The literature on vertical disparity is complicated by the fact that several different definitions of the term “vertical disparity” are in common use, often without a clear statement about which is intended or a widespread appreciation of the properties of the different definitions. Here, we examine two definitions of retinal vertical disparity: elevation-latitude and elevation-longitude disparities. Near the fixation point, these definitions become equivalent, but in general, they have quite different dependences on object distance and binocular eye posture, which have not previously been spelt out. We present analytical approximations for each type of vertical disparity, valid for more general conditions than previous derivations in the literature: we do not restrict ourselves to objects near the fixation point or near the plane of regard, and we allow for non-zero torsion, cyclovergence, and vertical misalignments of the eyes. We use these expressions to derive estimates of the latitude and longitude vertical disparities expected at each point in the visual field, averaged over all natural viewing. Finally, we present analytical expressions showing how binocular eye position—gaze direction, convergence, torsion, cyclovergence, and vertical misalignment—can be derived from the vertical disparity field and its derivatives at the fovea.

*η*; it is called inclination by Bishop, Kozak, and Vakkur (1962) and is analogous to the Helmholtz coordinates of Figure 1A. Equivalently, one can project the hemispherical retina onto a plane and take the vertical Cartesian coordinate on the plane,

*y*(Figure 2A). This is usual in the computer vision literature. Since there is a simple one-to-one mapping between these two coordinates,

*y*= tan

*η,*we shall not need to distinguish between them in this paper. The natural definition of vertical disparity within this coordinate system is then the difference between the elevation longitude of the images in the two eyes. Papers that have defined vertical disparity to be differences in either

*η*or

*y*include Garding et al. (1995), Hartley and Zisserman (2000), Longuet-Higgins (1982), Mayhew (1982), Mayhew and Longuet-Higgins (1982), and Read and Cumming (2006).

*latitude*on the sphere ( Figures 2CE and 3BD). This is analogous to the Fick coordinates of Figure 1B. We shall refer to the corresponding vertical coordinate as elevation latitude,

*κ*(see Table A2 in 1 for a complete list of symbols used in this paper). Studies defining vertical disparity as the difference in elevation latitude

*κ*include Barlow, Blakemore, and Pettigrew (1967), Bishop et al. (1962), and Howard and Rogers (2002).

*I*

_{L}and

*I*

_{R}in the left and right retinas, respectively ( Figure 3AB). Figures 3C and 3D show the left and right retinas aligned and superimposed, so that the positions of the images

*I*

_{L}and

*I*

_{R}can be more easily compared. The left (AC) and right-hand panels (BD) of Figure 3 are identical apart from the vertical coordinate system drawn on the retina: Figure 3AC shows elevation longitude

*η,*and Figure 3BD shows elevation latitude

*κ*. The vertical disparity of point P is the difference between the vertical coordinates of its two half-images. For this example, the elevation-longitude vertical disparity of P is

*η*

_{Δ}= −8°, while the elevation-latitude disparity is

*κ*

_{Δ}= −6°.

*H,*elevation

*V,*and torsion

*T*( Figure 4; 1). Thus, in total the two eyes have potentially 6 degrees of freedom. It is convenient to represent these by the mean and the difference between the left and right eyes. Thus, we shall parametrize eye position by the three coordinates of an imaginary cyclopean eye ( Figure 5),

*H*

_{ c},

*V*

_{ c}, and

*T*

_{ c}, and the three vergence angles,

*H*

_{Δ},

*V*

_{Δ}, and

*T*

_{Δ}, where

*H*

_{ c}= (

*H*

_{R}+

*H*

_{L})/2, and

*H*

_{Δ}=

*H*

_{R}−

*H*

_{L}, and so on ( Tables A1 and A2). When we refer below to convergence, we mean the horizontal vergence angle

*H*

_{Δ}. We shall refer to

*V*

_{Δ}as vertical vergence error or vertical vergence misalignment. We call

*V*

_{Δ}a misalignment because, in order for the two eyes' optic axes to intersect at a single fixation point,

*V*

_{Δ}must be zero, and this is empirically observed to be usually the case.

*T*

_{Δ}the cyclovergence. Non-zero values of

*T*

_{Δ}mean that the eyes have rotated in opposite directions about their optic axes. This occurs when the eyes look up or down: if we specify the eyes' position in Helmholtz coordinates, moving each eye to its final position by rotating through its azimuth

*H*about a vertical axis and then through the elevation

*V*about the interocular axis, we find that in order to match the observed physical position of each eye, we first have to apply a rotation

*T*about the line of sight. If

*V*> 0, so the eyes are looking down, this initial torsional rotation will be such as to move the top of each eyeball nearer the nose, i.e., incyclovergence. Note that the sign of the cyclovergence depends on the coordinate system employed; if eye position is expressed using rotation vectors or quaternions, converged eyes excycloverge when looking downward (Schreiber, Crawford, Fetter, & Tweed, 2001).

*T*

_{ c}as cycloversion. Non-zero values of

*T*

_{ c}mean that the two eyes are both rotated in the same direction. This happens, for example, when the head tilts to the left; both eyes counter-rotate slightly in their sockets so as to reduce their movement in space, i.e., anti-clockwise as viewed by someone facing the observer (Carpenter, 1988).

