Cyclopean Geometry of Binocular Vision

Miles Hansard and Radu Horaud

INRIA Rhône-Alpes, 655 Avenue de l’Europe,
38330 Montbonnot, France.

###### Abstract

The geometry of binocular projection is analyzed, with reference to the
primate visual system. In particular, the effects of coordinated eye movements
on the retinal images are investigated. An appropriate oculomotor
parameterization is defined, and is shown to complement the classical version
and vergence angles. The midline horopter is identified, and subsequently used
to construct the epipolar geometry of the system. It is shown that the
Essential matrix can be obtained by combining the epipoles with the projection
of the midline horopter. A local model of the scene is adopted, in which depth
is measured relative to a plane containing the fixation point. The binocular
disparity field is given a symmetric parameterization, in which the unknown
scene-depths determine the location of corresponding image-features. The
resulting Cyclopean depth-map can be combined with the estimated oculomotor
parameters, to produce a local representation of the scene. The recovery of
visual direction and depth from retinal images is discussed, with reference to
the relevant psychophysical and neurophysiological literature.

OCIS codes: 330.1400, 330.2210

##  1 Introduction

Information about the 3-d structure of a scene can be recovered from binocular
image-pairs, owing to the spatial separation of the two viewpoints. If the
projections of corresponding points can be identified in the left and right
images, then in principle, the scene can be reconstructed from the resulting
binocular disparity field. In addition to the image information, the
reconstruction process requires knowledge (or estimation) of the projection
parameters; in particular, the relative orientation of the eyes must be known.
If some, or all, of these calibration parameters remain unknown, then
reconstruction process is limited to affine or projective properties of the
scene[11, 23]. Hence, in order to facilitate the recovery of as much
information as possible from the disparity field, it would be desirable to
keep the orientation of the eyes fixed in the head. Despite this fact, human
vision involves frequent eye movements, which serve a variety of purposes[6].
For example, the eyes may be moved in order to direct the field of view, or to
foveate an object of interest. Eye movements are also used to stabilize the
retinal image with respect to head movements[38], and to track moving visual
targets. It would be undesirable to suspend these functions, which are
essentially monocular, during the binocular analysis of a scene.

The geometry of binocular stereopsis is complicated by movements of the eyes,
as described above. However, the two eyes typically move in a coordinated
fashion, such that a single point in the scene is fixated. This can be
achieved, in particular, by vergence eye-movements, which are _driven_ by
binocular disparity.[34] These coordinated eye movements benefit steropsis, as
they align the two retinal images at the respective foveas. It follows that
the amount of disparity around the fixation point tends to be reduced,
assuming that the scene is locally smooth. This is important, given the
relatively short range of biological disparity detectors.[12, 33]

There may also be ethological reasons for the existence of binocular eye-
movements, despite the resulting complication of stereopsis. It has been
suggested that the evolution of binocular vision was motivated by the ability
to detect camouflaged prey, by segmentation in depth, with respect to the
background [19]. Another impetus may have been the improvement in image
quality that can be achieved by combining two views, especially in nocturnal
conditions [31]. Both of these applications would benefit from binocular eye
movements, which allow the scene to be scanned without moving the head, and
help to register the two images. Because neither application requires
geometric reconstruction of the scene, the disadvantages of moving the eyes
are negligible. It should, however, be noted that stereopsis also exists in
animals that do not move their eyes significantly, such as owls.[30]

It is clear that the binocular vision of humans (and other primates) has
evolved beyond simple tasks such as camouflage-breaking. Psychophysical
evidence shows that at least some geometric properties of a 3-d layout can be
estimated by stereopsis, and that these estimates can be combined, as the eyes
fixate successive points in the visible scene[13]. Furthermore, it is clear
that most types of eye movement are binocularly coordinated[6].

The combination of eye movements and stereopsis raises important questions
about the oculomotor parameterization, the processing of the binocular
disparity field, and the representation of the scene. These three questions
are developed in more detail below, in sections 1.A, 1.B and 1.C,
respectively.

###  1.A Oculomotor Parameterization

Firstly, what is the appropriate parameterization of binocular eye movements?
This an important issue, because it determines the complexity of the control
problem that the oculomotor system must solve. In particular, it is desirable
to establish the minimal number of parameters that are compatible with the
observed range of oculomotor behaviour. The combination of two sensors, each
of which can rotate in space, results in a system which has six angular
degrees of freedom. However, if Donder’s Law is obeyed[6], then the rotation
of each eye _around_ the corresponding line of sight is determined by the
_direction_ of that line of sight. This removes one degree of freedom from
each eye. Furthermore, binocular fixation implies co-planarity of the visual
axes, which removes one elevation angle from the parameterization. This leaves
three degrees of freedom, which can be conveniently assigned to the elevation,
azimuth and distance of the fixation point. These variables are most naturally
specified in relation to the ‘Cyclopean point’ [16], which, in the present
work, is defined as the midpoint of the inter-ocular axis. The trigonometry of
this Cyclopean parameterization is defined in section 3, and its relationship
to the classical version/vergence model[18] is stated.

###  1.B Disparity Processing

The second question that is addressed here concerns the relationship between
the orientation of the eyes, and the structure of the binocular disparity
field. The difference of position between the left and right projections of a
given scene point is, by convention, called the ‘absolute disparity’ of the
point[4]. This is the quantity that can be measured most directly, by
disparity-sensitive mechanisms[9]. It is important to note that the response
of such a mechanism must depend on the orientation of the eyes. Indeed, for a
typical scene, and a typical fixation point, it may be hypothesized that the
relative orientation of the eyes will be the dominant source of absolute
disparity.

It is important, for the reasons given above, to establish exactly how the
disparity field is affected by eye movements. The question is approached in
section 4, in which the horopter of the fixating system is defined; this is
the set of scene points that project to the same location in each image.[16]
The horopter is used in section 5 to construct the epipolar geometry[22] of
the system, which is effectively parametrized by the version and vergence
angles. If a projected point is identified in one image, then the epipolar
constraint restricts the location of the corresponding point to a _line_ in
the other image. This important relationship can be expressed for any
configuration of the eyes. In principle, the epipolar geometry could be used
to ‘rectify’ the retinal images, thereby removing the effect of eye movements
on the disparity field. However, this would not be consistent with the
observed dependence of early binocular processing on absolute retinal
disparity[9]. Hence it is desirable to parameterize the disparity field, with
respect to the orientation of the eyes. The epipolar geometry is the basis of
such a parameterization.

###  1.C Scene Representation

The two questions described above are part of a third, more general question:
How can the geometric structure of the scene be represented by the visual
system? This issue is complicated by the fact that the early mechanisms of
primate binocular vision are sensitive to a quite limited range of
disparities[12, 33]. The region of space that can be resolved in depth
depends, consequently, on the relative orientation of the eyes. Specifically,
only those points in Panum’s area (which is centred on the fixation point) can
be fused[28, 4]. It follows that any global representation of the scene must
be assembled piecewise, over a series of fixations. It is natural to formulate
this process as the measurement of scene-structure with respect to a reference
surface, followed by an integration of the resulting local models[3]. The
plane that passes through the fixation point, and that is orthogonal to the
Cyclopean visual direction, is a convenient local model for binocular scene
representation, as will be shown in section 6.

###  1.D Scope and Assumptions

The word ‘Cyclopean’ has, in the present context, several possible meanings.
As described in section 1.A, the ‘Cyclopean point’ is a notional centre of
projection, located at the midpoint of the inter-ocular axis (cf.
Helmholtz[16]). It is convenient to use this point as the origin of the
binocular coordinate system, although there is a useful alternative, as will
be shown in section 3. The word ‘Cyclopean’ is used elsewhere in a more
general sense, with reference to visual information that is intrinsically
binocular, such as the ‘edges’ that can be perceived in a random-dot
stereogram (cf. Julesz[19]). The phrase ‘Cyclopean geometry’, as used here,
refers to the fact that the binocular configuration of a fixating visual
system can be parameterized by the state of a _single_ eye (cf. Hering[18]),
as will be explained in section 3.

It will be assumed, in this paper, that the retinal projections can be
described by the usual pin-hole camera equations, and that these projections
are ‘internally calibrated’. This means that the visual system is able to
relate the monocular retinal separation of any two points to the angle between
the corresponding optical rays. It is unlikely that the visual system actually
achieves such a Euclidean representation [36]. However, the present work is
primarily a _description_ of binocular geometry, which need not necessarily
correspond to the encoding that is used by the visual system. The main
constructions of this paper, including the horopter and the epipolar geometry,
can be obtained (more naturally) in the projective setting, which does not
require retinal calibration [11, 23]. However, the present work is much
concerned with the effects of the oculomotor configuration on binocular
vision, and these effects are more readily studied by Euclidean methods.

A distinction should be made between _descriptions_ and _models_ of binocular
vision. The present work aims to describe binocular geometry in the most
convenient way. This leads to Cyclopean parameterizations of visual direction
and binocular disparity. Whether these parameterizations are actually used by
the visual system is a further question[20]. In particular, it is not
_assumed_ here that the Cyclopean representation has any biological reality.
Discussion of the psychophysical and physiological evidence that can be used
to make such claims is confined to section 7.

