# A Riemannian Geometry Theory of Three-Dimensional Binocular Visual Perception

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

[Listing] reduced the eye model to a single refracting surface, the vertex of which corresponds to the principal plane and the nodal point of which lies at the centre of curvature. The justification for this model is that the two principal points that lie midway in the anterior chamber are separated only by a fraction of a millimetre and hardly shift during accommodation. Similarly, the two nodal points lie equally close together and remain fixed near the posterior surface of the lens. In the reduced model the two principal points and the two nodal points are combined into a single principal point and a single nodal point. Retinal image sizes may be determined very easily because the nodal point is at the centre of curvature of this single refractory surface. A ray from the tip of an object directed toward the nodal point will go straight to the retina without bending, therefore object and image subtend the same angle. The retinal image size is found by multiplying the distance from the nodal point to the retina (17.2 mm) by the angle in radians subtended by the object [42] (see Figure 18).

## 2. Preliminaries

#### 2.1. Retinal Coordinates

#### 2.2. Hyperfields

#### 2.3. Retinotopic Connections between Hyperfields and Hypercolumns

#### 2.4. Hypercolumns

#### 2.5. Visual Features Extracted by Cortical Columns

#### 2.6. Gaze and Focus Control

#### 2.7. Singular Value Decomposition as a Model for Visual Feature Extraction

**V**

_{L}over all the left ocular dominance minicolumns in the hypercolumns of V1. Similarly, the image-point vectors ${\Sigma}_{R}$ from all the corresponding right hyperfields form a 30-dimensional vector field

**V**

_{R}over all the right ocular dominance minicolumns in the hypercolumns of V1. Due to the retinotopic projections between retinal hyperfields and cortical hypercolumns, the vector fields

**V**

_{L}and

**V**

_{R}can also be thought of as vector fields over the left and right retinal hyperfields, respectively.

**V**

_{L}and

**V**

_{R}over hypercolumns and over retinal hyperfields in this way facilitates a mathematical framework appropriate for development of a Riemannian geometry theory of binocular vision. But this requires a mechanism for quantifying the depth of objects perceived. These depth measures then provide a coordinate system for the Riemannian manifold on which the above vector fields are defined.

#### 2.8. Depth Perception

#### 2.9. Cyclopean Gaze Coordinates

_{E}and the angle $\alpha $ are anatomical parameters that change with growth of the head and eye. Since these parameters influence the geometrical optics of images projected on to the retinas it does not seem unreasonable to suggest that the nervous system is able to model them adaptively through experience, for example, by modelling the relationship between the depth of an object and the size of its image on the retina, and by sensing the change in place of the head required to match the image on one retina with the memorized image on the other. The place and orientation of the head in the environment are encoded by neural activity in the hippocampus and parahippocampus so, referring to Figure 1, the angle ${\theta}_{H}$ of the head relative to the translated external coordinates (${\mathrm{X}}^{\prime}$,${\mathrm{Y}}^{\prime}$) is known, and the angles of rotation ${\theta}_{L}$ and ${\theta}_{R}$ of the left and right eye within the head are sensed proprioceptively. Using the geometry of Figure 1, it can be shown that these known variables ${\theta}_{H}$, ${\theta}_{L}$, ${\theta}_{R}$, d, r

_{E}and $\alpha $ completely determine the Euclidean distance and angle from each eye to the gaze point as well as the length and direction $\left(r,\theta \right)$ of the cyclopean gaze vector OQ. This can be demonstrated by basic trigonometry (sine rule and cosine rule) of the three triangles N

_{L}LC

_{L}, N

_{R}LC

_{R}, and LQR. Importantly, this is not to say that the nervous system ‘does’ trigonometry in the same way we do. It is simply to establish that the information available to it is sufficient to determine uniquely the length and direction of the cyclopean gaze vector.