*V*

_{Δ}is usually zero. It is also observed that for a given elevation, gaze azimuth, and convergence, the torsion of each eye takes on a unique value, which is small and proportional to elevation (Tweed, 1997c). Thus, out of the 6 degrees of freedom, it is a reasonable approximation to consider that the visual system uses only 3:

*H*

_{c},

*V*

_{c}, and

*H*

_{Δ}, with

*V*

_{Δ}= 0, and cycloversion

*T*

_{c}and cyclovergence

*T*

_{Δ}given by functions of

*H*

_{c},

*V*

_{c}, and

*H*

_{Δ}. Most treatments of physiological vertical disparity have assumed that there is no vertical vergence misalignment or torsion:

*V*

_{Δ}=

*T*

_{Δ}=

*T*

_{c}= 0. We too shall use this assumption in subsequent sections, but we start by deriving the most general expressions that we can. The expressions given in Table C2 assume that all three vergence angles are small but not necessarily zero. This enables the reader to substitute in realistic values for the cyclovergence

*T*

_{Δ}at different elevations (Minken & Van Gisbergen, 1994; Somani, DeSouza, Tweed, & Vilis, 1998; Van Rijn & Van den Berg, 1993). The expressions in Table C2 also assume that the interocular distance is small compared to the distance to the viewed object. If the eyes are fixating near object P, then the small vergence approximation already implies this small baseline approximation, since if P is far compared to the interocular separation, then both eyes need to take up nearly the same posture in order to view it. While Porrill et al. (1990) extended the results of Mayhew and Longuet-Higgins (1982) to include cyclovergence, we believe that this paper is the first to present explicit expressions for two-dimensional retinal disparity that are valid all over the visual field and which allow for non-zero vertical vergence misalignment and cycloversion as well as cyclovergence.

*I*/

*S, H*

_{Δ},

*T*

_{Δ}, and

*V*

_{Δ}are all small, while the vertical disparity expressed as the difference in elevation-latitude coordinates is

*I*/

*R, H*

_{Δ},

*T*

_{Δ}, and

*V*

_{Δ}are all small.

*α*

_{ c},

*η*

_{ c}) represent the visual direction of the viewed object P in the azimuth-longitude/elevation-longitude coordinate system shown in Figure 3AC, while (

*α*

_{ c},

*κ*

_{ c}) represent visual direction in the azimuth-longitude/elevation-latitude system of Figure 3BD. (

*α*

_{ c},

*η*

_{ c}) or (

*α*

_{ c},

*κ*

_{ c}) specify the position of P's image on an imaginary cyclopean retina midway between the two real eyes, with gaze azimuth

*H*

_{c}, elevation

*V*

_{c}, and torsion

*T*

_{ c}.

*S*and

*R*both represent the distance to the viewed object P.

*R*is the distance of P from the cyclopean point midway between the eyes.

*S*is the length of the component along the direction of cyclopean gaze ( Figure 5). These are simply related by the following equation:

*I*/

*S, I*/

*R, H*

_{Δ},

*V*

_{Δ}, and

*T*

_{Δ}are all small, and they are correct to first order in these terms. However, they make no assumptions about

*α*

_{ c},

*η*

_{ c},

*κ*

_{ c},

*H*

_{ c},

*V*

_{ c}, and

*T*

_{ c}. They are thus valid over the entire retina, not just near the fovea, and for all cyclopean eye positions. Under this small-vergence approximation, the total vertical disparity is the sum of four terms, respectively proportional to one of four possible sources of disparity: (i) the

*interocular separation*as a fraction of object distance,

*I*/

*R*or

*I*/

*S,*(ii) the

*horizontal vergence H*

_{Δ}, (iii)

*vertical vergence error V*

_{Δ}, and (iv)

*cyclovergence T*

_{Δ}. Each source of disparity is multiplied by a term that depends on one or both of the components of visual direction (

*α*

_{ c}and

*η*

_{ c}or

*κ*

_{ c}), the gaze azimuth

*H*

_{ c}and the overall torsion

*T*

_{ c}. For example, cyclovergence

*T*

_{Δ}is multiplied by

*α*

_{ c}, and so makes no contribution to vertical disparity on the vertical retinal meridian. None of the four disparity terms explicitly depends on elevation

*V*

_{ c}, although elevation would affect the disparity indirectly, because it determines the torsion according to Donders' law (Somani et al., 1998; Tweed, 1997a).

*H*

_{Δ}is itself a function of object distance

*R*. We shall make this assumption ourselves in the next section. However, this section does

*not*assume that the eyes are fixating the object P, so the three vergence angles

*H*

_{Δ},

*V*

_{Δ}, and

*T*

_{Δ}are completely independent of the object's distance

*R*. Thus, object distance affects disparity only through the explicit dependence on

*R*(or

*S*) in the first term (i). The contribution of the three vergence terms (ii–iv) is independent of object distance, provided that the visual direction and eye posture is held constant. That is, if we move the object away but also increase its distance from the gaze axis such that the object continues to fall at the same point on the cyclopean retina, then the contribution of the three vergence terms to the disparity at that point are unchanged. (If the vergence changed to follow the object as it moved away, then of course this contribution would change.)

*H*

_{c}=

*T*

_{c}=

*T*

_{Δ}=

*V*

_{Δ}= 0), except for vergence,

*H*

_{Δ}, which varies between 0 and 40°, so that elevation-longitude disparity is

*η*

_{Δ}≈ 0.5

*H*

_{Δ}tan(

*α*

_{c})sin(2

*η*

_{c}). Under these circumstances, 10° to the left and 10° up from the fovea, vertical disparity would always be positive, running from 0° to +0.9°. On the opposite side of the retina, 10° right and 10° up, vertical disparity would always be negative, running from 0° to −0.9°. Along the retinal meridians, the vertical disparity would always be zero. Read and Cumming's analysis would lump all these together to report that the range of possible vertical disparity is from −0.9° to +0.9°. In other words, the results of Read and Cumming (2004), like those of Hibbard (2007) and Liu et al. (2008), confound variation in the vertical disparity that is possible at a given retinal location with variation across different locations. Similarly, physiological studies that have investigated tuning to vertical disparity have not reported where in the visual field individual neurons were, making it impossible to relate the tuning of these neurons to ecological statistics. For example, one would expect the range of vertical disparity tuning to be narrower for neurons located directly above or below the fovea than for neurons to the “northwest.” The published physiological literature does not make it possible to examine this prediction.