It will be assumed here that the left and right visual axes intersect at a
point in space. This is approximately true in practice. In the absence of an
intersection, it would be possible to define an appropriate chord between the
left and right visual axes, and to choose a notional fixation point on this
segment. Finally, it is assumed that each eye rotates in accordance with
Donders’ law, meaning that the cyclotorsion of the eyes can be estimated from
the gaze direction.[6]

###  1.E Novel Contributions

The results that are presented here are related to established approaches in
computer vision [11, 23, 2, 35]. The derivations however, are novel, and the
details are specific to the biological context. The following results are of
particular interest: I. The Cyclopean parameterization of binocular
orientation (equations 5,6,9); II. The identification of the midline horopter
as an axis that passes through the pole of the visual plane (17,18); III. The
construction of the Essential matrix from the epipoles and midline horopter
(24); IV. The symmetric parameterization of binocular correspondence (28) and
V. the parameterization of binocular parallax as a function of deviation from
the fixation plane (47).

##  2 Projection Model

The notation and coordinate systems used in this paper are described here.
Points and vectors will be shown in bold type, for example $\mathitbf{q}$,
$\mathitbf{v}$. The transpose of $\mathitbf{v}$ is a row-vector
$\mathitbf{v}^{\scriptscriptstyle\top}$, and the Euclidean length is
$|\mathitbf{v}|$. The notation $(\mathitbf{v})_{3}$ will be used to indicate
the third component of the vector $\mathitbf{v}$. Matrices are represented by
upper-case letters, for example $\mathitbf{M}$.

The 3-d Euclidean coordinates of a point will be distinguished by a bar, e.g.
$\bar{\mathitbf{q}}$. Note that the difference of two Euclidean points results
in a vector, e.g. $\mathitbf{v}=\bar{\mathitbf{q}}-\bar{\mathitbf{p}}$. The
homogeneous image-coordinates of points and lines are written without a bar;
for example, a point at retinal location $(x,y)^{\scriptscriptstyle\top}$ is
represented by $\mathitbf{q}=(\mu x,\mu y,\mu)^{\scriptscriptstyle\top}$, with
$\mu\neq 0$. Note that the inhomogeneous coordinates can be recovered from
$\mathitbf{q}/\mu$. Scalar multiples of the homogeneous coordinates represent
the same image point. For example, if $\mathitbf{p}=(\lambda x,\lambda
y,\lambda)^{\scriptscriptstyle\top}$ then
$\mathitbf{q}/\mu=\mathitbf{p}/\lambda$; this relationship will be written as
$\mathitbf{p}\sim\mathitbf{q}$.

A line in the image plane has homogeneous coordinates
$\mathitbf{n}=(a,b,c)^{\scriptscriptstyle\top}$, such that $\mathitbf{q}$ is
on $\mathitbf{n}$ if $\mathitbf{n}^{\scriptscriptstyle\top}\mathitbf{q}=0$.
Scalar multiples represent the same line; if $\mathitbf{m}=(\kappa a,\kappa
b,\kappa c)^{\scriptscriptstyle\top}$, then $\mathitbf{m}\sim\mathitbf{n}$,
with $\mathitbf{m}^{\scriptscriptstyle\top}\mathitbf{q}=0$, as before. If
$\mathitbf{n}$ is defined as $\mathitbf{n}=\mathitbf{p}\times\mathitbf{q}$,
then
$\mathitbf{n}^{\scriptscriptstyle\top}\mathitbf{p}=\mathitbf{n}^{\scriptscriptstyle\top}\mathitbf{q}=0$;
hence $\mathitbf{n}$ is the line through the two points. Similarly, if
$\mathitbf{q}=\mathitbf{m}\times\mathitbf{n}$ then
$\mathitbf{m}^{\scriptscriptstyle\top}\mathitbf{q}=\mathitbf{n}^{\scriptscriptstyle\top}\mathitbf{q}=0$;
hence $\mathitbf{q}$ is the intersection point of the two lines.

The left and right optical centres are labeled $\bar{\mathitbf{c}}_{\ell}$ and
$\bar{\mathitbf{c}}_{r}$, respectively. The Cyclopean point
$\bar{\mathitbf{c}}_{b}$ is fixed halfway between, and it will also be useful
to define the ‘baseline’ vector $\mathitbf{b}$;

$\displaystyle\mathitbf{b}$
$\displaystyle=\bar{\mathitbf{c}}_{r}-\bar{\mathitbf{c}}_{\ell}$ (1)
$\displaystyle\bar{\mathitbf{c}}_{b}$
$\displaystyle=\tfrac{1}{2}(\bar{\mathitbf{c}}_{\ell}+\bar{\mathitbf{c}}_{r}).$
(2)

Only the _ratio_ of the scene size to the baseline length can be recovered
from the images, in the absence of other information. For this reason it is
helpful to define the distance between the two optical centres as
$|\mathitbf{b}|=1$, so that Euclidean coordinates are measured units of
interocular separation. The location of the scene coordinate-system is
immaterial, so it will be convenient to put the origin at the Cyclopean point
$\bar{\mathitbf{c}}_{b}$. The coordinates of the optical centres, with
reference to figure 1, will be

$\bar{\mathitbf{c}}_{\ell}=-\bigl{(}\tfrac{1}{2},0,0\bigr{)}^{\scriptscriptstyle\top}\quad\text{and}\quad\bar{\mathitbf{c}}_{r}=\bigl{(}\tfrac{1}{2},0,0\bigr{)}^{\scriptscriptstyle\top}.$
(3)

The baseline vector (1) is therefore parallel to the $\mathitbf{x}$ axis, and
a perpendicular axis $\mathitbf{z}=(0,0,1)^{\scriptscriptstyle\top}$ will be
taken as the head-centric outward direction. These two vectors define
Cartesian coordinates in the horizontal plane. The downward normal of this
plane is $\mathitbf{y}=(0,1,0)^{\scriptscriptstyle\top}$, so that the axes
$\mathitbf{x}$, $\mathitbf{y}$ and $\mathitbf{z}$ form a right-handed system,
as shown in figure 1.

The orientations of the left, right and Cylcopean eyes are expressed by
$3\times 3$ rotation matrices $\mathitbf{R}_{\ell}$, $\mathitbf{R}_{r}$ and
$\mathitbf{R}$, respectively. A view of the scene is obtained by expressing
each point $\bar{\mathitbf{q}}$ relative to an optical centre, and applying
the corresponding rotation. Hence the homogeneous perspective projection into
the left image is

$\mathitbf{p}_{\ell}\sim\mathitbf{R}_{\ell}(\bar{\mathitbf{q}}-\bar{\mathitbf{c}}_{\ell}).$
(4)

If both eyes are looking straight ahead (with zero cyclo-rotation),
$\mathitbf{R}_{\ell}=\mathitbf{R}_{r}=\mathitbf{I}$, then the projections of
$\bar{\mathitbf{q}}=(x,y,z)^{\scriptscriptstyle\top}$ can be computed easily;
they are
$\mathitbf{q}_{\ell}\sim\bigl{(}x+\tfrac{1}{2},y,z\bigr{)}^{\scriptscriptstyle\top}$
and
$\mathitbf{q}_{r}\sim\bigl{(}x-\tfrac{1}{2},y,z\bigr{)}^{\scriptscriptstyle\top}$.
The inhomogeneous coordinates of these points are
$x_{\ell}=(x+\tfrac{1}{2})\big{/}z$, $x_{r}=(x-\tfrac{1}{2})\big{/}z$, and
$y_{\ell}=y_{r}=y/z$, corresponding to the ordinary perspective equations. It
is straightforward, in this case, to evaluate the _binocular disparity_
$(x_{\ell},y_{\ell})^{\scriptscriptstyle\top}-(x_{r},y_{r})^{\scriptscriptstyle\top}=(1/z,0)^{\scriptscriptstyle\top}$.
Note that, in this case of parallel visual axes, the disparity vector is
confined to the horizontal direction. For general orientations of the eyes,
disparity equations are more complicated, as will be seen in section 6.

##  3 Binocular Orientation

An angular parameterization of binocular eye movements is introduced in this
section. As was noted in section 1.A, the degrees of freedom of the binocular
system can be reduced from six, to three. The reduction is achieved by
imposing the fixation constraint, together with Donders’ Law. An appropriate
parameterization will be developed, based on the Cyclopean azimuth, elevation
and distance of the fixation point. It will be shown that this representation
complements the classical version/vergence coordinates[18].

Suppose that the point $\bar{\mathitbf{p}}_{0}$ is to be fixated. The scene
coordinates of this point can be specified by a head-fixed direction
$\mathitbf{v}$ from the Cyclopean origin, in conjunction with a distance,
$\rho$, along the corresponding ray;

$\bar{\mathitbf{p}}_{0}=\rho\mathitbf{v}.$ (5)

The direction $\mathitbf{v}$ is a unit-vector, and the positive scalar $\rho$
will be called the _range_ of the fixation point $\bar{\mathitbf{p}}_{0}$. The
Cyclopean direction may be written in terms of the elevation and azimuth
angles $\alpha$ and $\beta$ respectively;

$\mathitbf{v}=(\sin\beta,\,-\sin\alpha\,\cos\beta,\,\cos\alpha\,\cos\beta)^{\scriptscriptstyle\top}$
(6)

where $\cos\beta$ is the projected length of $\mathitbf{v}$ in the mid-
sagittal plane $x=0$, which divides one side of the head from the other. Note
that the elevation $\alpha$ is positive for points above the horizontal plane
$(y<0)$, and that the azimuth $\beta$ is positive for points to the right of
the mid-sagittal plane $(x>0)$. These visual angles will each be in the range
$[-\pi/2,\pi/2]$, so that any point with $z\geq 0$ can be identified, as shown
in figure 1. If the fixation point
$\bar{\mathitbf{p}}_{0}=(x,y,z)^{\scriptscriptstyle\top}$ is given in
Cartesian coordinates, then the corresponding range and direction are

$\displaystyle\rho$ $\displaystyle=|\bar{\mathitbf{p}}_{0}|$ (7)
$\displaystyle\mathitbf{v}$ $\displaystyle=\bar{\mathitbf{p}}_{0}/\rho$ (8)

respectively. The elevation and Cyclopean azimuth angles can be obtained from
the equations $\tan\alpha=-y/z$ and $\sin\beta=x/\rho$ respectively. The
vector $(\alpha,\beta,\rho)^{\scriptscriptstyle\top}$ contains the _Helmholtz
coordinates_ of the point $\bar{\mathitbf{p}}_{0}$.

