proof
Suppose θ _{ fin }=π then on the one hand by Eq. (61) we have \(\dot{z}(\ell)=\dot{z}_{0}\) whereas on the other hand by Eq. (52) we have \(\dot{z}_{\ell}\sqrt {1z_{0}^{2}}\dot{z}_{0}\sqrt{1z(\ell)^{2}}=0\) from which we deduce z(ℓ)=z _{0}=1. Suppose z _{0}^{2}=z(ℓ)^{2}=1 and \(\dot{z}_{0}=\dot{z}(\ell)\) then z(0)≠−z(ℓ) and we obtain x _{ fin }=0 and y _{ fin }≠0 by Eq. (52). Finally, suppose x _{ fin }=0 and y _{ fin }≠0 then D=ψ=R _{2}=0 and ρ=R _{1}=−α in Eq. (63) and thereby we obtain cos(2θ _{ fin })=1 and the result follows □
6.3 The Cones of Reachable Angles
Theorem 9
For a direct graphical validation of Theorem 9 see Fig. 11 (in particular the top view along θ), where we note that the bound in (65) relates to the spatial projection of the curve that arises by taking the intersection of the blue and red surface on \(\partial\mathcal{R}\) at θ=0 (the thick black line in Fig. 11 at θ=0). For more details on the proof see Appendix E.
As already mentioned in Sect. 3.1, it does not matter if one considers problem P _{curve} on the projective line bundle \(\mathbb{R}^{2} \rtimes P^{1}\) or on \(\mathbb{R}^{2} \rtimes S^{1} \equiv\mathrm {SE}(2)\). This is due to the following theorem.
Theorem 10
Proof
From Theorem 3 we have \(\dot {\tilde{y}}(s) \geq0\) from which we deduce condition \(\sin(\theta _{fin}\overline{\theta}_{0}) \leq0\) implying the result. □
7 Solving the Boundary Value Problem
In order to explicitly solve the boundaryvalue problem for P _{curve} for admissible boundary conditions (Eq. (3)) we can apply leftinvariance (i.e. rotation and translation invariance) of the problem and consider the case g _{ in }=e=(0,0,0) and \(g_{fin} \in \mathcal{R}\).
We invert the boundary value problem for a very large part analytically, yielding a novel very fast and highly accurate algorithm to solve the boundary value problem. In comparison to previous work on this topic [45], we have less parameters to solve (and moreover, our proposed optimization algorithm involves less parameters).
However, before we can formulate this formally there is a technical issue to be solved first, which is the choice of sign in Eq. (64).
Lemma 7
Proof
The \(\widetilde{\mathrm{Exp}}\) is a (global) homeomorphism and its orbits \(s \mapsto\widetilde{\mathrm{Exp}}(p_{0},s)\) are analytic for each \(p_{0} \in T^{*}_{e}(\mathrm{SE}(2))\). Thereby the sign cannot switch along orbits (unless D=0, which only occurs at θ _{ fin }=±π at \(\partial\mathcal{R}\)). Furthermore, since \(\widetilde{\mathrm{Exp}}\) is a homeomorphism sign switches (in Eq. (64)) between neighboring orbits are not possible unless it happens across an orbit \(s\mapsto(z(s),\dot {z}(s))\) with \(\dot{z}_{0}=0\). Now from the phase portrait it is clear that orbits in phase space \(s \mapsto(z(s), \dot{z}(s))\) with \(\dot{z}(s)>0\) and \(\mathfrak{c}>1\), i.e. orbits in \(C^{+}_{2}\) need a plus sign, whereas orbits in \(C^{}_{2}\) need a minus sign in Eq. (64). The line \(\dot{z}_{0}=0\) splits the phase portrait in two parts, and by the results in Theorem 6 this means that the surface \(\mathcal{V}\) splits the set \(\mathcal{R}\) into two parts. Now \(\widetilde{\mathrm{Exp}}\) maps \(C^{+}_{2}\) onto \(\mathcal{C}^{+}_{2}\) and it maps \(C^{}_{2}\) onto \(\mathcal{C}^{}_{2}\), and \(\mathcal{C}^{}_{2}\) lies beneath V and \(\mathcal{C}^{+}_{2}\) lies above V, from which the result follows. □
Remark 7.1
The surface \(\mathcal{V}\) is depicted in Fig. 15. Lemma 7 is depicted in Fig. 16, where we used Theorem 3 to compute for each point in \((z_{0},\dot{z}_{0}) \in[1,1] \times[2,2]\) in phase space the sign of \(2a \dot{z}_{0}+b\) at respectively \(s=0, \frac{1}{2}s_{max}(z_{0},0), \frac{3}{4}s_{max}(z_{0},0)\) and s=s _{ max }(z _{0},0). We see that the black points (where the sign is positive) lies above the orbits family of orbits with z _{0}∈[−1,1] and \(\dot{z}_{0}=0\).