*T*

_{Δ}, the cyclovergence. According to the extended binocular versions of Listing's law (Minken & Van Gisbergen, 1994; Mok, Ro, Cadera, Crawford, & Vilis, 1992; Somani et al., 1998; Tweed, 1997c), this term depends on elevation and more weakly on convergence. Relative to zero Helmholtz torsion, the eyes twist inward (i.e., top of each eye moves toward the nose) on looking down from primary position and outward on looking up, and this tendency is stronger when the eyes are converged:

*T*

_{Δ}=

*V*

_{c}(Λ +

*MH*

_{Δ}), where Λ and

*M*are constants < 1 (Somani et al., 1998). Humans tend to avoid large elevations: if we need to look at something high in our visual field, we tilt our head upward, thus enabling us to view it in something close to primary position. Thus, cyclovergence remains small in natural viewing, and since both positive and negative values occur, the average is likely to be smaller still. Thus, we can reasonably approximate the mean cyclovergence as zero: 〈

*T*

_{Δ}〉 = 0. This means that the last term in the expressions for both kinds of vertical disparity vanishes.

*V*

_{Δ}. We assume that this is on average zero and independent of gaze azimuth or torsion, so that terms like

*V*

_{Δ}cos

*H*

_{ c}cos

*T*

_{ c}all average to zero. This assumption may not be precisely correct, but vertical vergence errors are likely to be so small in any case that neglecting this term is not likely to produce significant errors in our estimate of mean vertical disparity.

*H*

_{Δ}. This is certainly not zero on average. However, part of its contribution depends on sin(

*T*

_{ c}), the sine of the cycloversion. Empirically, this is approximately

*T*

_{ c}∼ −

*V*

_{ c}

*H*

_{ c}/2 (Somani et al., 1998; Tweed, 1997b). So, although cycloversion can be large at eccentric gaze angles, provided we assume that gaze is symmetrically distributed about primary position, then 〈

*V*

_{c}

*H*

_{c}〉 = 0 and so the mean torsion is zero. Again, in the absence of a particular asymmetry, e.g., that people are more likely to look up and left while converging and more likely to look up and right while fixating infinity, we can reasonably assume that 〈

*H*

_{Δ}sin

*T*

_{c}〉 = 0. Equation 1 also contains a term in

*H*

_{Δ}cos

*T*

_{c}. This does not average to zero, but under the assumption that convergence and cycloversion are independent, and that cycloversion is always small, the mean value of this term will be approximately 〈

*H*

_{Δ}〉. Thus, under the above assumptions, Equations 1 and 2 become

*S,*the distance to the surface: 〈sin(

*T*

_{c})cos(

*H*

_{c})/

*S*〉 and 〈sin(

*H*

_{c})/

*S*〉. We now make the reasonable assumption that, averaged across all visual experience, gaze azimuth is independent of distance to the surface. This assumes that there are no azimuthal asymmetries such that nearer surfaces are systematically more likely to be encountered when one looks left, for example. Under this assumption, the term 〈sin(

*H*

_{c})/

*S*〉 averages to zero. Similarly we assume that 〈sin(

*T*

_{c})cos(

*H*

_{c})/

*S*〉 = 0. Thus, the entire term in

*I*/

*S*averages to zero. The vast array of different object distances encountered in normal viewing makes no contribution to the

*mean*elevation-longitude disparity at a particular place on the retina. The mean elevation-longitude disparity encountered at position (

*α*

_{c},

*η*

_{c}) is simply

*H*

_{Δ}〉, but simply left it as an unknown. It does not affect the pattern of expected vertical disparity across the retina but merely scales the size of vertical disparities. Convergence is the only eye-position parameter that we cannot reasonably assume is zero on average, and thus it is the only one contributing to mean vertical disparity measured in elevation longitude.

*S*does not immediately average out. Again, we assume that the terms 〈sin(

*H*

_{c})/

*R*〉 and 〈sin(

*T*

_{c})cos(

*H*

_{c})/

*R*〉 are zero, but this still leaves us with

*δ*:

*R*

_{0}is the radial distance from the origin to the fixation point (or to the point where the optic axes most nearly intersect, if there is a small vertical vergence error), i.e., the distance OF in Figure 5. This is

*H*

_{Δ}(1 −

*δ*)cos

*T*

_{c}〉 averages to 〈

*H*

_{Δ}(1 −

*δ*)〉. In natural viewing, the distributions of

*H*

_{Δ}and

*δ*will not be independent (cf. Figure 6 of Liu et al., 2008). For example, when

*H*

_{Δ}is zero, its smallest value, the fixation distance is infinity, and so

*δ*must be negative or zero. Conversely when the eyes are converged on a nearby object (large

*H*

_{Δ}), perhaps most objects in the scene are usually further away than the fixated object, making

*δ*predominantly positive. In the absence of accurate data, we assume that the average 〈

*H*

_{Δ}

*δ*〉 is close to zero. We then obtain

*κ*

_{Δ}〉 as a function of cyclopean elevation latitude,

*κ*

_{ c}, whereas Equation 6 gave the expected elevation-longitude disparity 〈

*η*

_{Δ}〉 as a function of cyclopean elevation longitude,

*η*

_{ c}. To make it easier to compare the two, we now rewrite Equation 6 to give the expected elevation-longitude disparity 〈

*η*

_{Δ}〉 as a function of cyclopean elevation

*latitude, κ*

_{c}. The expected vertical disparity at (

*α*

_{ c},

*κ*

_{ c}) is thus, in the two definitions,

*H*

_{Δ}〉, and viewing a spherical surface centered on the cyclopean point and passing through fixation. The much more general expressions we have considered reduce to this, because vertical-disparity contributions from eccentric gaze, from the fact that objects may be nearer or further than fixation, from cyclovergence and from vertical vergence all cancel out on average. Thus, they do not affect the

*average*vertical disparity encountered at different points in the visual field (although of course they will affect the

*range*of vertical disparities encountered at each position).

*ξ*is defined as the angle

*E*Ĉ

*V,*where

*E*is the point on the retina whose retinal eccentricity is being calculated,

*C*is the center of the eyeball, and

*V*is the center of the fovea ( Figure 6).

*H*

_{Δ}〉, is not known but must be positive. It does not affect the pattern of vertical disparity expected at different points in the visual field but simply scales it. Figure 7 shows the pattern expected for both types of vertical disparity. In our definition,

*α*

_{ c}and

*κ*

_{ c}represent position on the cyclopean retina, and their signs are thus inverted with respect to the visual field (bottom of the retina represents upper visual field). However, conveniently Equation 13 is unchanged by inverting the sign of both

*α*

_{ c}and

*κ*

_{ c}, meaning that Figure 7 can be equally well interpreted as the pattern across either the cyclopean retina or the visual field.