Fig. 1: A visual plane $\mathcal{V}_{\alpha}$ is defined by the optical
centres $\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$, together
with the fixation point $\bar{\mathitbf{p}}_{0}$. The visual directions
$\mathitbf{v}$, $\mathitbf{v}_{\ell}$ and $\mathitbf{v}_{r}$ lie in this
plane, which has an elevation angle $\alpha$. The scene coordinates are
located at the Cyclopean point
$\bar{\mathitbf{c}}_{b}=(0,0,0)^{\scriptscriptstyle\top}$, such that
$\mathcal{V}_{0}$ coincides with the $x,z$ plane.

In addition to the Cyclopean visual axis $\mathitbf{v}$, defined above (8),
there exist left and right axes $\mathitbf{v}_{\ell}$ and $\mathitbf{v}_{r}$,
respectively. If the eyes are fixating the point $\bar{\mathitbf{p}}_{0}$, as
described above, then $\mathitbf{v}_{\ell}$ and $\mathitbf{v}_{r}$ can be
derived from $\mathitbf{v}$ and $\rho$, as will be shown below. The optical
centres $\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$, together
with the fixation point $\bar{\mathitbf{p}}_{0}$ define a _visual plane_ ,
$\mathcal{V}_{\alpha}$ as shown in fig. 1. The three visual axes intersect at
$\bar{\mathitbf{p}}_{0}$, and so $\mathitbf{v}_{\ell}$, $\mathitbf{v}_{r}$ and
$\mathitbf{v}$ lie in $\mathcal{V}_{\alpha}$. All of the possible visual
planes contain the baseline $\mathitbf{b}$, and may be parameterized by the
dihedral angle $\alpha$ between $\mathcal{V_{\alpha}}$ and the horizontal
plane $\mathcal{V}_{0}$. The azimuth angles $\beta$, $\beta_{\ell}$ and
$\beta_{r}$ will now be defined _in_ the visual plane $\mathcal{V}_{\alpha}$.

Firstly it will be shown that, if the eyes are fixating, then the left and
right visual directions can be simultaneously parameterized by the Cyclopean
direction and distance of the fixation point. It is convenient to begin by
assuming that the fixation point is in the horizontal plane, such that
$\alpha=0$. The removal of this assumption will be discussed subsequently. It
can be seen, with reference to figure 2, that if the baseline separation is
$|\mathitbf{b}|=1$, then $\tan\beta_{\ell}$ and $\tan\beta_{r}$ are equal to
$(\rho\sin\beta\pm\tfrac{1}{2})/(\rho\cos\beta)$. Some re-arrangement leads to
the definitions

$\tan\beta_{\ell}=\tan\beta+\frac{\sec\beta}{2\rho}\quad\text{and}\quad\tan\beta_{r}=\tan\beta-\frac{\sec\beta}{2\rho}.$
(9)

It is clear from these equations that, for a given Cyclopean azimuth $\beta$,
the visual directions become more equal as the fixation distance, $\rho$,
increases. It may also be noted that if $\beta=0$, then the fixation is
symmetric, with left and right azimuths $\pm\tan^{-1}(\tfrac{1}{2}/\rho)$, as
is commonly assumed in the literature.

If the fixation point $\bar{\mathitbf{p}}_{0}$ is in $\mathcal{V}_{\alpha}$,
then the matrices representing the orientation of the eyes are easily
constructed. For example, the matrix $\mathitbf{R}_{\ell}$ in (4) is

$\mathitbf{R}_{\ell}=\begin{pmatrix}\cos\beta_{\ell}&0&-\sin\beta_{\ell}\\\
0&1&0\\\ \sin\beta_{\ell}&0&\cos\beta_{\ell}\end{pmatrix}.$ (10)

The analogous definitions are made for the matrices $\mathitbf{R}$ and
$\mathitbf{R}_{r}$, with angles $\beta$ and $\beta_{r}$, respectively.

Fig. 2: Angular coordinates of a point $\bar{\mathitbf{p}}_{0}$ in the visual
plane $\mathcal{V}_{\alpha}$. The optical centres are indicated by
$\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$. The azimuth angles
$\beta$ and $\beta_{\ell}$ are positive in this example, whereas $\beta_{r}$
is negative. The Cyclopean range of the fixation point is $\rho$.

Although the Helmholtz coordinates are convenient for specifying visual
directions, the eyes do not, in general, rotate around the corresponding axes.
One characteristic of actual eye movements is that, for general fixation
points, each eye will be cyclo-rotated around the corresponding visual
direction. Although the cyclo-rotation angles $\gamma_{\ell}$ and $\gamma_{r}$
are non-zero, Donders’ law states that they are completely determined by the
corresponding visual directions; hence there exist functions
$\gamma_{\ell}(\alpha,\beta_{\ell})$ and $\gamma_{r}(\alpha,\beta_{r})$. The
definition of these functions can be obtained from, for example, Listing’s law
and its extensions.[17]

The effects of cyclo-rotation are relevant to the distribution of
physiological disparity mechanisms across the visual field [21]. However,
cyclo-rotation is of limited interest with respect to the analysis of
binocular _geometry_ ; this is because the parameters $\gamma_{\ell}$ and
$\gamma_{r}$ simply determine rotations of the respective _images_ ,
independently of any 3-d information. For this reason, the effects of cyclo-
rotation will not be incorporated into the following derivations, meaning that
Donders’ law is obeyed in the trivial sense;

$\gamma_{\ell}(\alpha,\beta_{\ell})=\gamma_{r}(\alpha,\beta_{r})=0.$ (11)

It is, at the end of all derivations, easy to substitute any other example of
Donders’ law, as will be explained in sections 4 and 6. The practical
advantage of the restriction (11) is that any dependence on the elevation
angle $\alpha$ is removed from the analysis. This makes it possible to study
the binocular geometry with respect to fixation points in a _single_ visual
plane, without loss of generality. Furthermore, Listing’s law agrees with (11)
when $\alpha=0$; hence it is natural to choose the horizontal plane
$\mathcal{V}_{0}$ for further investigation.

The remainder of this section will be concerned with the classical binocular
_vergence_ and _version_ parameters, and their relation to the Helmholtz
coordinates. This discussion will lead to the definition of binocularly
_fixed_ points. Section 4 will develop this idea, which will subsequently be
used to define the epipolar geometry.

The vergence angle, $\delta$, will be defined as the angle between the lines
of sight at the fixation point; the version angle, $\epsilon$, will be defined
as the average gaze azimuth. In relation to the Helmholtz coordinates, this
means that

$\displaystyle\delta$ $\displaystyle=\beta_{\ell}-\beta_{r}$ (12)
$\displaystyle\epsilon$ $\displaystyle=\tfrac{1}{2}(\beta_{\ell}+\beta_{r}).$
(13)

The vergence angle $\delta$ is non-negative, due to the inequality
$\beta_{r}\leq\beta_{\ell}$, which follows from the signs and limits of
$\beta_{\ell}$ and $\beta_{r}$ as defined above. The equality
$\beta_{\ell}=\beta_{r}$ occurs for infinitely distant fixation points, for
which $\delta=0$. These definitions are illustrated in figure 3.

Fig. 3: The Vieth-Müller circle is defined by the positions of the optical
centres $\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$, together
with the fixation point, $\bar{\mathitbf{p}}_{0}$. The forward $(z>0)$ arc of
the circle intersects the mid-sagittal plane at the point
$\bar{\mathitbf{c}}_{a}$. The vergence angle, $\delta$, is inscribed at
$\bar{\mathitbf{p}}_{0}$ by $\bar{\mathitbf{c}}_{\ell}$ and
$\bar{\mathitbf{c}}_{r}$. The same angle is inscribed at all other points on
the circle, including $\bar{\mathitbf{c}}_{a}$.

The properties of the vergence and version parameters can be understood with
reference to the Vieth-Müller circle[37], which is defined by the two optical
centres $\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$, together
with the fixation point $\bar{\mathitbf{p}}_{0}$. The vergence, $\delta$, is
the inscribed angle at $\bar{\mathitbf{p}}_{0}$, being opposite the inter-
ocular axis $\mathitbf{b}$. The law of sines gives the diameter of the
circumcirle as $1/\sin\delta$, with $|\mathitbf{b}|=1$, as usual. The angle
subtended by $\mathitbf{b}$ from the centre of the circle is $2\delta$, being
twice the inscribed angle. The isosceles triangle formed by
$\bar{\mathitbf{c}}_{\ell}$, $\bar{\mathitbf{c}}_{r}$ and the centre of the
circle can be split into two right-angled triangles, such that
$\tan\delta=\frac{1}{2}/\zeta$, where $\zeta$ is the $z$ coordinate of the
centre. It follows that the Vieth-Müller circle is centred at the point
$(0,0,\zeta)^{\scriptscriptstyle\top}$, and has radius $\eta$, where

$\displaystyle\zeta$ $\displaystyle=\tfrac{1}{2}\cot\delta$ (14)
$\displaystyle\eta$ $\displaystyle=\tfrac{1}{2}\csc\delta.$ (15)

The optical centres, $\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$,
divide the Vieth-Müller circle into two arcs, according to the sign of $z$.
The forward ($z\geq 0$) arc contains the fixation point,
$\bar{\mathitbf{p}}_{0}$, with inscribed angle $\delta$. Furthermore, the
inscribed angles at all other points
$\bar{\mathitbf{q}}_{\mathrm{\scriptscriptstyle VM}}$ on this arc must be
equal; hence the Vieth-Müller circle contains the locus of iso-vergence.