#### 2.10. Cyclopean Coordinates of Peripheral Image Points

## 3. The Three-Dimensional Perceived Visual Space

#### 3.1. Gaze-Based Visuospatial Memory

**V**

_{L}and

**V**

_{R}of image-point vectors ${\Sigma}_{L}\left({r}_{{a}_{Li}},{\theta}_{{a}_{Li}},\text{}{\phi}_{{a}_{Li}}\right)$ and ${\Sigma}_{R}\left({r}_{{a}_{Ri}},{\theta}_{{a}_{Ri}},\text{}{\phi}_{{a}_{Ri}}\right)$ over the hypercolumns described in Section 2.10 are replaced by new image-point vectors and by new vector fields associated with the next gaze point in the scanning sequence. To build a visuospatial memory of an environment through scanning we argue that the information encoded by the vector fields

**V**

_{L}and

**V**

_{R}during a current interval of fixed gaze must be stored before the gaze is shifted and the information lost. Such memory is accumulated over time and scanning of an environment from a fixed place does not have to occur in one continuous sequence. Images associated with different gaze points from a fixed place can be acquired (and if necessary overwritten) in a piecemeal fashion every time the person passes through that given place.

#### 3.2. A Riemannian Metric for the G-Memory

## 4. Quantifying the Geometry of the Perceived Visual Manifold

#### 4.1. The Relationship Between Perceived Depth and Euclidean Distance

#### 4.2. The Geodesic Spray Field

#### 4.3. Covariant Derivatives

#### 4.4. Christoffel Symbols

#### 4.5. The Riemann Curvature Tensor

## 5. Geodesics of the Perceived Visual Manifold

#### 5.1. Simulations

#### 5.2. Initial Planes II Passing through the Egocentre

#### 5.3. Initial Planes II Normal to the Radial line from the Egocentre to the Initial Point

#### 5.4. Initial Planes II Not Normal to the Radial Line from the Egocentre to the Initial Point and Not Passing Through the Egocentre

#### 5.5. Interpreting Geodesic Simulations

#### 5.6. Euclidean Coordinates versus Perceptual Coordinates

## 6. Binocular Perception of the Size and Shape of Objects

#### 6.1. Seeing the Size of an Object

#### 6.2. Seeing the Outline of an Object

#### 6.3 Seeing the Shape of an Object

## 7. A Geometric Representation of Visuospatial Memory

#### 7.1. The Geometric Structure of G-Memory for a Fixed Place

**V**

_{L}and

**V**

_{R}in $\Gamma E$ (consisting of all the left and right image-point vectors ${\Sigma}_{L}\left(r,\theta ,\text{}\phi \right)$ and ${\Sigma}_{R}\left(r,\theta ,\text{}\phi \right)$ over all the image-points $q=\left(r,\theta ,\text{}\phi \right)$ in $\left(G,g\right)$ accumulated within a single vector bundle $\pi :\mathrm{E}\to G$ through visual scanning) thus encode a visual image of the entire 3D environment as seen from the given fixed place. However, while we describe vector fields

**V**

_{L}and

**V**

_{R}as being defined over all image points $q=\left(r,\theta ,\text{}\phi \right)$ in $\left(G,g\right)$, it must be kept in mind from Section 6 that the image-point vectors ${\Sigma}_{L}\left(r,\theta ,\text{}\phi \right)$ and ${\Sigma}_{R}\left(r,\theta ,\text{}\phi \right)$ are only non-zero at those points $q$ that are located on the surfaces of objects. We also note again here that if the visual environment includes reflections then the orientation of the reflected images is reversed and the vector bundle is said to be twisted [114].

**V**

_{L}$\left(U\right)$ and

**V**

_{R}$\left(U\right)$ confined to open subsets $U$ in $\left(G,g\right)$ can be defined and all the fibres within these can be parallel processed as a unit. Indeed, it is this point processing (i.e., within fibre) nature of computations in Riemannian geometry that makes this geometry so well suited for describing parallel processing in the nervous system.

**V**

_{L}and

**V**

_{R}in $\Gamma E$ can be regarded as fused into a single binocular vector field

**V**over $\left(G,g\right)$. For simplicity of description, in subsequent sections we assume that a sufficient number of gaze points have been accumulated through visual scanning for the vector fields

**V**

_{L}and

**V**

_{R}in $\Gamma E$ to be fused into a single binocular vector field

**V**over $\left(G,g\right)$ in the vector bundle $\pi :\mathrm{E}\to G$.

#### 7.2. The Geometric Structure of Visuospatial Memory with Place Encoding

#### 7.3. Fibre Bundles and Vector-Bundle Morphisms

#### 7.4. Removing Occlusions

#### 7.5. A Geometric Description of Vector-Bundle Morphisms

## 8. Discussion

#### 8.1. Size Perception

#### 8.2. Shape Perception

#### 8.3. Warped Geometry

#### 8.4. IIlusions

#### 8.5. Measuring the Geometry of Perceived Visual Space

#### 8.6. 2D versus 3D Representations

#### 8.7. Visuospatial Memory

#### 8.8. Visuospatial Representation as a Philosophical Issue

## 9. Future Directions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Extraction of Non-Linear Orthogonal Visual Image Features Using Singular Value Decomposition (SVD)