*H*

_{Δ}〉

*α*

_{ c}

*κ*

_{ c}. Throughout the visual field, the sign of vertical disparity depends on the quadrant. Points in the top-right or bottom-left quadrants of the visual field experience predominantly negative vertical disparity in normal viewing, while points in the top-left or bottom-right quadrants experience predominantly positive vertical disparities. Points on the vertical or horizontal meridian experience zero vertical disparity on average, although the range would clearly increase with vertical distance from the fovea. To our knowledge, no physiological studies have yet probed whether the tuning of disparity-sensitive neurons in early visual areas reflects this retinotopic bias.

*T*

_{ c}=

*T*

_{Δ}=

*V*

_{Δ}= 0) and demonstrate the consequences of this assumption on the properties of the two types of vertical disparity. In this case, the only degrees of freedom that affect disparity are the horizontal rotation of each eye, expressed as the convergence

*H*

_{Δ}and the gaze angle

*H*

_{ c}. From Equation 1, elevation-longitude vertical disparity in the absence of torsion and vertical vergence is

*H*

_{ c}=

*H*

_{Δ}= 0. Elevation-latitude vertical disparity is not in general zero when the eyes are in primary position, except for objects on the midline or at infinite distance. Rotating the eyes into primary position does not affect elevation-latitude disparity because, as noted in the Introduction, horizontal rotations of the eyes cannot alter which line of elevation latitude each point in space projects to; they can only alter the azimuthal position to which it projects. Thus,

*κ*

_{Δ}is independent of convergence, while gaze azimuth simply sweeps the vertical disparity pattern across the retina, keeping it constant in space.

*κ*

_{Δ}depends on gaze azimuth

*H*

_{ c}only through the difference (

*α*

_{ c}−

*H*

_{ c}), representing azimuthal position in

*head-centric*space. As a consequence, elevation-latitude vertical disparity is zero for all objects on the midline (

*X*= 0, meaning that

*α*

_{ c}=

*H*

_{ c}). The elevation-latitude disparity

*κ*

_{Δ}at each point in the cyclopean retina is scaled by the reciprocal of the distance to the viewed object at that point. In contrast, elevation-longitude disparity,

*η*

_{Δ}, is

*independent*of object distance when fixation is on the mid-sagittal plane; it is then proportional to convergence. In Table 1, we summarize the different properties of elevation-longitude and elevation-latitude vertical disparities, under the conditions

*T*

_{ c}=

*T*

_{Δ}=

*V*

_{Δ}= 0 to which we are restricting ourselves in this section.

Vertical disparity defined as: | Properties in the absence of vertical vergence error and torsion ( T _{ c} = T _{Δ} = V _{Δ} = 0) |
---|---|

Difference in retinal elevation
longitude, η _{Δ} | Is zero for objects in plane of gaze. |

Is zero when the eyes are in primary position, for objects at any distance anywhere on the retina. | |

Increases as eyes converge. | |

May be non-zero even for objects at infinity, if the eyes are converged. | |

Is proportional to sine of twice the elevation longitude. | |

Is not necessarily zero for objects on the midsagittal plane. | |

For fixation on midline, is independent of object distance for a given convergence angle. | |

Difference in retinal elevation
latitude, κ _{Δ} | Is zero for objects in plane of gaze. |

Is zero for objects at infinity. | |

Is inversely proportional to object's distance. | |

Is independent of convergence for objects at a given distance. | |

May be non-zero even when eyes are in primary position. | |

Is proportional to sine of elevation latitude. | |

Is zero for objects on the mid-sagittal plane. |

*η*

_{Δ}and

*κ*

_{Δ}depend very differently on convergence and object distance. For midline fixation,

*η*

_{Δ}is proportional to convergence and is independent of object distance, whereas

*κ*

_{Δ}is independent of convergence and is inversely proportional to object distance. However, if we consider only objects close to fixation, then object distance and convergence convey the same information. Under these circumstances the two definitions of vertical disparity become similar. This is shown in Figure 8, which plots the vertical disparity field for a frontoparallel plane. The two left panels show elevation-longitude vertical disparity; the two right panels show elevation-latitude vertical disparity. In the top row, the eyes are viewing a frontoparallel plane at a distance of 60 cm, and in the bottom row, a plane at 10 m. In each case, the eye position is the same: looking 15° to the left, and converged so as to fixate the plane at 60 cm.

*Y*= 0, i.e.,

*η*

_{ c}=

*κ*

_{ c}= 0) and also along a vertical line whose position depends on the gaze angle. For elevation-latitude disparity

*κ*

_{Δ}, this line is simply the line of azimuth longitude

*α*

_{ c}=

*H*

_{ c}, here 15°. This is the retinal projection of the mid-sagittal plane,

*X*= 0. That is, in the absence of torsion or vertical vergence error, elevation-latitude vertical disparity is zero for objects on the midline, independent of their distance or of the convergence angle. For elevation-longitude vertical disparity

*η*

_{Δ}, no such simple result holds. The locus of zero vertical disparity (vertical white line in Figure 8AC) depends on object distance and the eyes' convergence, as well as gaze angle. However, for objects relatively near fixation, these differences are minor, so the locus of zero

*η*

_{Δ}is also close to 15°.

*H*

_{Δ}. Thus, for

*Z*= 10 m, it is already close to zero across the whole retina ( Figure 8D). Elevation-longitude vertical disparity does not have this property. It is zero for objects at infinity only if the eyes are also fixating at infinity, i.e.,

*H*

_{Δ}= 0. Figure 8C shows results for

*H*

_{Δ}= 5.7°, and here the second term in Equation 15 gives non-zero vertical disparity everywhere except along the two retinal meridians.

*epipolar line*(Hartley & Zisserman, 2000).

