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Edited by: James Elder, York University, Canada

Reviewed by: Carlos Vazquez, École de technologie supérieure (ÉTS), Canada

Johannes Burge, University of Pennsylvania, United States

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Classical good continuation for image curves is based on 2

Binocular vision is the ability of the visual system to provide information about the three-dimensional environment starting from two-dimensional retinal images. Disparities are among the main cues for depth perception and stereo vision but, in order to extract them, the brain needs to determine which features coming from the right eye correspond to those from the left eye, and which do not. This generates a coupling problem, which is usually referred to as the

Good continuation in the plane (retinotopic coordinates) is one of the foundational principles of Gestalt perceptual organization. It enjoys an extensive history (Wagemans et al.,

...a perspective drawing, even when viewed monocularly, does not give the same vivid impression of depth as the same drawing if viewed through a stereoscope with binocular parallax... for in the stereoscope the tri-dimensional force of the parallax co-operates with the other tri-dimensional forces of organization; instead of conflict between forces, stereoscopic vision introduces mutual reinforcement.

Our specific goal in this paper is to develop a neurogeometrical model of stereo vision, based on the functionality of binocular cells. The main application will be a good continuation model in three dimensions that is analogous to the models of contour organization in two dimensions. We will develop

Although only one dimension higher than contours in the plane, contours extending in depth raise subtle new issues; this is why a geometric model can be instructive. First among the issues is the choice of coordinates which, of course, requires a mathematical framework for specifying them. In the plane, position and orientation are natural; smoothness is captured by curvature or the relationship between nearby orientations along a contour. For stereo, there is monocular structure in the left eye and in the right. Spatial disparity is a standard variable relating them, and it is well-known that primate visual systems represent this variable directly (Poggio,

Hubel and Wiesel reported disparity-tuned neurons in early, classic work (Hubel and Wiesel,

The classical model for expressing the left/right-eye receptive field combination is the

Many other mathematical models for stereo vision based on neural models have been developed. Some claim (e.g., Marr and Poggio,

We propose a sub-Riemannian model for the cortical-inspired geometry underlying stereo vision based on the encoding of positional disparities and orientation differences in the information coming from the two eyes. We build on neuromathematical models, starting from the work of Koenderink and van Doorn (

To motivate our mathematical approach, it is instructive to build on an abstraction of visual cortex. We start with monocular information, segregated into ocular dominance bands (LeVay et al.,

Cartoon of visual cortex, V1, superficial layers.

Since much of the paper is technical, we here specify, informally, the main ingredients of the model and the results. We first list several of the key points, then illustrate them directly.

Stereo geometry enjoys a mathematical structure that is a formal extension of plane curve geometry. In the plane, points belonging to a curve are described by an orientation at a position, and these are naturally represented as elements (orientation, position) of columns. In our model, these become abstract fibers. The collection of fibers across position is a fiber bundle. Elements of the (monocular) fiber can be thought of as neurons.

Our geometrical model is based on tangents and curvatures. Tangents naturally relate to orientation selectivity, and are commonly identified with “edge” elements in the world. We shall occasionally invoke this relationship, for intuition and convenience, but some caution is required. While edge elements comprising, e.g., a smooth bounding contour are tangents, the converse is not necessarily true (e.g., elongated attached highlights or hair textures). Instead, our model should be viewed as specifying the constraints relevant to understanding neural circuitry; see Section 1.3.

To elaborate the previous point: the tangents in our model need not be edges in the world; they are neural responses. The constraints in our model can be used to determine whether these responses should be considered as “edges.” This is why the model is built from the geometry of idealized space curves: to support such inferences.

For stereo, we shall need fibers that are a “product” of the left and right-eye monocular columns. This is the reason why we choose position, positional disparity and orientations from the left and right eyes respectively, as the natural variables that describe the stereo fiber over each position. We stress that these fibers are not necessarily explicit in the cortical architecture.

Curvature provides a kind of “glue” to enable transitions from points on fibers to nearby points on nearby fibers. These transitions specify “integral curves” through the stereo fiber bundle.