#### A1. Extraction of Linear Orthogonal SVD Image Features

#### A2. Extraction of Non-Linear Orthogonal SVD Image Features

## Appendix B. Computing Curvatures

#### B1. Gaussian Curvatures and Sectional Curvatures

#### B2. Principal Curvatures, Principal Directions and Perceived Curvatures

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**Figure 1.**A schematic 2D diagram based on the reduced model of the human eye [42]. C

_{L}and C

_{R}are the centres of rotation of the left and right eye, respectively, and N

_{L}and N

_{R}are the nodal points. The optic axis for each eye connects the centre of rotation to the nodal point. The visual axis for each eye connects the fovea to the nodal point. The distance between C

_{L}and C

_{R}is known as d. Its midpoint O is known as the egocentre and marks the position of a hypothetical cyclopean eye. The distances C

_{L}N

_{L}and C

_{R}N

_{R}are the same for each eye and are known as r

_{E}. The angle $\alpha $ between the optic axis and the visual axis is also the same for each eye and is typically about 5 degrees in adults. ${\theta}_{H}$ gives the angle of the head relative to a translated external reference frame (${\mathrm{X}}^{\prime}$, ${\mathrm{Y}}^{\prime}$ ) and ${\theta}_{L}$ and ${\theta}_{R}$ give the angles of the left and right eyes relative to the head when gaze is fixed on a surface point Q in the environment. The diagram shows the cyclopean gaze vector OQ in relation to the above geometry.

**Figure 2.**A schematic 2D diagram illustrating the angle of the head relative to a translated external reference frame (${\mathrm{X}}^{\prime}$, ${\mathrm{Y}}^{\prime}$ ) and the angles of the left and right eyes relative to the head when gaze is fixed on a surface point Q in the environment. The left and right eye visual axes are straight lines connecting the fovea through the nodal point of the eye to the gaze point Q. The fan-shaped grids of straight lines passing through the nodal point of each eye connect corresponding left and right retinal hyperfields to points ${a}_{Li}$ and ${a}_{Ri}$, respectively, on the surface. The image point ${a}_{Li}$ projecting to a left retinal hyperfield is translated by a small amount relative to the image point ${a}_{Ri}$ projecting to the corresponding right retinal hyperfield. Thus the points ${a}_{Li}$ and ${a}_{Ri}$ induce a disparity between the images projected to the corresponding left and right retinal hyperfields. The diagram also includes the hypothetical surface known as an horopter. This contains the points which induce no disparity between the images projected to corresponding left and right hyperfields.

**Figure 3.**A graph showing the logarithmic relationship between the perceived distance $r$ and the actual Euclidean distance $\overline{r}$. See text for details.

**Figure 4.**A block diagram for the Matlab/Simulink simulator used to generate geodesic trajectories in the 3D Euclidean outside world given initial conditions $\alpha \left(0\right)=\left(r\left(0\right),\theta \left(0\right),\phi \left(0\right)\right)$ and $\dot{\alpha}\left(0\right)=(\dot{r}\left(0\right),\dot{\theta}\left(0\right),\dot{\phi}\left(0\right))$ set equal to (r$\left(0\right)$,theta$\left(0\right)$,phi$\left(0\right)$ ) and (dr$\left(0\right)$,dtheta$\left(0\right)$,dphi$\left(0\right)$ ) in the diagram. The MATLAB Function block was programmed to evaluate the expression for ${f}_{2}\left(\alpha \left({t}_{i}\right),\dot{\alpha}\left({t}_{i}\right)\right)$ in Equation (17). For each run the geodesic trajectory alpha = (r,theta,phi) was stored in the workspace, converted to Cartesian coordinates and plotted as shown in Figure 5, Figure 6 and Figure 7 below.