*line of possible disparities*at a given point in the cyclopean visual field. Figures 9B– 9D show how this differs from an epipolar line. As one moves along an epipolar line ( Figure 9B), not only the two-dimensional disparity, but also the cyclopean position, varies. We shall consider how disparity varies while keeping cyclopean position constant ( Figure 9D).

*T*

_{ c}=

*T*

_{Δ}=

*V*

_{Δ}= 0. Under these circumstances, azimuth-longitude horizontal disparity is ( 2; Table C5)

*α*

_{ c}or (

*α*

_{ c}−

*H*

_{ c}) is 90°, since then horizontal disparity is independent of object distance ( Equation 17). So for example if we are considering an azimuthal direction of 45° (

*α*

_{ c}= 45°) and the eyes are looking off 45° to the right (

*H*

_{ c}= −45°), this expression fails. Apart from this relatively extreme situation, it is generally valid.

*H*

_{Δ}. For objects closer than infinity, the horizontal disparity is smaller, becoming negative for objects nearer than the fixation point. Thus, the green line in Figure 10 terminates at

*α*

_{Δ}=

*H*

_{Δ}. The elevation-latitude vertical disparity at this point in the visual field thus has only one possible sign, either negative or positive depending on the sign of

*κ*

_{ c}(

*H*

_{ c}−

*α*

_{ c}) (since (

*α*

_{Δ}−

*H*

_{Δ}) is always negative). For elevation-latitude vertical disparity, the eye-position parameters have a particularly simple effect on the line of possible disparities. The convergence angle

*H*

_{Δ}controls the intercept on the abscissa, i.e., the horizontal disparity for which the vertical disparity is zero. The gradient of the line is independent of convergence, depending only on the gaze angle. To avoid any confusion, we emphasize that this “disparity gradient” is the rate at which vertical disparity would change if an object slid nearer or further along a particular visual direction, so that its horizontal disparity varied while its position in the cyclopean visual field remained constant. Thus, we are considering the set of two-dimensional disparities that can be produced by a real object for a given binocular eye position. This might theoretically be used by the visual system in solving the stereo correspondence problem if eye position were known. This “disparity gradient” is

*not*the same as the disparity derivatives discussed below (see Discussion section) in the context of deriving eye position given the solution of the correspondence problem, which concern the rate at which vertical disparity changes as a function of visual direction in a given scene.

*κ*

_{ c}and (

*H*

_{ c}−

*α*

_{ c}) are both small (i.e., for objects near the midline and near the plane of regard), the gradient is close to zero. Even quite large changes in horizontal disparity produce very little effect on vertical disparity. In these circumstances, it is reasonable to approximate vertical disparity by its value at the chosen distance, ignoring the gradient entirely. We go through this in the next section.

*R,*or along the gaze direction,

*S,*Figure 5). Figure 8AB versus Figure 8CD, which differ only in the object distance, show how both types of vertical disparity depend on this value. Elevation-latitude disparity does not even depend on the convergence angle

*H*

_{Δ}, making it appear impossible to reconstruct vergence from measurements of elevation-latitude disparity alone.

*R,*as a fraction of the distance to fixation,

*R*

_{0}:

*H*

_{ c}= 90°. This is the case where the eyes are both directed along the interocular axis. Then, the distance to the fixation point is undefined, and we cannot express

*R*as a fraction of it. The case

*H*

_{ c}= 90° is relevant to optic flow, but not to stereo vision. Our analysis holds for all gaze angles that are relevant to stereopsis.

*R*=

*S*sec

*α*

_{ c}sec

*κ*

_{ c}, the two definitions of vertical disparity then become

*δ,*the fractional distance from fixation. However, by assumption, this is much smaller than 1. The vertical disparities are dominated by terms independent of distance; to an excellent approximation, we have

*κ*= tan

*η*cos

*α*to substitute for elevation latitude

*κ*in the expression for elevation-longitude vertical disparity,

*η*

_{Δ}.

*α*

_{ c}and

*κ*

_{ c}).

*δ*as being small in comparison with 1, meaning that horizontal disparity is also independent of object distance. If

*H*

_{ c},

*α*

_{ c}, and

*κ*

_{ c}are large, this is correct. Under these “extreme” conditions (far from the fovea, large gaze angles), horizontal disparity behaves just like vertical disparity. It is dominated by eye position and location in the visual field, with object distance making only a small contribution. However, the conditions of most relevance to stereo vision are those within ∼10° of the fovea, where spatial resolution and stereoacuity is high. In this region, a key difference now emerges between horizontal and vertical disparities: Vertical disparity becomes independent of scene structure, whereas horizontal disparity does not. The terms in Equation 25 that are independent of object distance

*δ*cancel out nearly exactly, meaning that the term of order

*δ*is the only one left. Thus, horizontal disparity becomes

*δ, α*

_{ c},

*κ*

_{ c}all small) and for gaze angles that do not approach 90° (where Equation 25 diverges). Near the fovea, elevation latitude and elevation longitude become indistinguishable (see lines of latitude and longitude in Figure 8). For the near-fixation objects we are considering, therefore, elevation-latitude and elevation-longitude definitions of vertical disparity we derived previously ( Equation 24) become identical and both equal to

*H*

_{ c}∼ 1 and sin

*H*

_{ c}∼

*H*

_{ c}. In Figure 12, we show that our results hold up well at least out to

*H*

_{ c}= 15°. This is likely to cover most gaze angles adopted during natural viewing. We work in the vicinity of the fovea, so retinal azimuth

*α*

_{ c}and elevation

*κ*

_{ c}are also both small. In this case, the distinction between latitude and longitude becomes immaterial. We shall write our expressions in terms of elevation latitude

*κ,*but in this foveal approximation, the same expressions would also hold for elevation longitude

*η*. We shall show how our equations for the vertical disparity field,

*κ*

_{Δ}, can be used to read off gaze angle, convergence, cyclovergence, cycloversion, and vertical vergence.