The integral curve viewpoint provides a direction of information flow (information diffuses through the bundle) thereby suggesting underlying circuits.

The integral curves formalize

Our formal theory addresses several conjectures in the literature. The first is the

Our formal theory provides a new framework for specifying the correspondence problem, by illustrating how good continuation in the 3-D world generalizes good continuation in the 2-D plane. This is the point where consistent binocular-binocular interactions are most important.

Our formal theory has direct implications for understanding torsional eye movements. It suggests, in particular, that the rotational component is not simply a consequence of development, but that it helps to undo inappropriate orientation disparity changes induced by eye movements. This role for Listing's Law will be treated in a companion paper (in preparation); see also the excellent paper (Schreiber et al.,

We now illustrate these ideas (^{3}, with a unit-length tangent at the point of fixation. Since the tangent is the derivative of a curve, the binocular cells naturally encode the unitary tangent direction ^{1}

^{3} in the left-eye innervated and right-eye innervated monocular orientation columns (Each short line denotes a neuron by its orientation preference.). Joint activity across the eyes, which denotes the space tangent, is illustrated by the binocular neuron (circle). Note the two similar but distinct monocular orientations. Connections from the actively stimulated monocular neurons to the binocular neuron are shown as dashed lines. ^{3} in the left and right retinal columns. Each space tangent projects to a different pair of monocular columns because of the spatial disparity. Consistency in the responses of these four columns corresponds to consistency between the space tangents attached to nearby positions along γ. This consistency is realized through the binocular neural connection (solid line).

Since space curves live in 3D, two angles are required to specify its space tangent at a point. In other words, monocular tangent angles span a circle in the plane; space tangent angles span a 2-sphere in 3D. In terms of the projections into the left-eye and the right-eye, the space tangent can be described by the parameters ^{2} (^{3} ⋊ 𝕊^{2}. Moving from one position in space to another, and changing the tangent orientation to the one at the new position, amounts to what is called a

_{L} and _{R}, the disparity and the orientations θ_{L} and θ_{R}. _{1} and _{2} in ℝ, but that will take some work to develop.).

We live in a 3

Moving now out to the world, we must be able to move between all points. Repeating the above metaphor more technically, we equip ℝ^{3} ⋊ 𝕊^{2} with a group action of the three-dimensional Euclidean group of rigid motions

There are many different ways to approach mathematical modeling in vision. One could, for example, ask what is the best an ideal observer could do for the stereo problem working directly on image data (Burge and Geisler,

Our working hypothesis is that there is a deep functional relationship between structure in the brain and structure in the world, and that geometry is the right language with which to capture this relationship, especially as regards connectivity between neurons and their functionality. The

At first blush this might seem completely unrelated to the statistics of natural images, and how these could be informative of neural connections, but we believe that there is a fundamental relationship. Consider, to start, the distribution of oriented edge elements in a small patch. Pairwise edge statistics are well-studied (August and Zucker, ^{2}

The geometric analysis that we shall use is continuous mathematics, and is essentially differential (Tu,

It has been observed that edge statistics for curves in the world depart from co-circularity. To quote (Geisler and Perry, ^{3}

The paper is organized as follows: in Section 2, we describe the geometrical and neuro-mathematical background underlying the problem of stereo vision. In particular, we review the standard stereo triangulation technique to relate the coordinate system of one retina with the other, and put them together in order to reconstruct the three-dimensional space. Then, we briefly review the classical neurogeometry of monocular simple cells selective for orientation and the underlying connections. The generalization of approximate co-circularity for stereo is also introduced. In Section 3, starting from binocular receptive profiles, we introduce the neuro-mathematical model for binocular cells. First we present the cortical fiber bundle of binocular cells. It follows the differential interpretation of the binocular profiles in terms of the neurogeometry of the simple cells, and we show how this is well in accordance with the results of the stereo triangulation. Then, we give a mathematical definition of the manifold ℝ^{3} ⋊ 𝕊^{2} with the sub-Riemannian structure. Finally, we study the integral curves and the suitable change of variables that allow us to switch our analysis from cortical to external space. In Section 4 we proceed to the validation of our geometry with respect to psychophysical experiments. We combine information about the psychophysics of 3^{4}