Fig. 4: The points $\bar{\mathitbf{c}}_{a}$ and $\bar{\mathitbf{p}}_{0}$
inscribe the version angle, $\epsilon$, at an optical centre,
$\bar{\mathitbf{c}}$, which is located on the backward $(z<0)$ arc of the
Vieth-Müller circle. The same angle is inscribed at
$\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$. It follows that as
$\bar{\mathitbf{p}}_{0}$ is fixated, $\bar{\mathitbf{c}}_{a}$ lies in the same
visual _direction_ from each eye. Furthermore, the triangle defined by
$\bar{\mathitbf{c}}_{\ell}$, $\bar{\mathitbf{c}}_{r}$, and
$\bar{\mathitbf{c}}_{a}$ is isosceles, and so the point
$\bar{\mathitbf{c}}_{a}$ is at the same _distance_ from each eye.

The version angle $\epsilon$ gives the azimuth of $\bar{\mathitbf{p}}_{0}$
from a Cyclopean point
$\bar{\mathitbf{c}}=(0,0,\zeta-\eta)^{\scriptscriptstyle\top}$, which lies at
the back of the Vieth-Müller circle, as shown in figure 4. It may be noted
that $\beta\approx\epsilon$, and that this approximation improves as $\delta$
decreases. Furthermore, it can be shown that $\rho\approx 2/\eta\cos\epsilon$.
Evidently the location of the point $\bar{\mathitbf{c}}$ varies according to
the vergence angle. This is one reason for _deriving_ the $(\delta,\epsilon)$
parameterization from the $(\beta,\rho)$ parameterization, as above: The
present analysis has a fixed reference point
$\bar{\mathitbf{c}}_{b}=(0,0,0)^{\scriptscriptstyle\top}$, meaning that visual
information can easily be combined as the eyes re-fixate. Furthermore, the
range parameter $\rho$ is directly related to the observer’s behaviour,
including listening and moving, whereas $\delta$ is specific to the oculomotor
system. Nonetheless, the vergence and version parameters are crucial, as will
be shown in the following sections.

##  4 Fixed Points

In this section it will be shown that, for a given vergence angle, certain
points in the scene project to the same location in each image. These points
constitute the geometric _horopter_ of the fixating system. The constructions
that are given here will be used to construct the epipolar geometry of the two
images, in section 5. For this purpose, it will be convenient to study the
horopter in the absence of any cyclo-rotation.

It was shown in section 3 that the Vieth-Müller circle is defined by the
optical centres $\bar{\mathitbf{c}}_{\ell}$ and $\bar{\mathitbf{c}}_{r}$,
together with the fixation point $\bar{\mathitbf{p}}_{0}$. Consider another
scene-point point, $\bar{\mathitbf{q}}_{\mathrm{\scriptscriptstyle VM}}$, that
lies on the forward section of the Vieth-Müller circle. This point is in the
visual plane $\mathcal{V}_{0}$, and therefore satisfies the equation $y=0$, as
well as the conditions

$\begin{split}\begin{gathered}x^{2}+(z-\zeta)^{2}=\eta^{2}\\\ z\geq
0.\end{gathered}\end{split}$ (16)

The two points $\bar{\mathitbf{p}}_{0}$ and
$\mathitbf{\bar{q}}_{\mathrm{\scriptscriptstyle VM}}$, both of which are on
the forward section of the Vieth-Müller circle, must inscribe equal angles at
the optical centres. The point $\bar{\mathitbf{p}}_{0}$ is being fixated, and
therefore appears in the fovea of each image. Hence the projected point
$\mathitbf{q}_{\mathrm{\scriptscriptstyle VM}}$ is fixed in the images; it
appears on the horizontal meridian of each retina, at a fixed angular offset
from the corresponding fovea.

The preceding argument has established that all scene points
$\bar{\mathitbf{q}}_{\mathrm{\scriptscriptstyle VM}}$ are fixed in the images.
This means, in camera coordinates, that the equations
$x_{\ell}/z_{\ell}=x_{r}/z_{r}$ and $y_{\ell}/z_{\ell}=y_{r}/z_{r}$ are
satisfied. The existence of points that are fixed in the scene, as well as in
the images, will now be investigated. Recall that the Euclidean coordinates of
$\bar{\mathitbf{q}}=(x,y,z)^{\scriptscriptstyle\top}$ in the left and right
camera frames are
$\bar{\mathitbf{q}}_{\ell}=\mathitbf{R}_{\ell}(\bar{\mathitbf{q}}-\bar{\mathitbf{c}}_{\ell})$
and
$\bar{\mathitbf{q}}_{r}=\mathitbf{R}_{r}(\bar{\mathitbf{q}}-\bar{\mathitbf{c}}_{r})$,
respectively. The point $\bar{\mathitbf{q}}$ is fixed in the scene if
$\bar{\mathitbf{q}}_{\ell}=\bar{\mathitbf{q}}_{r}$, which in turn implies that
$|\bar{\mathitbf{q}}_{\ell}|^{2}=|\bar{\mathitbf{q}}_{r}|^{2}$. The squared-
lengths are preserved by the rotation matrices $\mathitbf{R}_{\ell}$ and
$\mathitbf{R}_{r}$, and so
$|\bar{\mathitbf{q}}-\bar{\mathitbf{c}}_{\ell}|^{2}=|\bar{\mathitbf{q}}-\bar{\mathitbf{c}}_{r}|^{2}$.
From the definition of $\bar{\mathitbf{c}}_{\ell}$ and
$\bar{\mathitbf{c}}_{r}$ in (3), this is equivalent, in scene-coordinates, to
the condition $(x+\tfrac{1}{2})^{2}=(x-\tfrac{1}{2})^{2}$. Hence it can be
seen that any fixed point in the scene must lie in the mid-sagittal plane
$x=0$, that divides one side of the head from the other. Substituting $x=0$
into (16) leads immediately to $z=\zeta+\eta$, leaving $y$ free to vary. In
general, $y_{\ell}=y_{r}$, because the axis of the vergence rotation is
perpendicular to the visual plane $\mathcal{V}_{\alpha}$. This argument has
established that there is an axis of fixed points $\bar{\mathitbf{q}}_{a}$ in
the scene. This axis intersects the Vieth-Müller circle at a point
$\bar{\mathitbf{c}}_{a}$, and is perpendicular to the visual plane. If,
without loss of generality, $\alpha=0$, then the coordinates of these points
are

$\displaystyle\bar{\mathitbf{c}}_{a}$
$\displaystyle=(0,0,\zeta+\eta)^{\scriptscriptstyle\top}$ (17)
$\displaystyle\bar{\mathitbf{q}}_{a}$
$\displaystyle=\bar{\mathitbf{c}}_{a}+(0,y,0)^{\scriptscriptstyle\top}.$ (18)

This axis of points, $\bar{\mathitbf{q}}_{a}$, which has been identified
elsewhere[37, 8], is the geometric _midline horopter_. The point
$\bar{\mathitbf{c}}_{a}$ is the _pole_ of the planar transformation induced by
the translation $\mathitbf{b}$ and vergence rotation
$\mathitbf{R}_{r}\mathitbf{R}_{\ell}^{\scriptscriptstyle\top}$. The points
$\bar{\mathitbf{q}}_{a}$ lie on the associated screw-axis[2].

It will be useful to compute the _image_ coordinates of the axis, which are
common to both eyes, as shown in fig. 4. The points $\bar{\mathitbf{p}}_{0}$
and $\bar{\mathitbf{c}}_{a}$ inscribe equal angles at
$\bar{\mathitbf{c}}_{\ell}$, $\bar{\mathitbf{c}}_{r}$ and
$\bar{\mathitbf{c}}$; moreover, the angle at $\bar{\mathitbf{c}}$ is, by
definition, the binocular version, $\epsilon$. Having established that the
angular direction of $\bar{\mathitbf{q}}_{a}$ from either optical centre is
$\epsilon$, the common distance of this point will also be computed. The
points $\bar{\mathitbf{c}}_{\ell}$, $\bar{\mathitbf{c}}_{r}$ and
$\bar{\mathitbf{c}}_{a}$ form an isosceles triangle, from which it can be seen
that $|\bar{\mathitbf{c}}_{a}|\sin(\delta/2)=\frac{1}{2}$. It follows that, in
the coordinates of either eye, the axis is specified by

$\displaystyle\mathitbf{c}_{a}$
$\displaystyle=\tfrac{1}{2}\csc(\delta/2)\,(-\sin\epsilon,0,\cos\epsilon)^{\scriptscriptstyle\top}$
(19) $\displaystyle\mathitbf{q}_{a}$
$\displaystyle=\mathitbf{c}_{a}+(0,y,0)^{\scriptscriptstyle\top}.$ (20)

These image points lie on a vertical line $\mathitbf{a}$, which has the same
coordinates in each eye. The equation of the line is
$\mathitbf{q}_{a}^{\scriptscriptstyle\top}\mathitbf{a}=0$, and so it follows
from equation (19) that the coordinates of the line are determined by the
version angle $\epsilon$;

$\mathitbf{a}\sim(\cos\epsilon,0,-\sin\epsilon)^{\scriptscriptstyle\top}.$
(21)

The results of this section can be summarized as follows. The geometric
horopter, in the absence of cyclo-rotation, consists of the forward part of
the Vieth-Müller circle, together with the midline component. If a scene point
$\bar{\mathitbf{q}}$ is on the horopter, then the image coordinates of the
corresponding points are equal, $\mathitbf{q}_{\ell}\sim\mathitbf{q}_{r}$.
Furthermore, if the point is on the midline component, then it is also true
that the eye-coordinates of $\bar{\mathitbf{q}}$ are equal,
$\bar{\mathitbf{q}}_{\ell}=\bar{\mathitbf{q}}_{r}$. Finally, the image
coordinates (21) of the midline are determined by the binocular version angle.
It should be noted that, in the presence of cyclo-rotation, the geometric
horopter takes the form of a twisted cubic curve[16]. This curve coincides
with the Vieth-Müller circle as it passes through the optical centres, and has
asymptotes at $\bar{\mathitbf{c}}_{a}\pm(0,y,0)^{\scriptscriptstyle\top}$.