*I*/

*R*in terms of horizontal vergence

*H*

_{Δ}and the fractional distance of an object relative to fixation,

*δ*. If there is a vertical vergence error

*V*

_{Δ}, then there will not be a fixation point, because gaze rays will not intersect. However, Equation 10 is still valid, with

*δ*interpreted as a fraction of the distance to the point where the gaze rays most closely approach each other. We substitute Equation 10 into our most general expression for vertical disparity, Equation 2, and make the additional approximation that the gaze azimuth

*H*

_{ c}and overall torsion

*T*

_{ c}are both small:

*α*

_{ c}and

*κ*

_{ c}are also small, we find that the lowest order terms are

*T*

_{ c}, vertical disparity now also depends on object distance, through

*δ*. However, this is a third-order term. To first order, the vertical disparity at the fovea measures any vertical vergence error. Thus, we can read off vertical vergence

*V*

_{Δ}simply from the vertical disparity measured at the fovea ( Figure 11):

*κ*

_{Δ α}indicates the first derivative of the vertical disparity

*κ*

_{Δ}with respect to azimuthal position in the visual field,

*α*

_{ c}, holding the visual field elevation

*κ*

_{ c}constant. Similarly,

*κ*

_{Δ ακ}is the rate at which this gradient itself alters as one moves vertically:

*δ,*the fractional difference between the distance to the surface and the distance to fixation. In the Average vertical disparity expected at different positions on the retina section, we were considering a single point in the cyclopean retina, and so

*δ*was just a number: the fractional distance at that point. Since we are now considering changes across the retina,

*δ*is now a function of retinal location,

*δ*(

*α*

_{ c},

*κ*

_{ c}). The first derivatives of

*δ*specify the surface's slant, its second derivatives specify surface curvature, and so on.

*δ*and its derivatives

*δ*

_{ α}, and so on, are assumed to remain small in the vicinity of the fovea.

*α*

_{ c},

*κ*

_{ c},

*δ, δ*

_{ α},

*δ*

_{ κ},

*δ*

_{ αα},

*δ*

_{ ακ},

*δ*

_{ κκ},

*H*

_{ c},

*T*

_{ c},

*H*

_{Δ},

*T*

_{Δ}, and

*V*

_{Δ}are all small.

*κ*

_{Δ κκ}yields

*T*

_{ c}≈ −2

*H*

_{ c}

*δ*

_{ κ}, instead of the correct value of zero. Now Equation 36 was derived assuming small

*H*

_{ c}and

*δ*

_{ κ}, so the misestimate will be small but nevertheless present. In Figure 12, we examine how well our approximations bear up in practice. Each panel shows the eye position parameters estimated from Equations 32– 36 plotted against their actual values, for 1000 different simulations. On each simulation run, first of all a new binocular eye posture was generated, by picking values of

*H*

_{ c},

*T*

_{ c},

*V*

_{ c},

*H*

_{Δ},

*T*

_{Δ}, and

*V*

_{Δ}randomly from uniform distributions. Torsion

*T*

_{ c}, cyclovergence

*T*

_{Δ}, and vertical vergence error

*V*

_{ c}are all likely to remain small in normal viewing and were accordingly picked from uniform distributions between ±2°. Gaze azimuth and elevation were picked from uniform distributions between ±15°. Convergence was picked uniformly from the range 0 to 15°, representing viewing distances from infinity to 25 cm or so. Note that it is not important, for purposes of testing Equations 32– 36, to represent the actual distribution of eye positions during natural viewing but simply to span the range of those most commonly adopted. A random set of points in space was then generated in the vicinity of the chosen fixation point. The

*X*and

*Y*coordinates of these points were picked from uniform random distributions, and their

*Z*coordinate was then set according to a function

*Z*(

*X, Y*), whose exact properties were picked randomly on each simulation run but which always specified a gently curving surface near fixation (for details, see legend to Figure 12). The points were then projected onto the two eyes, using exact projection geometry with no small baseline or other approximations, and their cyclopean locations and disparities were calculated. In order to estimate derivatives of the local vertical disparity field, the vertical disparities of points within 0.5° of the fovea, of which there were usually 200 or so, were then fitted with a parabolic function:

*c*

_{ i}were then used to obtain estimates of vertical disparity and its gradients at the fovea (

*κ*

_{Δ α}=

*c*

_{1}, and so on). Finally, these were used in Equations 32– 36 to produce the estimates of eye position shown in Figure 12.

*T*

_{ c}, which is recovered quite accurately (say to within 10 arcmin) about half the time, but the rest of the time is widely scattered, for the reasons discussed around Equation 38. Nevertheless, overall performance is good, with a median error of <0.3°. This shows that our simple intuitive analytical expressions relating eye position to vertical disparity ( Equations 32– 36) are reliable under most circumstances.

*α*

_{Δ}. In the domain we are considering, horizontal disparity differs from vertical disparity in that it is affected by the viewed scene as well as eye position (recall that Equation 29 showed that, to lowest order, vertical disparity is independent of scene structure). The horizontal disparity itself depends on the distance of the viewed surface relative to fixation,

*δ,*while its first derivatives reflect the surface's slant. Again retaining terms to lowest order, it can be shown that

*δ*

_{ α}is the rate of change of

*δ*as we move horizontally in the visual field,

*δ*

_{ α}= ∂

*δ*/∂

*α*|

_{ κ}. It is a measure of surface slant about a vertical axis, and

*δ*

_{ κ}, defined analogously, reflects surface slant about a horizontal axis.

*δ*is a dimensionless quantity, the fractional distance from the fixation point, but the derivative

*δ*

_{ α}is approximately equal to

*R*

_{ α}/

*R,*where

*R*is the distance to the fixated surface and

*R*

_{ α}is the rate at which this distance changes as a function of visual field azimuth. Thus,

*δ*

_{ α}is the tangent of the angle of slant about a vertical axis, while

*δ*

_{ κ}represents the tangent of the angle of slant about a horizontal axis. We can invert Equation 40 to obtain estimates of surface distance and slant in terms of horizontal disparity and eye position, and then substitute in the eye position parameters estimated from vertical disparity. Note that the estimates of surface slant are unaffected by small amounts of cycloversion,

*T*

_{ c}. This is convenient for us, since cycloversion was the aspect of eye position captured least successfully by our approximate expressions ( Figure 12).