In this subsection, we briefly recall the geometrical configuration underlying 3

We consider the global reference system (^{3}, with _{1}, _{2}, _{3}). We introduce the optical centers _{L} = (−_{R} = (_{L}, _{L}, _{L}), (_{R}, _{R}, _{R}), the reference systems of the retinal planes _{L}, _{R}, _{3} =

Reconstruction of the 3_{L} the retinal plane _{R} in

Remark 2.1. If we know the coordinate of a point ^{3}, then it is easy to project it in the two planes via perspective projection, having _{L} and Π_{R}, respectively, for the left and right retinal planes:

^{3} such that _{L}, _{L} and

Analogously, considering _{R} and _{R}, we get:

□

In a standard way, the

up to a scalar factor. Moreover, it is also possible to define the coordinate

where the set of coordinates (

Corresponding points in the retinal planes allow to project back into ℝ^{3}. An analogous reasoning can be done for the tangent structure: if we have tangent vectors of corresponding curves in the retinal planes, it is possible to project back and recover an estimate of the 3

Remark 2.2. Let γ_{L} and γ_{R} be corresponding left and right retinal curves; i.e., perspective projections of a curve γ ∈ ℝ^{3} through optical centers _{L} and _{R} with focal length

^{3}, we project it in the two retinal planes obtaining γ_{L} = Π_{L}(γ) and γ_{R} = Π_{R}(γ) from Equation (1). The retinal tangent vectors are obtained through the Jacobian matrix^{5}

Extending the tangent vectors and the points into ℝ^{3}, we get

Then _{tR} × _{tL} is a vector parallel to the tangent vector

□

Although this section has been based on the geometry of space curves and their projections, we observe that related geometric approaches have been developed for planar patches and surfaces; see, e.g., Li and Zucker,

We now provide background on the geometric modeling of the monocular system, and good continuation in the plane. Our goal is to illustrate the role of sub-Riemannian geometry in the monocular system, which will serve as the basis for generalization to the stereo system.

We model the activation map of a cortical neuron's receptive field (RF) by its receptive profile (RP) φ. A classical example is the receptive profiles of simple cells in V1, centered at position (_{{x, y, θ}}. RPs are mathematical models of receptive fields; they are operators which act on a visual stimulus.

Formally, it is possible to abstract the primary visual cortex as ℝ^{2} × 𝕊^{1}, or position-orientation space, thereby naturally encoding the Hubel/Wiesel hypercolumnar structure (Hubel and Wiesel,

^{2} × 𝕊^{1} generated by the sub-Riemannian model (Citti and Sarti,

Following the model of Citti and Sarti (^{T} and rotation around angle θ from a unique “mother” profile φ_{0}(ξ, η):

This RP is a Gabor function with even real part and odd imaginary part (

where _{(x, y, θ)} denotes the action of the group of rotations and translations ^{2}. This group operation associates to every point (ξ, η) a new point

and this represents the action of the group

The retinal plane ^{2} plane, whose coordinates are (

where the function

For (

We will then say that the point (

We shall now recall a model of the long range connectivity which allows propagation of the visual signal from one cell in a column to another cell in a nearby column. This is formalized as a set of directions for moving in the cortical space

To begin, in the right hand side of the Equation (11) the integral of the signal with the real and imaginary part of the Gabor filter is expressed. The two families of cells have different shapes, hence they detect (or play a role in detecting) different features. Since the odd-symmetry cells suggest boundary detection, we concentrate on them, but this is a mathematical simplification. The output of a simple cell can then be locally approximated as _{3,p}(_{σ})(_{σ} is a smoothed version of

is the directional derivative in the direction ^{6}

Now, think of vector fields as defining a coordinate system at each point in cortical space. Then, in addition to above, the vector fields orthogonal to _{3,p} are:

and they define a 2-dimensional admissible tangent bundle^{7}^{2} × 𝕊^{1}. One can define a scalar product on this space by imposing the orthonormality of _{1,p} and _{2,p}: this determines a sub-Riemannian structure on ℝ^{2} × 𝕊^{1}.