##  5 Epipolar Geometry

It was established, in the preceding section, that certain scene points have
the same coordinates in both images. The related _epipolar constraint_ is
weaker, but much more useful, as it applies to _all_ scene points. The
epipolar geometry of the fixating system will now be described; in particular,
the image of the midline horopter (21), will be used to construct the
appropriate ‘Essential matrix’.[22]

The epipolar constraint is as follows: Given an image point
$\mathitbf{q}_{\ell}$ in $\mathcal{I}_{\ell}$, the corresponding point
$\mathitbf{q}_{r}$ in $\mathcal{I}_{r}$ must lie on a known epipolar line
$\mathitbf{u}_{r}$, such that
$\mathitbf{u}_{r}^{\scriptscriptstyle\top}\\!\mathitbf{q}_{r}=0.$ The
geometric interpretation of this is that the scene point $\bar{\mathitbf{q}}$
must be located on the ray defined by the optical centre
$\bar{\mathitbf{c}}_{\ell}$ and the image point $\mathitbf{q}_{\ell}$; the ray
projects to a line $\mathitbf{u}_{r}$ in the other view, and so
$\mathitbf{q}_{r}$, being another image of $\bar{\mathitbf{q}}$, must lie on
the line. Furthermore, note that the optical centre
$\bar{\mathitbf{c}}_{\ell}$ is common to all such rays, and so the resulting
lines $\mathitbf{u}_{r}$ must intersect at a single point in
$\mathcal{I}_{r}$. This point is the right _epipole_ , $\mathitbf{e}_{r}$.
Similar arguments can be used to introduce the left epipole
$\mathitbf{e}_{\ell}$, as well as the associated lines $\mathitbf{u}_{\ell}$
in $\mathcal{I}_{\ell}$.

Suppose that the point $\mathitbf{q}_{\ell}$ is given; then, with reference to
fig. 5, $\mathitbf{u}_{\ell}=\mathitbf{e}_{\ell}\times\mathitbf{q}_{\ell}$.
Furthermore, this line intersects the projection $\mathitbf{a}$ of the midline
horopter (21) at the image point
$\mathitbf{q}_{a}=\mathitbf{a}\times\mathitbf{u}_{\ell}$. Any point on
$\mathitbf{u}_{\ell}$ must also be on $\mathitbf{u}_{r}$, as the two lines are
images of the same ray. Hence $\mathitbf{u}_{r}$ can be constructed from
$\mathitbf{e}_{r}$ and the point in $\mathcal{I}_{r}$ that corresponds to
$\mathitbf{q}_{a}$. Furthermore, $\mathitbf{q}_{a}$ is a fixed point (being on
$\mathitbf{a}$), and so its coordinates are unchanged in $\mathcal{I}_{r}$. It
follows that $\mathitbf{u}_{r}=\mathitbf{e}_{r}\times\mathitbf{q}_{a}$. The
preceding construction may be summarized as

$\mathitbf{u}_{r}\sim\mathitbf{e}_{r}\times\bigl{(}\mathitbf{a}\times(\mathitbf{e}_{\ell}\times\mathitbf{q}_{\ell})\bigr{)}.$
(22)

This equation will now be put into a more useful form. Suppose that
$\mathitbf{w}=(x,y,z)^{\scriptscriptstyle\top}$; then the cross product
$\mathitbf{w}\times\mathitbf{p}$ can be expressed as a matrix-vector
multiplication, $(\mathitbf{w}\times)\mathitbf{p}$, where

$\bigl{(}\mathitbf{w}\times\bigr{)}=\begin{pmatrix}0&-z&y\\\ z&0&-x\\\
-y&x&0\end{pmatrix}$ (23)

is a $3\times 3$ antisymmetric matrix, constructed from the components of
$\mathitbf{w}$. Consider the part of equation (22) that does not depend on the
particular choice of point $\mathitbf{q}_{\ell}$; the equivalence (23) can be
used to express this as a transformation

$\mathitbf{E}\sim\bigl{(}\mathitbf{e}_{r}\times\bigr{)}\bigl{(}\mathitbf{a}\times\bigr{)}\bigl{(}\mathitbf{e}_{\ell}\times\bigr{)}$
(24)

which is the $3\times 3$ _Essential matrix_[22]. Given a point
$\mathitbf{q}_{\ell}$, the corresponding point $\mathitbf{q}_{r}$ must be on a
certain epipolar line $\mathitbf{u}_{r}$, as described above. This constraint
is expressed via the Essential matrix as

$\mathitbf{q}_{r}^{\scriptscriptstyle\top}\mathitbf{E}\mathitbf{q}_{\ell}=0$
(25)

where $\mathitbf{u}_{r}\sim\mathitbf{E}\mathitbf{q}_{\ell}$. The analogous
constraint,
$\mathitbf{q}_{\ell}^{\scriptscriptstyle\top}\mathitbf{E}^{\scriptscriptstyle\top}\mathitbf{q}_{r}=0$
applies in the opposite direction, the epipolar line being
$\mathitbf{u}_{\ell}\sim\mathitbf{E}^{\scriptscriptstyle\top}\\!\mathitbf{q}_{r}$
in this case. The epipoles, as described above, are each the image of the
‘other’ camera centre. This means that
$\mathitbf{e}_{\ell}\sim\mathitbf{R}_{\ell}(\mathitbf{b})$, and
$\mathitbf{e}_{r}\sim\mathitbf{R}_{r}(-\mathitbf{b})$, where $\mathitbf{b}$ is
the vector between the optical centres. Equations (1), (3) and (4) can be used
to show that the epipoles are simply

$\mathitbf{e}_{\ell}\sim(\cos\beta_{\ell},\,0,\,\sin\beta_{\ell})^{\scriptscriptstyle\top}\quad\text{and}\quad\mathitbf{e}_{r}\sim(-\cos\beta_{r},\,0,\,-\sin\beta_{r})^{\scriptscriptstyle\top}.$
(26)

These equations can be combined with the definition of the geometric midline
horopter (21), to give a parametric structure to the Essential matrix. The
non-zero terms in the matrix product (24) are found to be
$E_{12}=-E_{r}\sin\beta_{r}$, $E_{21}=E_{\ell}\sin\beta_{\ell}$,
$E_{23}=-E_{\ell}\cos\beta_{\ell}$ and $E_{22}=E_{r}\cos\beta_{r}$, where
$E_{\ell}=\cos\beta_{r}\cos\epsilon+\sin\beta_{r}\sin\epsilon$ and
$E_{r}=\cos\beta_{\ell}\cos\epsilon+\sin\beta_{\ell}\sin\epsilon$. The factors
$E_{\ell}$ and $E_{r}$ are seen to be the angle-difference expansions of
$\cos(\beta_{\ell}-\epsilon)$ and $\cos(\beta_{r}-\epsilon)$, respectively.
Furthermore, by reference to (12,13) the arguments $\beta_{\ell}-\epsilon$ and
$\beta_{r}-\epsilon$ are equal to $\pm\delta/2$, and so it follows from the
even-symmetry of the cosine function that $E_{\ell}=E_{r}=\cos(\delta/2)$. The
Essential matrix is defined here as as a homogeneous transformation (cf. 25),
and so this common scale-factor can be disregarded, which leaves

$\mathitbf{E}\sim\begin{pmatrix}0&-\sin\beta_{r}&0\\\
\sin\beta_{\ell}&0&-\cos\beta_{\ell}\\\ 0&\cos\beta_{r}&0\end{pmatrix}.$ (27)

It is straightforward to verify that $\mathitbf{E}$ is indeed an Essential
matrix, having one singular-value equal to zero, and two identical non-zero
singular-values (here equal to unity).