*δ*and its derivatives in terms of horizontal and vertical disparities and their derivatives:

*δ*

_{ α}, Equation 41, is a version of the well-known expressions deriving surface slant from horizontal and vertical size ratios (Backus & Banks, 1999; Backus et al., 1999; Banks et al., 2002; Banks et al., 2001; Kaneko & Howard, 1996, 1997b; Koenderink & van Doorn, 1976; Rogers & Bradshaw, 1993, 1995; Rogers & Cagenello, 1989). “Horizontal size ratio” or HSR is closely related to the rate of change of horizontal disparity as a function of horizontal position in the visual field, whereas “vertical size ratio” reflects the gradient of vertical disparity as a function of vertical position. In the notation of Backus and Banks (1999), for example, which defines HSR and VSR around the fixation point,

*S*= surface slant, so our

*δ*

_{α}= tan(

*S*), and convergence, our

*H*

_{Δ}≈

*κ*

_{Δακ}, is

*μ*. Thus if there is no vertical disparity at the fovea, Equation 41 becomes

*m*about the fixation point, thus adding a term

*mκ*

_{ c}to the vertical disparity field. The vertical disparity at the fovea is still zero, and the only vertical disparity derivative to be affected is

*κ*

_{Δ κ}, which gains a term

*m*. This causes a misestimate of surface slant about a vertical axis:

- Previous derivations have often been couched in terms of head-centric disparity or have assumed that the surfaces viewed have special properties such as being oriented vertically. Our derivations are couched entirely in terms of retinal images and do not assume the viewed surface has a particular orientation. We feel this may provide a more helpful mathematical language for describing the properties of disparity encoding in early visual cortex.
- We present analytical expressions for both elevation-longitude and elevation-latitude vertical disparities that are valid across the entire retina, for arbitrary gaze angles and cycloversion, and for non-zero vertical vergence and cyclovergence. Much previous analysis has relied on parafoveal approximations and has assumed zero vertical vergence, cycloversion, and cyclovergence.
- We present analytical expressions for the average vertical disparity expected at each position in the visual field, up to a scale factor representing the mean convergence.
- Explanations relating the perceptual effects of vertical disparity to disparity gradients have sometimes been contrasted with those based on explicit estimates of eye position (Garding et al., 1995; Longuet-Higgins, 1982; Mayhew & Longuet-Higgins, 1982). This paper is the first to give explicit (though approximate) expressions for 5 binocular eye position parameters in terms of retinal vertical disparity at the fovea. The way in which all 5 eye position parameters can be derived immediately from vertical disparity derivatives has not, as far as we are aware, been laid out explicitly before. Thus, this paper clarifies the underlying unity of gaze-angle and vertical-size-ratio explanations of vertical-disparity illusions such as the induced effect.

L | left eye |
---|---|

R | right eye |

Δ | difference between left and right eye values, e.g.,
convergence angle H _{Δ} = H _{R} − H _{L} |

δ | half-difference between left and right eye values, e.g., half-convergence H _{ δ} = ( H _{R} − H _{L})/2 |

c | cyclopean eye (mean of left and right eye values), e.g., cyclopean gaze angle H _{ c} = ( H _{R} + H _{L})/2 |

I | interocular distance |
---|---|

i | half-interocular distance, i = I/2 |

k, l | integer counters taking on values 1, 2, 3 |

M _{L}, M _{R} | rotation matrix for left and right eyes, respectively |

M _{c} | cyclopean rotation matrix, M _{c} = ( M _{R} + M _{L})/2 |

M _{ δ} | half-difference rotation matrix, M _{ δ} = ( M _{R} − M _{L})/2 |

m | vectors m_{ j} are the three columns of the corresponding rotation matrix M, e.g., m_{c1} = [ M _{c} ^{11} M _{c} ^{21} M _{c} ^{31}]; m_{ δ2} = [ M _{ δ} ^{12} M _{ δ} ^{22} M _{ δ} ^{32}] ( Equation A6) |

H _{L,R,c} | gaze azimuth in Helmholtz system for left, right, and cyclopean eyes |

V _{L,R,c} | gaze elevation in Helmholtz system for left, right, and cyclopean eyes |

T _{L,R,c} | gaze torsion in Helmholtz system for left, right, and cyclopean eyes |

H _{Δ} | horizontal convergence angle |

V _{Δ} | vertical vergence misalignment (non-zero values indicate a failure of fixation) |

T _{Δ} | cyclovergence |

X, Y, Z | position in space in Cartesian coordinates fixed with respect to the head ( Figure A1) |

X ^ | unit vector parallel to the X-axis |

P | vector representing position in space in head-centered coordinates: P = ( X, Y, Z) |

U, W, S | position in space in Cartesian coordinates fixed with respect to the cyclopean gaze. The S-axis is the optic axis of the cyclopean eye (see Figure 5) |

R | distance of an object from the origin. R ^{2} = X ^{2} + Y ^{2} + Z ^{2} = U ^{2} + W ^{2} + S ^{2} (see Figure 5) |

R _{0} | distance of the fixation point from the origin (or distance to the point where the gaze rays most nearly intersect, if the eyes are misaligned so that no exact intersection occurs) |

δ | fractional difference between the fixation distance, R _{0}, and the distance to the object under consideration, R. That is, δ = ( R − R _{0})/ R _{0} |

x | horizontal position on the retina in Cartesian coordinate system ( Figure 2A) |

y | vertical position on the retina in Cartesian coordinate system ( Figure 2A) |

α | azimuth-longitude coordinate for horizontal position on the retina ( Figures 2B and 2C) |

η | elevation-longitude coordinate for vertical position on the retina ( Figures 2B and 2D) |

β | azimuth-latitude or declination coordinate for horizontal position on the retina ( Figures 2D and 2E) |

κ | elevation-latitude or inclination coordinate for vertical position on the retina ( Figures 2C and 2E) |

ξ | retinal eccentricity ( Equation 14) |

*Head-centered coordinate system (X, Y, Z) for object position in space*

*X*-axis points left, the

*Y*-axis upward, and the

*Z*-axis straight ahead of the observer. By definition, the nodal point of the left eye is at (

*X, Y, Z*) = (

*i,*0, 0) and the nodal point of the right eye is at (

*X, Y, Z*) = (−

*i,*0, 0), where

*i*represents half the interocular distance

*I*. The position of a point in space can be described as a vector,

**P**= (

*X, Y, Z*).