The visual signal propagates, in an anisotropic way, along cortical connectivity and connects more strongly cells with comparable orientations. This propagation has been expressed by the geometry just developed and 2-dimensional contour integration. This is the neural explanation of the Gestalt law of good continuation (Koffka,

Based on these findings, Citti and Sarti (_{1} and _{2}, namely curves γ:[0, ^{2} × 𝕊^{1} described by the following differential equation:

obtained by varying the parameter

A related model has been proposed by Duits et al. (

The concept of co-circularity in ℝ^{2} has been developed by observing that a bidimensional curve γ can be locally approximated at 0 via the osculating circle.^{8}

While in the two-dimensional case the approximation of the curve using the Frenet 2_{3}-helices to improve stability in terms of camera calibration. In this way the orientation disparity is encoded in the behavior of the helix in the 3

In Li and Zucker (

Proposition 2.1. Given two perspective views of a 3

Hence, using the knowledge of the Frenet basis together with the fundamental addition of the curvature variable, Zucker et al. applied the concept of

Remark 2.3. The model that we propose in this paper is related to, but differs from, what has just been stated. In particular, to remain directly compatible with the previous neuro-geometric model, we will work only with the monocular variables of position and orientation. Rather than using curvature directly, we shall assume that these variables are encoded within the connections; mathematically they appear as parameters. A theoretical result of our model is that the heuristic assumption regarding the _{3}-helix can now be established rigorously.

Let us also mention the paper (Abbasi-Sureshjani et al.,

Here, we do not want to directly impose a co-circularity property: our scope is to model the behavior of binocular cells, and deduce properties of propagation, which will ultimately induce a geometry of 3D good continuation laws.

Binocular neurons receive inputs from both the left and right eyes. To facilitate calculations, we assume these inputs are first combined in simple cells in the primary visual cortex, a widely studied approach (Anzai et al.,

This binocular model allows us to define disparity and frontoparallel coordinates as

perfectly in accordance with the introduction of cyclopean coordinates in (4). In this way (_{1}, _{2}, _{3}), via the change of variables (5).

The hypercolumnar structure of monocular simple cells (orientation selective) has been described as a jet fiber bundle in the works of Petitot and Tondut (^{2} × 𝕊^{1}, with fiber 𝕊^{1}; see, e.g., Ben-Shahar and Zucker,

In our setting, the binocular structure is based on monocular ones; recall the example illustrations from the Introduction. In particular, for each cell on the left eye there is an entire fiber of cells on the right, and vice versa, for each cell on the right there is an entire fiber of cells on the left. This implies that the binocular space is equipped with a symmetry that involves the left and right structures, allowing us to use the cyclopean coordinates (

Hence, we define the cyclopean retina ^{2}, endowed with coordinates (_{L}, θ_{R}) = (

A schematic representation can be found in

To simplify calculations, as stated in the Introduction, we follow the classical binocular energy model (Anzai et al., _{L} + _{R} of the inputs from the left and right eye is positive, the firing rate of the binocular neuron is proportional to the square of the sum, and it vanishes, if the sum of the inputs is negative:

with _{B} the binocular output.

If _{L} + _{R} > 0, then the output of the binocular simple cell can be explicitly written as _{L}_{R} can be interpreted as the binocular interaction term.

The activity of a cell is then measured from the output and will be strongest at points that have a higher probability of matching each other. The maximum value over

It is worth noting that neurophysiological computations of binocular profiles displayed in

Remark 3.1 (Orientation matters). In 2001, the authors of Bridge et al. (

The binocular energy model is a type of minimal model. It serves as a starting point, allowing the combination of monocular inputs. But is not sufficient to solve the stereo-matching problem.