Remark 7.2
The next theorem reduces the boundary value problem to finding the unique root of a single positive realvalued function.
Theorem 11
Proof
By Theorem 1 there is a unique stationary curve connecting e and \(g_{fin} \in\mathcal{R}\). The exponential map of P _{curve} is a homeomorphism by Theorem 6 and thereby the continuous function F has a unique zero, since ℓ and \(\dot{z}_{0}\) are already determined by z _{0} and g _{ fin } via Theorem 3 and Lemma 7. □
Remark 7.3
Theorem 11 allows fast and accurate computations of subRiemannian geodesics, see Fig. 12 where the computed geodesics are instantly computed with an accuracy of relative \(\mathbb{L}_{2}\)errors in the order of 10^{−8}. Finally, we note that Theorem 6 implies that (our approach to) solving the boundaryvalue problem is wellposed (i.e. the solutions are both unique and stable).
8 Modeling Association Fields with Solutions of P _{curve}
Contact geometry plays a major role in the functional architecture of the primary visual cortex (V1) and more precisely in its pinwheel structure, cf. [52]. In his paper [52] Petitot shows that the horizontal corticocortical connections of V1 implement the contact structure of a continuous fibration π:R×P→P with base space the space of the retina and P the projective line of orientations in the plane. This model was refined by Citti and Sarti [22], who formulated the model as a contact structure within SE(2) producing problem P _{curve} given by Eq. (11).
Petitot applied his model to the Field’s, Hayes’ and Hess’ physical concept of an association field, to several models of visual hallucinations [32] and to a variational model of curved modal illusory contours [42, 48, 65].
In their paper, Field, Hayes and Hess [34] present physiological speculations concerning the implementation of the association field via horizontal connections. They have been confirmed by Jean Lorenceau et al. [43] via the method of apparent speed of fast sequences where the apparent velocity is overestimated when the successive elements are aligned in the direction of the motion path and underestimated when the motion is orthogonal to the orientation of the elements. They have also been confirmed by electrophysiological methods measuring the velocity of propagation of horizontal activation [37].
There exist several other interesting lowlevel vision models and psychophysical measurements that have produced similar fields of association and perceptual grouping [39, 49, 68], for an overview see [52, Chaps. 5.5, 5.6].
8.1 Three Models and Their Relation
Subsequently, we discuss three models of the association fields: horizontal exponential curves, Legendrian geodesics, and cuspless subRiemannian geodesics (which for many boundary conditions coincide with Petitot’s circle bundle model, as we will explain below).
For the explicit connections between each of the 3 mathematical models we refer to Appendix G.
8.2 SubRiemannian Geodesics Versus Cocircularity
In Fig. 8 we have modeled the association field with subRiemannian geodesics (ξ=1) and horizontal exponential curves (Eq. (72) as proposed in [9, 57]). Horizontal exponential curves are circular spirals and thereby rely on “cocircularity”, a wellknown basic principle to include orientation context in image analysis, cf. [35, 46].
On the one hand, a serious drawback arising in the cocircularity model for association fields is that the only the spatial part (x _{ fin },y _{ fin }) of the endcondition can be prescribed (the angular part is imposed by cocircularity), whereas with geodesics one can prescribe (x _{ fin },y _{ fin },θ _{ fin }) (as long as the ending condition is contained within \(\mathcal{R}\)). This drawback is clearly visible in Fig. 8, where the association field (see a) in Fig. 8) typically ends in points with almost vertical tangent vectors.
On the other hand, the subRiemannian geodesic model has more difficulty describing the association field by Field and coworkers in the almost circular connections to the side (where the cocircularity model is reasonable). To this end we note that circles are not subRiemannian geodesics as the ODE \(\ddot{z}=\xi z\) does not allow z to be constant.
This difficulty, however, can be tackled by variation of ξ in Problem P _{curve}. Our algorithm explained in Sect. 5, combined with the scaling homothety described in Remark 1.2, is wellcapable of reconstructing the almost circular field line cases as well. This can be observed in Fig. 17.