Fig. 5: Construction of the epipolar geometry. Point $\mathitbf{q}_{\ell}$ is
given, so the epipolar line in $\mathcal{I}_{\ell}$ is
$\mathitbf{u}_{\ell}=\mathitbf{q}_{\ell}\times\mathitbf{e}_{\ell}$. This line
intersects the image $\mathitbf{a}$ of the midline horopter in
$\mathcal{I}_{\ell}$ at
$\mathitbf{q}_{a}=\mathitbf{a}\times\mathitbf{u}_{\ell}$. The point
$\mathitbf{q}_{a}$ is on $\mathitbf{a}$, and is therefore fixed, having the
same coordinates $\mathitbf{a}$ in $\mathcal{I}_{r}$. It follows that the
epipolar line in $\mathcal{I}_{r}$ is
$\mathitbf{u}_{r}=\mathitbf{e}_{r}\times\mathitbf{q}_{a}$. The location of
$\mathitbf{q}_{r}$, which corresponds to $\mathitbf{q}_{\ell}$, is unknown,
but it must lie on $\mathitbf{u}_{r}$. The Vieth-Müller circle is shown in the
figure, as is the midline-horopter, which passes through
$\bar{\mathitbf{c}}_{a}$ and $\bar{\mathitbf{q}}_{a}$.

The same result can be obtained from a variant of the more traditional
definition of the essential matrix,[22, 5]
$\mathitbf{E}\sim\mathitbf{R}_{r}(\mathitbf{b}\times)\mathitbf{R}_{\ell}^{\scriptscriptstyle\top}$.
This definition is, however, not specific to the case of visual fixation, and
offers correspondingly less insight into the present configuration.

##  6 Binocular Disparity

It was established in the preceding section that projected points in the left
and right images must obey the epipolar constraint. In particular it was shown
that, given a point $\mathitbf{q}_{\ell}$ in $\mathcal{I}_{\ell}$, the
corresponding point $\mathitbf{q}_{r}$ must lie on a line
$\mathitbf{u}_{r}\sim\mathitbf{E}\mathitbf{q}_{\ell}$ in the other image,
$\mathcal{I}_{r}$. The structure of the scene displaces the left and right
image points along the corresponding lines, which pass through the left and
right epipoles, respectively. These image displacements are quantified in this
section. In particular, it is shown that a scene-point $\bar{\mathitbf{q}}$
which is in Cyclopean direction $\mathitbf{p}_{c}$ will be projected to
corresponding image-points $\mathitbf{q}_{\ell}$ and $\mathitbf{q}_{r}$, where

$\mathitbf{q}_{\ell}=\mathitbf{p}_{\ell}+t_{\ell}(s)\,\mathitbf{d}_{\ell}\quad\text{and}\quad\mathitbf{q}_{r}=\mathitbf{p}_{r}+t_{r}(s)\,\mathitbf{d}_{r}.$
(28)

The unit-vectors $\mathitbf{d}_{\ell}$ and $\mathitbf{d}_{r}$ point towards
the respective epipoles $\mathitbf{e}_{\ell}$ and $\mathitbf{e}_{r}$. The
relation to the epipolar geometry developed in section 5 is that
$\mathitbf{q}_{\ell}$ and $\mathitbf{p}_{\ell}$ are on the same epipolar line;
hence, in addition to
$\mathitbf{q}_{r}^{\scriptscriptstyle\top}\mathitbf{E}\mathitbf{q}_{\ell}=0$,
as in (25), it is true that
$\mathitbf{q}_{r}^{\scriptscriptstyle\top}\mathitbf{E}\mathitbf{p}_{\ell}=0$.
The formulation (28) makes an approximate correspondence between points
$\mathitbf{p}_{\ell}$ and $\mathitbf{p}_{r}$, which is corrected by parallax
functions $t_{\ell}(s)$ and $t_{r}(s)$. The common parameter $s$ is the signed
orthogonal distance of $\bar{\mathitbf{q}}$ from the plane $\mathcal{P}$ which
passes through the fixation point $\bar{\mathitbf{p}}_{0}$. This
representation makes it possible, in principle, to estimate a ‘local’
Cyclopean depth map $S(\mathitbf{p}_{c};\beta,\rho)$ at each fixation point,
where $\mathitbf{p}_{c}$ ranges over the Cyclopean field of view.

The decomposition (28) has three important properties. Firstly, the unknown
parallax variables are scalars; there is no need to consider horizontal and
vertical disparities separately. Secondly, each image correspondence is
parameterized by a single variable $s$, which has a direct interpretation as a
Euclidean distance in the scene. Thirdly, for points close to the fixation
plane, the predictions $\mathitbf{p}_{\ell}$ and $\mathitbf{p}_{r}$ will be
close to $\mathitbf{q}_{\ell}$ and $\mathitbf{q}_{r}$ respectively. In
particular, the predicted correspondence will be exact if the point
$\bar{\mathitbf{q}}$ lies in the fixation plane; $t_{\ell}(0)=t_{r}(0)=0$. The
decomposition (28) will now be described in detail.

The fixation plane $\mathcal{P}$, by definition, passes through the fixation
point $\bar{\mathitbf{p}}_{0}$, and is perpendicular to the Cyclopean gaze
direction. Hence the plane has an outward normal vector $\mathitbf{v}$, as
defined in (8). The plane consists of scene points in the set

$\mathcal{P}=\Bigl{\\{}\bar{\mathitbf{p}}:\mathitbf{v}^{\scriptscriptstyle\top}(\bar{\mathitbf{p}}-\bar{\mathitbf{p}}_{0})=0\Bigr{\\}}.$
(29)

The orthogonal distance from $\mathcal{P}$ to the Cyclopean origin is equal to
the range, $\rho$, of the fixation point, as defined in (7). The orthogonal
distance from $\mathcal{P}$ to the scene point $\bar{\mathitbf{q}}$ will be
$s$; hence

$\displaystyle\rho$
$\displaystyle=\mathitbf{v}^{\scriptscriptstyle\top}\bar{\mathitbf{p}}_{0}$
(30) $\displaystyle s$
$\displaystyle=\mathitbf{v}^{\scriptscriptstyle\top}(\bar{\mathitbf{q}}-\bar{\mathitbf{p}}_{0}).$
(31)

The range, $\rho$, is strictly positive, whereas $s$ is negative, positive or
zero, according to whether $\bar{\mathitbf{q}}$ is closer than, further than,
or in the plane $\mathcal{P}$, respectively. Note that $s$ represents the
_structure_ of the scene with respect to $\mathcal{P}$. Equations (30) and
(31) can now be used to decompose the Cyclopean depth $z_{c}$ of the point
$\bar{\mathitbf{q}}$ as follows;

$z_{c}=\mathitbf{v}^{\scriptscriptstyle\top}\bar{\mathitbf{q}}=\rho+s.$ (32)

The Cyclopean ray through $\bar{\mathitbf{q}}$ intersects the fixation plane
at $\bar{\mathitbf{p}}$. Hence the Cyclopean coordinates of
$\bar{\mathitbf{q}}$ can be expressed as
$z_{c}\mathitbf{p}_{c}=\mathitbf{R}\bar{\mathitbf{q}}$, where
$\mathitbf{p}_{c}$ has been normalized such that $(\mathitbf{p}_{c})_{3}=1$,
and $\mathitbf{R}$ encodes the orientation of the Cyclopean eye (defined by
the angle $\beta$, cf. equation 10). The scene-coordinates of points on the
corresponding visual ray, parameterized by $z_{c}$, can be obtained by
inverting this equation. In particular, the intersection $\bar{\mathitbf{p}}$
of the ray with the fixation plane $\mathcal{P}$ can be obtained, as can the
original scene-point $\bar{\mathitbf{q}}$;

$\displaystyle\bar{\mathitbf{p}}$
$\displaystyle=\rho\mathitbf{R}^{\scriptscriptstyle\top}\\!\mathitbf{p}_{c}$
(33) $\displaystyle\bar{\mathitbf{q}}$
$\displaystyle=z_{c}\mathitbf{R}^{\scriptscriptstyle\top}\\!\mathitbf{p}_{c}.$
(34)

These two points will now be projected into the left image,
$\mathcal{I}_{\ell}$, and the difference $t_{\ell}(s)\mathitbf{d}_{\ell}$
between the two projections will be evaluated. The analogous derivation
applies to the other image, $\mathcal{I}_{r}$, with subscripts ‘$\ell$’ and
‘$r$’ exchanged. The left coordinates are

$\displaystyle\rho_{\ell}\mathitbf{p}_{\ell}$
$\displaystyle=\rho\mathitbf{R}_{\ell}\mathitbf{R}^{\scriptscriptstyle\top}\\!\mathitbf{p}_{c}+\tfrac{1}{2}\mathitbf{e}_{\ell}$
(35) $\displaystyle z_{\ell}\mathitbf{q}_{\ell}$
$\displaystyle=z_{c}\mathitbf{R}_{\ell}\mathitbf{R}^{\scriptscriptstyle\top}\\!\mathitbf{p}_{c}+\tfrac{1}{2}\mathitbf{e}_{\ell}$
(36)

where
$\tfrac{1}{2}\mathitbf{e}_{\ell}=\mathitbf{R}_{\ell}(-\mathitbf{c}_{\ell})$,
as the left, right and Cyclopean optical centres are collinear. Note that the
image-points are normalized such that
$(\mathitbf{p}_{\ell})_{3}=(\mathitbf{q}_{\ell})_{3}=1$, as in the case of
$\mathitbf{p}_{c}$. It is necessary to consider the third components of the
vectors
$\mathitbf{R}_{\ell}\mathitbf{R}^{\scriptscriptstyle\top}\\!\mathitbf{p}_{c}$
and $\tfrac{1}{2}\mathitbf{e}_{\ell}$ in (35) and (36) respectively; these are