Remark 3.2 (Connections). It is argued in Samonds et al. (

It is possible to write the interaction term _{L}_{R} coming from (17), in terms of the left and right receptive profiles:

If we fix _{L,R} = φ_{θR, xR, y}φ_{θL, xL, y} as the product of monocular left and right profiles. This is in accordance with the measured profiles of

Proposition 3.1. The binocular interaction term can be associated with the cross product of the left and right directions defined through (13), namely

To better understand the geometrical idea behind Proposition 3.1, we recall that the retinal coordinates can be expressed in terms of cyclopean coordinates (4) as _{R} = _{L} =

We define

Remark 3.3. The vector ω_{bin} of Equation (21) can be interpreted as the intersection of the orthogonal spaces defined with respect to

then

The result of the intersection of these monocular structures identifies a direction, as shown in

_{bin} through the intersection of left and right planes generated by _{bin}. _{L} and γ_{R}. Their three dimensional counterpart ^{3} by the cross product

We earlier showed that the result of the action of a monocular odd simple cell is to select directions for the propagation of information. We now combine these, for the two eyes, to show that in the three-dimensional case the binocular neural mechanisms also lead to a direction. We will see in the next sections that this direction is the direction of the tangent vector to the 3

We consider the direction characterizing the binocular structure ω_{bin} defined in (21) and we show that it can be associated with the 3

Precisely, we consider the normalized tangent vector _{L} and _{R} on retinal planes

to the points (_{R}, _{L}, ^{3}, then we associate to these points the correspondents in ℝ^{3}, namely

and the tangent direction is recovered by

If we define

and the corresponding 2 form

up to a scalar factor. See Appendix C (

In this way, the disparity binocular cells couple in a natural way positions, identified with points in ℝ^{3}, and orientations in 𝕊^{2}, identified with three-dimensional unitary tangent vectors. As already observed in Remark 3.2, the geometry of the stereo vision is not solved only with these punctual and directional arguments, but there is the need to take into accounts suitable type of connections. In Alibhai and Zucker (^{3} ⋊ 𝕊^{2}, in line with the theoretical toolbox proposed in Miolane and Pennec (

We now derive the objects in ^{3}, and orientations in 𝕊^{2}. An element ξ of the space ℝ^{3} ⋊ 𝕊^{2} it is defined by a point ^{3} and an unitary vector ^{2}. Since the topological dimension of this geometric object is 2, we introduce the classical spherical coordinates (θ, φ) such that

with θ ∈ [0, 2π] and φ ∈ (0, π). The ambiguity that arises using local coordinate chart is overcome by the introduction of a second chart, covering the singular points.

Translations and rotations are expressed using the group law of the three-dimensional special Euclidean group

with (^{3} ⋊ 𝕊^{2}, (^{3}.

The emergence of a privileged direction in ℝ^{3} (associated with the tangent vector to the stimulus) is the reason why we endow ℝ^{3} ⋊ 𝕊^{2} with a sub-Riemannian structure that favors the direction in 3_{bin}.

Formally, we consider admissible movements in ℝ^{3} ⋊ 𝕊^{2} described by vector fields:

with ξ ∈ ℝ^{3} ⋊ 𝕊^{2} for φ ≠ 0, φ ≠ π. The admissible tangent space^{9}

encodes the coupling between position and orientations, as remarked by Duits and Franken (^{3}, while _{θ} and _{φ} allow changing this direction, involving just orientation variables of 𝕊^{2}. The vector fields ^{3} ⋊ 𝕊^{2} in a point, allowing to connect every point of the manifold using privileged directions (^{2} the distance inherited from the immersion in ℝ^{3} with the Euclidean metric.

We have already expressed the change of variable in the variables (_{1}, _{2}, _{3}) in Equation (5). However, the cortical coordinates also contain the angular variables θ_{R} and θ_{L} which involve the introduction of the spherical coordinates θ, φ.

To identify a change of variable among these variables, we first introduce the function

where the retinal right angle

Analogously, it is possible to define the change of variable

where the angles

The connectivity of the space is described by admissible curves of the vector fields spanning ^{3} ⋊ 𝕊^{2} is said to be ^{10}

where

with _{1} and _{2} varying in ℝ.

These curves can be thought of in terms of trajectories in ℝ^{3} describing a movement in the

It is worth noting that in the case described by coefficients _{1} and _{2} equal to zero, the 3^{3}; by varying the coefficients _{1} and _{2} in ℝ, we allow the integral curves to follow curved trajectories, twisting and bending in all space directions.