8.3 Variation of ξ and Association Field Modeling
Varying of ξ ^{2}>0 also takes into account a wellknown parameter in completion; namely the area of the completed figures (see e.g. [52]). This area equals A=(x _{ fin }−x _{ in })(y _{ fin }−y _{ in }). By Remark 1.1 we can as well set x _{ in }=y _{ in }=θ _{ in }=0 and then as explained in Remark 1.2 solving P _{curve} with ξ>0 amounts to solving P _{curve} with ξ=1 with scaled endconditions (x _{ fin } ξ,y _{ fin } ξ). In fact, such rescaling of endconditions rescales the area as follows A↦Aξ ^{2}.
8.4 A Conjecture and Its Motivation
Apparently, both the shape of the association field lines and their ending is wellexpressed by the subRiemannian geodesics model P _{curve}, which was proposed by Citti and Sarti [22]. Therefore, following the general idea of Petitot’s work [50] (in particular, his circle bundle model) and the results in this article on the existence set \(\mathcal{R}\) this puts the following conjecture:
Conjecture 1
The criterium in our visual system to connect two local orientations, say g _{0}=(x _{0},y _{0},θ _{0})=(0,0,0) and g _{ fin }=(x _{ fin },y _{ fin },θ _{ fin })∈SE(2), could be modeled by checking whether g _{ fin } is within the range \(\mathcal{R}\) of the exponential map.
Here we recall that from the results in [16] (summarized in Theorem 1) it follows that the set \(\mathcal{R}\) consists precisely of those points in SE(2) that are connected to the origin by a unique global minimizer of P _{curve}. This conjecture needs further investigation by psychophysical and neurophysiological experiments. In any case, within the model P _{curve} (relating to Petitot’s circle bundle model [52] and the subRiemannian model by Citti and Sarti [22]) a curve is optimal if and only if it is stationary. Furthermore, the subRiemannian geodesics strongly deviate from horizontal exponential curves even if the end condition is chosen such that the cocircularity condition is satisfied (this can be observed in item c) of Fig. 8). This discrepancy between horizontal exponential curves and cuspless subRiemannian geodesics in \((\mathrm{SE}(2), \Delta,\mathcal{G}_{\xi})\) is also intruiging from the differential geometrical viewpoint: see Theorem 12 in Appendix C.
In the remainder of this section we will mathematically underpin our observation that endpoints of association fields are close to cusps.
Lemma 8
 1.
\(\dot{z}_{0}=0\).
 2.
γ ends with a cusp in γ(ℓ)=(x(ℓ),y(ℓ),θ(ℓ)).
 3.
\(\theta(\ell)=\frac{\pi}{2}\).
Proof
Remark 8.1
The curves in the association field have \(\theta_{fin}=\frac{\pi }{2}\) and relatively small initial curvature so that \(\dot{z}_{0}\ll 1\) and therefore they end very close to cusps, i.e. ℓ≈s _{max}.
9 Conclusion and Future Work
As the derivation of these cuspless geodesics is much less trivial than it seems (many conflicting results have appeared in the imaging literature on this topic), we derived them via 3 different mathematical approaches producing the same results from different perspectives. There are two ways to reasonably parameterize such curves, via spatial arclength and subRiemannian arclength and in this article we explicitly relate these parameterizations. The phase portrait in momentum space induced by subRiemannian arclength parametrization corresponds to (a strip within) the phase portrait of the mathematical pendulum, whereas the phase portrait in momentum space induced by spatial arclength parametrization is a hyperbolic phase portrait associated to a linear ODE for normalized curvature \(z=\kappa /\sqrt{\kappa^{2}+\xi^{2}}\). Using the latter approach we have analyzed and computed the existence set \(\mathcal{R}\) for P _{curve} (where every stationary curve is globally minimizing!). We have also solved the boundary value problem, where the numerics is reduced to finding the unique root of a continuous explicit realvalued function on a small subset of [−1,1].
As such cuspless subRiemannian geodesics provide a suitable alternative to (involved and not necessarily optimal) elastica curves in computer vision. Moreover, they seem to provide a very adequate model for association fields and they are the solutions to Petitot’s circle bundle model. They also relate to previous models for association fields based on horizontal exponential curves (i.e. “cocircularity”) via the Cartan connection: Along horizontal exponential curves tangent vectors are parallel transported, whereas along subRiemannian geodesics momentum is parallel transported.