$\displaystyle\lambda_{\ell}$
$\displaystyle=(\mathitbf{R}_{\ell}\mathitbf{R}^{\scriptscriptstyle\top}\mathitbf{p}_{c})_{3}=x_{c}\sin(\beta_{\ell}-\beta)+\cos(\beta_{\ell}-\beta)$
(37) $\displaystyle\mu_{\ell}$
$\displaystyle=(\tfrac{1}{2}\mathitbf{e}_{\ell})_{3}=\tfrac{1}{2}\sin\beta_{\ell}.$
(38)

The actual image point $\mathitbf{q}_{\ell}$ will now be decomposed into a sum
of the predicted point $\mathitbf{p}_{\ell}$, plus a scalar _parallax_ in the
direction of a unit vector $\mathitbf{d}_{\ell}$. The vector
$\mathitbf{d}_{\ell}$ should be in the direction of $\mathitbf{q}_{\ell}$ with
respect to the epipole $\mathitbf{e}_{\ell}$. Furthermore, the vector
$\mathitbf{d}_{\ell}$ must lie in the image plane,
$(\mathitbf{d}_{\ell})_{3}=0$. However, it is desirable to avoid defining
$\mathitbf{d}_{\ell}$ from
$\mathitbf{p}_{\ell}-\tfrac{1}{2}\mathitbf{e}_{\ell}/\mu_{\ell}$, because
$\mu_{\ell}=0$ whenever $\beta_{\ell}=0$, as is clear from equation (38).
Hence it is better to use

$\mathitbf{d}_{\ell}=\bigl{(}\mu_{\ell}\mathitbf{p}_{\ell}-\tfrac{1}{2}\mathitbf{e}_{\ell}\bigr{)}/\kappa_{\ell}$
(39)

where
$(\mu_{\ell}\mathitbf{p}_{\ell})_{3}=(\tfrac{1}{2}\mathitbf{e}_{\ell})_{3}=\mu_{\ell}$,
and hence $(\mathitbf{d}_{\ell})_{3}=0$. The scalar
$\kappa_{\ell}=\bigl{|}\mu_{\ell}\mathitbf{p}_{\ell}-\tfrac{1}{2}\mathitbf{e}_{\ell}\bigr{|}$
has been introduced, so that $\mathitbf{d}_{\ell}$ is a unit vector. This is
not strictly necessary, but has the advantage of imposing the original unit of
measurement, $|\mathitbf{b}|$, on the parallax function $t_{\ell}(s)$ that is
associated with each scene point. The function $t_{\ell}(s)$ will now be
derived. It follows from (35) and (36), along with the requirement
$(\mathitbf{p}_{\ell})_{3}=(\mathitbf{q}_{\ell})_{3}=1$, that the depth
variables $\rho_{\ell}$ and $z_{\ell}$ can be expressed as affine functions of
the corresponding Cyclopean parameters $\rho$ and $z_{c}$;

$\displaystyle\rho_{\ell}$ $\displaystyle=\lambda_{\ell}\rho+\mu_{\ell}$ (40)
$\displaystyle z_{\ell}$ $\displaystyle=\lambda_{\ell}z_{c}+\mu_{\ell}$ (41)

where $\lambda_{\ell}$ and $\mu_{\ell}$ are the scalars identified by (37) and
(38). A solution for $\lambda_{\ell}$ can be obtained from either of these
equations, and substituted into the other. The resulting expression can then
be solved for $\mu_{\ell}$;

$\mu_{\ell}=\frac{\rho_{\ell}z_{c}-\rho z_{\ell}}{z_{c}-\rho}.$ (42)

The equation (42) is independent of the point $\bar{\mathitbf{q}}$ that is
associated with depths $z_{c}$ and $z_{\ell}$. The result
$\mathitbf{q}_{\ell}=\mathitbf{p}_{\ell}+t_{\ell}(s)\,\mathitbf{d}_{\ell}$, as
in (28), can now be derived in full. Equations (35) and (36) are used to
express the actual projection $\mathitbf{q}_{\ell}$ as a function of the
predicted projection $\mathitbf{p}_{\ell}$;

$\rho
z_{\ell}\mathitbf{q}_{\ell}=\rho_{\ell}z_{c}\mathitbf{p}_{\ell}-(z_{c}-\rho)\tfrac{1}{2}\mathitbf{e}_{\ell}.$
(43)

The quantity $\rho z_{\ell}\mathitbf{p}_{\ell}$ is now subtracted from both
sides of (43), and the resulting equation is re-arranged as follows;

$\displaystyle\rho z_{\ell}(\mathitbf{q}_{\ell}-\mathitbf{p}_{\ell})$
$\displaystyle=(\rho_{\ell}z_{c}-\rho
z_{\ell})\mathitbf{p}_{\ell}-(z_{c}-\rho)\tfrac{1}{2}\mathitbf{e}_{\ell}$
$\displaystyle=(z_{c}-\rho)\left(\frac{\rho_{\ell}z_{c}-\rho
z_{\ell}}{z_{c}-\rho}\,\mathitbf{p}_{\ell}-\tfrac{1}{2}\mathitbf{e}_{\ell}\right)$
$\displaystyle=(z_{c}-\rho)\bigl{(}\mu_{\ell}\mathitbf{p}_{\ell}-\tfrac{1}{2}\mathitbf{e}_{\ell}\bigr{)},$
(44)

where the substitution of $\mu_{\ell}$ has been made with reference to
equation (42). Both sides of (44) are now divided by $\rho z_{\ell}$, and
comparison with (39) leads to

$\displaystyle\mathitbf{q}_{\ell}-\mathitbf{p}_{\ell}$
$\displaystyle=\frac{z_{c}-\rho}{\rho
z_{\ell}}\,\bigl{(}\mu_{\ell}\mathitbf{p}_{\ell}-\tfrac{1}{2}\mathitbf{e}_{\ell}\bigr{)}$
$\displaystyle=\frac{\kappa_{\ell}(z_{c}-\rho)}{\rho
z_{\ell}}\,\mathitbf{d}_{\ell}$ $\displaystyle=\frac{\kappa_{\ell}s}{\rho
z_{\ell}}\,\mathitbf{d}_{\ell}$ (45)

where (32) has been used above, to make the substitution $s=z_{c}-\rho$. The
practical problem with (45) is that in addition to the free parameter $s$, the
variable $z_{\ell}$ is apparently unknown. This is resolved by making the
substitution $z_{\ell}=\lambda_{\ell}(\rho+s)+\mu_{\ell}$, which follows from
(32) and (41). Therefore, if $\mathitbf{p}_{\ell}$ is added to both sides of
(45), then the result is

$\displaystyle\mathitbf{q}_{\ell}=\mathitbf{p}_{\ell}+t_{\ell}(s)\,\mathitbf{d}_{\ell}$
(46) with $\displaystyle
t_{\ell}(s)=\frac{\kappa_{\ell}(s/\rho)}{\lambda_{\ell}(\rho+s)+\mu_{\ell}}.$
(47)

From this it can be seen that the function $s\mapsto t_{\ell}$ given by (47)
is a 1-d projective transformation, from which it may be inferred that
$t_{\ell}$ and $t_{r}$ are also related by a projective transformation. The
analogous definitions are made for $\mathcal{I}_{r}$, with subscripts ‘$\ell$’
and ‘$r$’ exchanged. Equations (45), (46) and (47) can be interpreted as
follows. It is assumed that the Cyclopean coordinates $(\mathitbf{v},\rho)$ of
the fixation point $\bar{\mathitbf{p}}_{0}$ are known. Then, given the
Cyclopean direction $\mathitbf{p}_{c}$ of another point, it is possible to
compute the predicted point $\mathitbf{p}_{\ell}$ (35), as well as the vector
$\kappa_{\ell}\mathitbf{d}_{\ell}$ (39). The scalars $\lambda_{\ell}$ and
$\mu_{\ell}$ are obtained from (37) and (38), respectively. The unknown
parallax, $t_{\ell}(s)$ is proportional to $s/z_{\ell}$; this ratio is the
depth of the scene point $\bar{\mathitbf{q}}_{\ell}$ with respect to the
fixation plane $\mathcal{P}$, divided by the depth of the point with respect
to the left viewing direction.

For points that are on the fixation plane, $s=0$, and therefore it is clear
from (47) that $t_{\ell}(s)=0$. It follows from (28) that
$\mathitbf{q}_{\ell}=\mathitbf{p}_{\ell}$ and
$\mathitbf{q}_{r}=\mathitbf{p}_{r}$. This makes it interesting to consider the
relationship between $\mathitbf{p}_{\ell}$ and $\mathitbf{p}_{r}$. It can be
shown that the points $\mathitbf{p}_{\ell}$ can be mapped onto the
corresponding points $\mathitbf{p}_{r}$ by a projective transformation

$\mathitbf{p}_{r}=\mathitbf{H}\mathitbf{p}_{\ell}.$ (48)

The $3\times 3$ matrix $\mathitbf{H}$ is the _homography_ induced by the
fixation plane $\mathcal{P}$. If $w_{\ell}$ is the perpendicular distance from
$\bar{\mathitbf{c}}_{\ell}$ to $\mathcal{P}$, then the transformation has the
form

$\mathitbf{H}=\mathitbf{R}_{r}\biggl{(}\mathitbf{I}-\frac{\mathitbf{b}\mathitbf{v}^{\scriptscriptstyle\top}}{w_{\ell}}\biggr{)}\mathitbf{R}_{\ell}^{\scriptscriptstyle\top}.$
(49)

In the general case, $s\neq 0$, equations (28), (48) and (49) can be combined,
leading to the well-known ‘plane plus parallax’ decomposition[35, 23]

$\mathitbf{q}_{r}=\mathitbf{H}\mathitbf{p}_{\ell}+t_{r}(s)\mathitbf{d}_{r}.$

The symmetric representation (28) is, however, preferable in the present
context. This is because it encodes the depth map directly in Cyclopean
coordinates, $S(\mathitbf{p}_{c};\beta,\rho)$.