Formally, the amount of “twisting and bending” in space is measured by introducing the notions of curvature and torsion. We then investigate how these measurements are encoded in the parameters of the family of integral curves, and what constraints have to be imposed to obtain different typologies of curves.

Remark 3.4. The 3

Using the explicit expression of the vector fields _{θ} and _{φ} in Equation (36), we get

from which it follows that:

Proposition 3.2. By varying the parameters _{1} and _{2} in (39) where we explicitly find solutions of (36), we have:

If _{3} = cost.

If φ = φ_{0}, with φ_{0} ≠ π/2, then _{1}cotanφ_{0}, and so the family of curves (36) are _{3}-helices.

If θ = θ_{0} then

Remark 3.5. If we know the value of the curvature _{2}, in the definition of the integral curves (36), then we are in the setting of Proposition 2.1. In fact, the coefficient _{1} is obtained by imposing

Examples of particular cases of the integral curves (36) according to Proposition 3.2 and Remark 3.5 are visualized in

Examples of integral curves obtained varying parameters _{1} and _{2}. _{3}-helices for φ = π/3.

Our sub-Riemannian model enjoys some consistency with the biological and psychophysical literature. We here describe some initial connections.

The foundation for building our sub-Riemannian model of stereo was a model of curve continuation, motivated by the orientation columns at each position. The connections between cells in nearby columns were, in turn, a geometric model of long-range horizontal connections in visual cortex (Bosking et al.,

The Grinwald group first established the presence of long-range connections between binocular cells (Malach et al.,

More precisely, Malach et al. (

In this section, we show that the connections described by the integral curves in our model can be related to the geometric relationships from psychophysical experiments on perceptual organization of oriented elements in ℝ^{3}; in other words, that our connections serve as a generalization of the concept of an association field in 3

The perception of continuity between two elements of position-orientation in ℝ^{3} has been studied experimentally. To start, Kellman, Garrigan, and Shipley (Kellman et al.,

Particularly, in a system of 3^{T}. For an initial edge _{0}, with application point on the origin of the coordinate system (0, 0, 0)^{T} and orientation lying on the _{1}-axis, described by θ = 0, φ = π/2, the range of possible orientations (θ, φ)^{11}_{0} is given by:

The bound on these equations identified with the quantity _{0} and an edge positioned at the arbitrary oriented point _{(r1, r2, r3)} so that its linear extension intersects _{0}; see Kellman et al. (

Numerical simulations allow us to visually represent an example of the 3_{0} with endpoints in _{1}- axis, we represent for an arbitrary point

_{0}.

Remark 4.1. By projecting on the retinal planes of the 3

Psychophysical studies, see Hess and Field (_{0} must meet the relations of the bi-dimensional association fields of Field et al. (

To model the associations underlying the 3

Importantly, these curves locally connect the association fan generated by the geometry of 3_{0} with 3

_{0}. Unrelatable 3_{0}. Horizontal integral curves with dotted lines do not connect 3_{1}-_{2} plane. _{1}-_{3} plane. _{0}. These curves (black lines) are not planar curves but helices. However, their projection (white lines) on the coplanar plane with initial edge satisfies the bidimensional constraints.

In analogy with the experiment of Field, Hayes, and Hess in Field et al. (_{0}, while on the right are 3_{0} and the ones on its right, while dotted lines connect the starting point _{0} with elements not correlated with it, as represented on the left part of the image.

Restricting the curves on the neighborhood of co-planar planes with an arbitrary edge _{1}-_{2} plane (fronto-parallel) and the _{1}-_{3} plane we have arcs of circle, as proved with Proposition 3.2. Furthermore, for an arbitrary plane in ℝ^{3} containing an edge

One final connection with the psychophysical literature concerns how depth discrimination thresholds increase exponentially with distance (Burge,

Although the goal of this paper is not to solve the stereo correspondence problem, we can show how our geometry is helpful in understanding how to match left and right points and features. These ideas are developed more fully in Bolelli (

Inspired by Hess and Field (^{3} ⋊ 𝕊^{2}. This set contains all the possible corresponding points, obtained by coupling left and right points which share the same

^{3} × 𝕊^{2}. _{ij} represents the value of curvature/ torsion for every couple of points ξ_{i}, ξ_{j}. The first eight points correspond to points of the curve γ while the others are random noise.