**Figure 5.**Cartesian plots in the Euclidean outside world of geodesics emanating from two different initial points in the horizontal $xy$-plane passing through the egocentre at $\left(0,0,0\right)$ with unit-length initial velocities at the initial point set in the $xy$-plane. These geodesics all remain confined to the $xy$-plane where they form radial, circular, or spiral lines from the initial point (see text for detail). The radial and circular geodesics have been slightly thickened in all four diagrams. Dots along geodesics mark 500 ms intervals of time. The same plots are obtained for any plane passing through the egocentre. (

**a**) A family of 36 geodesics emanating from initial point $\left(x=0\mathrm{m},y=1\mathrm{m},z=0\mathrm{m}\right)$ generated from 36 unit-length initial velocity vectors set in the $xy$-plane and equally spaced in all directions from the initial point. (

**b**) A family of 36 geodesics emanating from initial point $\left(x=0\mathrm{m},y=5\mathrm{m},z=0\mathrm{m}\right)$ generated from 36 unit-length initial velocity vectors as in (a). (

**c**) A magnified view of the initial point in (a) showing the family of 36 unit-length initial velocity vectors set in the $xy$-plane at that point together with their corresponding geodesics. (

**d**) A magnified view of the initial point in (b) showing the family of 36 unit-length initial velocity vectors set in the $xy$-plane at that point together with their corresponding geodesics.

**Figure 6.**Cartesian plots in the Euclidean outside world of geodesics emanating from two different initial points with 36 unit-length initial velocity vectors set in a plane at the initial point normal to the radial line (indicated by the arrow) connecting it to the egocentre $\left(0,0,0\right)$ (indicated by the dot). Small dots along the geodesics mark 500 ms intervals of time. (

**a**) A family of 36 geodesics emanating from initial point $\left(x=0\mathrm{m},y=5\mathrm{m},z=0\mathrm{m}\right)$ (same as initial point in Figure 5b) generated from 36 unit-length initial velocity vectors set in the $xz$-plane passing through the initial point and equally spaced in all directions from that point in the plane. The resulting 36 geodesics do not remain in the $xz$-plane but become constant tangential speed longitude lines emanating from the initial point to form a 5 m radius sphere centred on the egocentre. The circle geodesic in the $xy$-plane in Figure 5b can be seen in Figure 6a and is slightly thickened. (

**b**) A family of 36 geodesics emanating from initial point $\left(x=0\mathrm{m},y=3.54\mathrm{m},z=3.54\mathrm{m}\right)$. The 36 unit-length equally spaced initial velocity vectors are set in the plane normal to the line connecting the egocentre and the initial point (a plane tilted by $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ rad to the horizontal and tangent to the sphere at that initial point). The resulting 36 geodesics do not remain in that plane but become constant tangential speed longitude lines emanating from the initial point to form a 5 m radius sphere centred on the egocentre with its axis between the egocentre and the initial point tilted by $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ rad. The spheres generated in (a) and (b) correspond to visual spheres centred on the egocentre.

**Figure 7.**Cartesian plots in the Euclidean outside world comparing geodesics with unit-length initial velocity vectors set in a plane at the initial point not normal to the radial line to the egocentre $\left(0,0,0\right)$ with those where the velocity plane is normal. The radial line is indicated by an arrow, the egocentre by a dot and small dots along the geodesics mark 500 ms intervals of time. (

**a**) Geodesics emanating from the initial point $\left(x=0\mathrm{m},y=3.54\mathrm{m},z=3.54\mathrm{m}\right)$ with a family of 36 unit-length initial velocity vectors set in the $xz$-plane passing through the initial point. This plane is tilted back by an angle of $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$ rad from the normal to the radial line. The resulting family of geodesics is comprised of a weighted sum of two parts: (i) Spherical geodesics associated with initial velocity vectors projected into the plane normal to the radial line as in Figure 6b. (ii) Spiral geodesics associated with initial velocity vectors projected into the plane containing the radial line through the egocentre and orthogonal to the normal plane. These geodesics resemble those in Figure 5b. (

**b**) For ease of comparison with part (a) the family of geodesics in Figure 6b is reproduced here but only half the sphere is plotted.