Our solutions, analysis and geometric control for the subRiemannain geodesics presented in this article form the venture point for datadependent active contour models in SE(2) (in combination with contourenhancement [1, 14, 22, 26, 29, 35, 36] and contour completion PDE’s [4, 8, 30, 48]) we are currently developing and applying in various applied imaging problems. Applications include extraction of the vascular tree in 2Dretinal imaging [10] and fibertracking in diffusion weighted magnetic resonance imaging [23, 62] (where we use subRiemannian geodesics in SE(3) solving the 3Dversion of P _{curve}). In these applications one replaces the constant measure on SE(2) in P _{curve} by a datadependent measure \(\tilde{C}:\mathrm{SE}(2) \to[1,\infty)\) in P _{curve}, producing external force terms in the EulerLagrange equations that pull the geodesics towards the data.
Finally, future work will include comparison of numerical algorithms for P _{MEC} and P _{curve}.
Footnotes
 1.
More precisely, the models coincide for cuspless subRiemannian geodesics that can be properly parameterized by their xcoordinate.
 2.
This fact has more or less been overlooked in the previous literature on this topic.
 3.
Although not considered here the third approach also includes local optimality via Jacobi operators appearing in 2nd order variations [20, Chap. 4.1, Prop. 4.4].
 4.
The choice of T>0 does not change the set of minimizers, but only their parametrization. For this reason, it can be useful to choose a T such that the minimizer is parametrized by subRiemannian arclength.
 5.
In this case the Lagrangian and Hamiltonian relate to each other by the Fenchel transform on the Lie algebra of horizontal leftinvariant vector fields akin to the 3Dcase [30].
 6.
Usually the minimization in Eq. (16) is made in the space of Lipschitz functions, to guarantee the existence of minimizers via PMP. However, a posteriori one verifies that these minimizers are indeed C ^{∞}.
 7.
For endcondition \((x_{fin},y_{fin},\theta_{fin}) \notin\mathcal{R}\) problem P _{MEC} has a minimizer with internal cusp (and thereby violating the natural settings of P _{curve}). Such a minimizer of P _{MEC} can be approximated by smooth curves satisfying the constraints of problem P _{curve}. In these cases P _{curve}does not allow local or global minimizers, nor does it allow a stationary curve [16].
 8.
This becomes even more apparent when considering the ddimensional extension of P _{curve}, see [31].
 9.
For the sake of simplicity we do not index \(\widetilde{\mathrm{Exp}}\) the exponential map with the initial condition g _{ in }, as throughout this article we set g _{ in }=e=(0,0,0).
 10.
As \(\mathcal{D}\) and \(\mathcal{R}\) are not open sets within the standard topologies on the embedding spaces \(T_{e}(\mathrm{SE}(2)) \times \mathbb{R}^{+}\) and \(\mathbb{R}^{2} \times S^{1}\). These subspace topologies do not coincide with the induced topology imposed by the embedding via the identity map, as such identity map is not continuous. However, with respect to the subspace topologies the set \(\mathcal{D}\), respectively \(\mathcal{R}\) are open sets and the homeomorphism \(\widetilde{\mathrm{Exp}}: \mathcal{D} \to\mathcal{R}\) is welldefined.
 11.
The dual basis in (SE(2))_{0} is equal to \(({\rm d}\theta, {\rm d}x, \theta\, {\rm d}x+ {\rm d}y)\) and thereby the subRiemannian metric on (SE(2))_{0} does not include the y′(x)^{2} term.
 12.
The corresponding minimization problem (and induced subRiemannian distance) is left invariant in (SE(2))_{0} and not leftinvariant in SE(2).
 13.
 14.
Within the association field model P _{curve} scaling of the endconditions amounts to scaling of ξ.
Notes
Acknowledgements
The authors wish to thank Tom Dela Haije and Arpan Ghosh, Eindhoven University of Technology, for fruitful discussion, fast numerical integration of the initial value problem and their assistance in creating Figs. 7, 11 and 14. The authors gratefully acknowledge European Research Council, ERC StG 2009 “GeCoMethods” contract number 239748, ERC StG 2013 “Lie Analysis” contract number 335555, the ANR “GCM” program “BlancCSD” project number NT09504490, the DIGITEO project “CONGEO” and the Russian Foundation for Basic Research (project no. 120100913a), and the Ministry of Education and Science of Russia within the federal program “Scientific and ScientificPedagogical Personnel of Innovative Russia” (contract no. 8209), for financial support.
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