We compute for every lifted point the binocular output _{B} of Equation (17). This output can be seen as a probability measure that gives information on the correspondence of the pair of left and right points. We then simply evaluate which are the points with the highest probability of being in correspondence, applying a process of suppression of the non-maximal pairs over the fiber of disparity. In this way, noise points are removed (_{1} and _{2} of integral curves (36) that connect all the remaining pairs of points, we can obtain for every pair the value of curvature and torsion using (39).

_{i}, ξ_{j} in the element _{ij}. In particular, we observe that random points are characterized by very high curvature and deviating torsion. So, by discarding these high values, we select only the three-dimensional points of the curve γ, which are well-connected by the integral curves, as shown in

In this artificial example we assumed that local edge elements have already been detected. Our goal was simply proof-of-concept. To apply this approach to realistic images, of course, stages of edge detection would have to be adopted, for which there is a huge literature well outside the scope of our theoretical study.

Understanding good continuation in depth, like good continuation for planar contours, can benefit from basic physiological constraints; from psychophysical performance measures, and from mathematical modeling. In particular, good continuation in the plane is supported by orientation selectivity and cortical architecture (orientation columns), by association field grouping performance, and by geometric modeling. We showed that the same should be true for good continuation in depth. However, while the psychophysical data may be comparable, the physiological data are weaker and the geometry of continuation is not as well-understood. In this paper, we introduced the neuro-geometry of stereo vision to fill this gap. It is strongly motivated by an analogical extension to 3

Technically, we proposed a sub-Riemannian model on the space of position and orientation ℝ^{3} ⋊ 𝕊^{2} for the description of the perceptual space of the neural cells involved. This geometrical structure favors the tangent direction of a 3

Although the goal of this paper is not to solve the stereo correspondence problem, we have seen how the geometry we propose is a good starting point to understand how to match left and right points and features. We used binocular receptive fields to prioritize orientation preferences and orientation differences under the assumption that neuronal circuitry has developed to facilitate the interpolation of contours in 3D space. On the other side, the neurogeometrical method has been coupled with a probabilistic methods for example in Sanguinetti et al. (

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

The author(s) declare financial support was received for the research, authorship, and/or publication of this article. MB, GC, and AS were supported by EU Project, GHAIA, Geometric and Harmonic Analysis with Interdisciplinary Applications, H2020-MSCA-RISE-2017. SZ was supported in part by US NIH EY031059 and by US NSF CRCNS 1822598.

The current article is part of the first named author's Ph.D. thesis (Bolelli,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

The Supplementary Material for this article can be found online at:

^{1}We are here being loose with language. By a tangent orientation in the left eye, we mean the orientation of a left-eye innervated column in V1.

^{2}Of course we need to take into account the difficulty of measurements of coupled position-orientation variables for small difference of angle and position. This is due to the well-known intrinsic uncertainty of measurement in the non-commutative group of position and orientation (Barbieri et al.,

^{3}The crucial point is that the curves demonstrate locally quadratic (not linear) behavior.

^{4}Portions of this material were presented at Bolelli et al. (

^{5}The Jacobian matrix (_{JΠ)p} evaluated at point ^{3}, then the matrix product ^{2}. In other words, the Jacobian matrix is the differential of Π at every point where Π is differentiable; common notation includes _{Π} or

^{6}The purpose of introducing this notation is also to motivate an implication of the mathematical model in Citti and Sarti (

^{7}as defined in Appendix A3 (

^{8}Locally, a curve can be approximated by its osculating circle and, at a slightly larger scale, by the integral (parabolic) curve through the first two Taylor terms. The first approximation is co-circularity; the second is a parabolic curve. The second is an accurate model over large distances; see discussion in Section 1.3. However, since in this paper we are working over small distances and with cortical sampling (

^{9}see Appendix A (

^{10}sometimes the term

^{11}The angle φ here has been modified to be compatible with our set of coordinates. The relationship between the angle