Fig. 6: Geometry of Cyclopean parallax. The fixation plane $\mathcal{P}$ is
defined by the fixation point $\bar{\mathitbf{p}}_{0}$, and is parallel to the
Cyclopean image plane. The point $\mathitbf{p}_{c}$ defines a Cyclopean ray,
which intersects the fixation plane at $\bar{\mathitbf{p}}$, and the scene at
$\bar{\mathitbf{q}}$. The scene point has depth $s$ with respect to
$\mathcal{P}$. The predicted image-projections of $\bar{\mathitbf{q}}$ are at
$\mathitbf{p}_{\ell}$ and $\mathitbf{p}_{r}$. The true projections,
$\mathitbf{q}_{\ell}$ and $\mathitbf{q}_{r}$, are displaced along the
corresponding epipolar lines, $\mathitbf{u}_{\ell}$ and $\mathitbf{u}_{r}$,
respectively. The displacement can be parameterized by $s$, as described in
the text.

##  7 Discussion

Three problems of binocular vision were identified in section 1, concerning
oculomotor parameterization, disparity processing, and scene representation. A
unified geometric treatment of these problems has been given in sections 3–6.
The main psychophysical and physiological findings that relate to each of the
proposed geometric solutions will now be reviewed.

###  7.A Oculomotor Parameterization

As described in section 3, the binocular orientation of the eyes can be
specified, in the visual plane, by the vergence and version angles, $\delta$
and $\epsilon$. Furthermore, these variables are convenient for the
specification of a coordinated eye movements from one fixation point to
another. It was suggested by Hering that the oculomotor system actually
encodes binocular eye movements in terms of vergence and version [18, 6].
Specifically, Hering’s law of _equal innervation_ states that the eyes are
oriented by signals $(\delta,\epsilon)$ where, according to (12,13),
$\beta_{\ell}=\epsilon+\delta/2$ and $\beta_{r}=\epsilon-\delta/2$. Each eye
moves according to the sum of the appropriate vergence and version components
(which may cancel; e.g. if $\epsilon+\delta/2=0$, then both the initial and
final fixation points lie along the visual direction $\mathitbf{v}_{\ell}$).
The existence of physiological mechanisms that encode pure vergence has been
demonstrated in the midbrain [26], and it has been suggested that the
vergence/version decomposition is used to represent the difference between the
current and target fixation points. However, the actual trajectories of large
binocular eye movements are not consistent with the simple vergence/version
decomposition [7]. Furthermore, it has been found that the visual axes can be
widely mis-aligned during rem sleep [40], which seems more consistent with the
independent parameterization of each eye.

It may be that the application of Hering’s law is limited to those situations
in which the existence of a 3-d fixation point can be ensured by the foveal
correspondence of the images. This condition would, for example, distinguish
the vergence response from the disconjugate component of a binocular saccade.
This is because vergence is driven by visual feedback[34] (alignment of the
retinal images), which is not generally available during the course of a
binocular saccade. Likewise, when the eyes are closed, there is no visual
information to ensure the existence of a 3-d fixation point. In summary, the
evidence for Hering’s law of equal innervation is mixed, but it seems clear
that there are situations in which it is not a satisfactory model.

###  7.B Disparity Processing

It is possible to separate the initial estimation of image correspondences
from the _interpretation_ of the resulting disparity field; this distinction
is made in several computational models of stereopsis[24, 13, 25]. It is
supposed, in these models, that the correspondence problem is first solved,
independently of the gaze parameters. The latter are then recovered from the
estimated disparity field[25], and the two types of information are combined,
leading to a 3-d (though not necessarily Euclidean) interpretation of the
scene[14]. This scheme is compatible with the physiological basis of
stereopsis; for example, it has been demonstrated that the initial binocular
mechanisms in area V1 are tuned to absolute disparity[9], as described in
section 1.B. This finding indicates that the low-level mechanisms of
stereopsis do not ‘compensate’ for any disparity that is imposed by the
relative orientation of the eyes.

The biological feasibility of a general solution to the binocular
correspondence problem will now be considered. Individual disparity-tuned
cells, in area V1, typically respond to a small range of absolute disparities,
centred on some preferred value. The distribution of preferred disparities,
over the population of cells, can also be inferred from the experimental
data[1]. This arrangement suggests the following difficulty: It seems likely
that, for any particular scene, and any particular fixation point, a large
proportion of the V1 disparity response may be spurious. This is because the
occurence of the preferred absolute disparity, for a given detector,
effectively depends on the orientation of the eyes, as well as on the scene
structure. Hence the possibility of detecting a ‘false match’ is exacerbated
by the fact that, owing to the variable orientation of the eyes, the true
match may not even be in the range of a given detector.

One way to address this problem is by imposing some prior knowledge about the
typical structure of the scene. For, example, it might be assumed that the
scene is approximately planar in the neighbourhood of the fixation point. Such
a model could then be used to define _sets_ of disparity detectors that are
effectively tuned to the same surface in depth. This is done by computing the
image-to-image mapping induced by the surface model, and comparing it to the
disparity response. Note that this process is entirely consistent with
detectors that respond to absolute disparity, as the scene model does not
influence the output of the individual mechanisms. Rather, the local scene
model is _imposed_ on the collective response.

If this approach is to be effective, then the induced image-to-image mapping
must be appropriately parameterized. It is important to note that an
appropriate parameterization would allow prior knowledge to be imposed on the
gaze parameters, as well as on the scene structure. For example, image-to-
image mappings associated with extreme configurations of the eyes might be
penalized, and some mappings might be excluded altogether (e.g. those
associated with non-intersecting visual axes). It is emphasized that this
approach does not require oculomotor information about the current gaze
parameters; rather, it assumes that a model of gaze-variation can be learned
from the image data. The geometric constraints described in sections 3 and 5
would seem to make this a biologically feasible task. For example, if the
scene is assumed to be approximately perpendicular to the Cyclopean gaze
direction, then the appropriate scene/gaze model has just three parameters;
$(\alpha,\beta,\rho)$. Furthermore, the form of the induced image-to-image
mapping has been exactly described in section 6.

###  7.C Scene Representation

It was shown in section 6 that binocular disparity can conveniently be
measured with respect to the fixation plane, $\mathcal{P}$. This plane passes
through the fixation point, and is orthogonal to the Cyclopean viewing
direction. This construction is commonly used in physiological studies of
stereopsis. In particular, the shape of the disparity tuning profile, with
respect to the depth of the fixation plane, has been used to classify
binocular neurons[32]. Subsequent experiments have suggested that there exists
a continuum of different disparity tuning curves [33], rather than a number of
distinct types. Nonetheless, the continuum of disparity tuning is clearly
organized around the fixation plane; for example, the majority of cells are
tuned to disparities close to that of the plane[10].

The importance of the fixation plane is also reflected in psychophysical
studies of stereopsis. As noted in section 1.C, only those points in space
that are in Panum’s area can be binocularly fused [28]. It has further been
demonstrated, using simple stimuli, that stereo acuity is highest for targets
located close to the fixation plane [28, 4]. However, the representation of
more complex binocular stimuli raises further questions. In particular, it has
been shown that judgment of the _relative_ depth between nearby targets is
much more accurate than judgment of the deviation of a singe target from the
fixation plane [39]. Furthermore, the surface that is perceived in depth is,
in some cases, an _interpolation_ of the point-wise disparity stimulus [27].
These observations suggest that the representation of binocular disparity may
depend on the local scene-structure, as well as on the gaze parameters [15].
The present model is compatible with this approach, and indeed, the fixation
plane $\mathcal{P}(\rho,\mathitbf{v})$ can be interpreted as the zeroth-order
approximation of the local scene structure. It is straightforward to
substitute the first-order model $\mathcal{P}(\rho,\mathitbf{v};\theta,\phi)$,
in which the angles $\theta$ and $\phi$ represent the local surface
orientation, and to repeat the derivation of binocular disparity in section 6.

The integration of the visual information across the larger scene will now be
considered. The representation described in section 6 allows the estimated
viewing distance to be combined directly with the Cyclopean depth map, because
both are measured along the Cyclopean gaze direction, as in equations (30) and
(31). The global structure of the scene could therefore be encoded in a
collection of local depth-maps, of the type described above. Although each
depth-map would be associated with a different fixation point, it would be
geometrically straightforward to combine the encodings, based on the
corresponding gaze parameters.

Finally, it can be argued that the Cyclopean representation is consistent with
the perception of visual space[18]. Human binocular vision results, at least
subjectively, in a single view of the scene. If this synthetic view has a
meaningful centre of projection, then it may be hypothesized that it is
located at the Cyclopean point[29].

###  7.D Conclusions

A Cyclopean parameterization of binocular vision has been developed in detail.
The parameterization has been used to construct the horopter and the epipolar
geometry of a fixating visual system. Furthermore, the effect of the
oculomotor parameters on the binocular disparity field has been described. It
is clear that the interpretation of the disparity field is complicated by the
variable orientation of the eyes. However, it has been argued here that this
complication is minimized by the binocular coordination of the eyes. The
geometric and computational appeal of the Cyclopean representation has been
emphasized, and the biological relevance of the model has been indicated.

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