**Figure 8.**(

**a**) A grid of circular and radial geodesics in the outside world in any plane passing through the egocentre represented by the dot • at the origin and (

**b**) its image under a conformal map $\mathsf{\Phi}$ between the plane in the Euclidean outside world and the corresponding plane in the perceived visual manifold with the egocentre again represented by •. The vectors $\xi $ are Killing vectors whose integral flows preserve the metric $g$. The vectors $\eta =\dot{\alpha}$ are velocity vectors tangent to the radial geodesics $\alpha \left(r\right)$. The conformal map $\mathsf{\Phi}$ maps circular geodesics $s\left(\theta \right)$ and radial geodesics $\alpha \left(r\right)$ intersecting at right angles in the Euclidean outside world to equivalent horizontal straight lines $s\left(\theta \right)$ and vertical straight lines $\alpha \left(r\right)$ intersecting at right angles in the perceived visual manifold. Notice that in (b) the intervals on the horizontal straight lines are equal while the equivalent circular arc-lengths $s=r\mathsf{\Delta}\theta $ in the outside world increase linearly. Also, the intervals on the vertical lines in (b) decrease logarithmically while the equivalent radial intervals in (a) are constant. The difference between the two coordinate systems illustrates the profound warping introduced by the visual system.

**Figure 9.**A table illustrating how the combination of the product $K={\kappa}_{1}{\kappa}_{2}$ and the mean $H=\raisebox{1ex}{$\left({\kappa}_{1}+{\kappa}_{2}\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ of the principal curvatures ${\kappa}_{1}$ and ${\kappa}_{2}$ is sufficient to encode the local shape of the submanifold uniquely at each point on the submanifold [113].

**Figure 10.**A schematic diagram illustrating the geometric vector bundle structure of $G$-memory for a fixed place. The cyclopean coordinates $q=\left(r,\theta ,\text{}\phi \right)$ of each image point in $\left(G,g\right)$ act as a memory accession code for the storage and retrieval of the image-point vectors ${\Sigma}_{L}\left(r,\theta ,\text{}\phi \right)\text{}$ and ${\Sigma}_{R}\left(r,\theta ,\text{}\phi \right)$. The union of all the image-point vectors ${\Sigma}_{L}$ and ${\Sigma}_{R}$ forms two 30-dimensional vector fields

**V**

_{L}and

**V**

_{R}in the space of vector fields $\Gamma E$ over $\left(G,g\right)$. $U$ represents an open subset in $\left(G,g\right)$ with ${V}_{\mathrm{L}}\left(U\right)$ and ${V}_{\mathrm{R}}\left(U\right)$ the vector fields over the open subset $U$.

**Figure 11.**A schematic diagram illustrating the geometric structure of a fibre bundle. The base manifold P encodes the place of the head in the Euclidean world. At each place ${p}_{i}\in P$ there exists a fibre containing a vector bundle. The vector fields ${V}_{{p}_{i}}$ and ${V}_{{p}_{j}}$ over the perceived visual manifolds $\left({G}_{{p}_{i}},g\right)$ and $\left({G}_{{p}_{j}},g\right)$ represent the encoded images of the environment seen from places ${p}_{i}$ and ${p}_{j}$ respectively. The map $H=\left[{H}_{1},{H}_{2}\right]$ between the two vector bundles illustrates a vector-bundle morphism. ${H}_{1}\left({p}_{i},{p}_{j}\right)$ depends only on the places ${p}_{i}$ and ${p}_{j}$ in the place map $P$ while ${H}_{2}\left({q}_{{p}_{i}},{q}_{{p}_{j}}\right)$ depends only on the positions of the image points ${q}_{{p}_{i}}$ and ${q}_{{p}_{j}}$ in the manifolds $\left({G}_{{p}_{i}},g\right)$ and $\left({G}_{{p}_{j}},g\right).$

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Neilson, P.D.; Neilson, M.D.; Bye, R.T.
A Riemannian Geometry Theory of Three-Dimensional Binocular Visual Perception. *Vision* **2018**, *2*, 43.
https://doi.org/10.3390/vision2040043

**AMA Style**

Neilson PD, Neilson MD, Bye RT.
A Riemannian Geometry Theory of Three-Dimensional Binocular Visual Perception. *Vision*. 2018; 2(4):43.
https://doi.org/10.3390/vision2040043

**Chicago/Turabian Style**

Neilson, Peter D., Megan D. Neilson, and Robin T. Bye.
2018. "A Riemannian Geometry Theory of Three-Dimensional Binocular Visual Perception" *Vision* 2, no. 4: 43.
https://doi.org/10.3390/vision2040043