*Z*-axis ( Figure A1). We define the torsion here to be zero. To move from this reference state in which all three coordinates are zero to a general posture with torsion, azimuth

*H,*and elevation

*V,*we start by rotating the eyeball about the optic axis by the torsion angle

*T*. Next rotate the eye about a vertical axis, i.e., parallel to the

*Y*-axis, through the gaze azimuth

*H*. Finally rotate the eye about a horizontal, i.e., interocular axis, through the gaze elevation

*V*. We define these rotation angles to be anti-clockwise around the head-centered coordinate axes. This means that we define positive torsion to be clockwise when viewed from behind the head, positive gaze azimuth to be to the observer's left, and positive elevation to be downward.

*V*

_{L}is the Helmholtz elevation of the left eye and

*V*

_{R}that of the right eye.

*V*

_{L}and

*V*

_{R}means that the eyes are misaligned. We refer to this as the vergence error,

*V*

_{R}−

*V*

_{L}. The difference in the Helmholtz gaze azimuths is the horizontal vergence angle,

*H*

_{R}−

*H*

_{L}. Negative values mean that the eyes are diverging.

*δ*:

*M*. So for example if we have a vector that is fixed with respect to the eye, then if the vector is initially

**r**in head-centered coordinates when the eye is in its reference position, it will move to

*M*

**r**when the eye adopts the posture specified by rotation matrix

*M*. An eye's rotation matrix

*M*depends on the eye's elevation

*V,*gaze azimuth

*H,*and torsion

*T*. As above, we use subscripts L and R to indicate the left and right eyes. For the left eye, the rotation matrix is

*M*

_{L}=

*M*

_{VL}

*M*

_{HL}

*M*

_{TL}, where

*V*

_{L},

*H*

_{L}, and

*T*

_{L}are the gaze elevation, gaze azimuth, and torsion of the left eye. The ordering of the matrix multiplication,

*M*

_{L}=

*M*

_{VL}

*M*

_{HL}

*M*

_{TL}, is critical, reflecting the definition of the Helmholtz eye coordinates. Obviously, analogous expressions hold for the right eye. Once again, it will be convenient to introduce the cyclopean rotation matrix, which is defined as the mean of the left- and right-eye rotation matrices:

**m**that are the columns of these matrices:

*M*

^{ kl}indicates the entry in the

*k*th row and

*l*th column of matrix

*M*.

**m**

_{c k}to define a new coordinate system for describing an object's position in space. As well as the head-centered coordinate system (

*X, Y, Z*), we introduce a coordinate system (

*U, W, S*) centered on the direction of cyclopean gaze, as specified by the three Helmholtz angles

*H*

_{ c},

*V*

_{ c}, and

*T*

_{ c}. Whereas

*Z*is the object's distance from the observer measured parallel to the “straight ahead” direction,

*S*is the object's distance parallel to the line of gaze ( Figure 5). The coordinates (

*U, W, S*) are defined by writing the vector

**P**= (

*X, Y, Z*) as a sum of the three

**m**

_{c}vectors:

*Z*-axis in Figure A1. By definition, our retinal coordinate systems are fixed with respect to the retina, not the head, so as the eye rotates in the head, the “horizontal” and “vertical” meridians will in general no longer be horizontal or vertical in space. For this reason we shall call the angle used to specify “horizontal” location the azimuth

*α,*and the angle used to specify “vertical” location, the elevation

*η*. Both azimuth and elevation can be defined as either latitude or longitude. This gives a total of 4 possible retinal coordinate systems ( Figures 2B– 2E). The azimuth-latitude/elevation-longitude coordinate system is the same Helmholtz system we have used to describe eye position (cf. Figure 1A). The azimuth-longitude/elevation-latitude coordinate system is the Fick system (cf. Figure 1B). One can also choose to use latitude or longitude for both directions. Such azimuth-longitude/elevation-longitude or azimuth-latitude/elevation-latitude systems have the disadvantage that the coordinates become ill-defined around the great circle at 90° to the fovea. However, this is irrelevant to stereopsis, since it is beyond the boundaries of vision. The azimuth-longitude/elevation-longitude coordinate system is very simply related to the Cartesian coordinate system, which is standard in the computer vision literature. We can imagine this as a virtual plane, perpendicular to the optic axis and at unit distance behind the nodal point ( Figure 2A). To find the image of a point

**P**, we imagine drawing a ray from point

**P**through the nodal point

**N**and see where this intersects the virtual plane (see Figure 3 of Read & Cumming, 2006). The ray has vector equation

**p**=

**N**+

*s*(

**P**−

**N**), where

*s*represents position along the ray. Points on the retina are given by the vector

**p**=

**N**−

*M*

*xM*

*yM*

*x*and

*y*are the Cartesian coordinates on the planar retina, and the rotation matrix

*M*describes how this plane is rotated with respect to the head. Equating these two expressions for

**p**, we find that

*M*by the unit vectors simply picks off a column of the matrix, e.g.,

*M*

**m**

_{1}. Using this plus the fact that

*M*

*M*

*M*

*y*

_{Δ}=

*y*

_{R}−

*y*

_{L}; (2) elevation-longitude disparity,

*η*

_{Δ}=

*η*

_{R}−

*η*

_{L}; and (3) elevation-latitude disparity,

*κ*

_{Δ}=

*κ*

_{R}−

*κ*

_{L}. In 2, we shall derive expressions for all 3 definitions.

*U, W, S*) in gaze-centered coordinates projects to on the cyclopean retina, in different retinal coordinate systems. Table A4 gives the relationships between location on the retina in different coordinate systems.

U ≈ − Sx _{ c} | W ≈ − Sy _{ c} | R ^{2} = U ^{2} + W ^{2} + S ^{2} = X ^{2} + Y ^{2} + Z ^{2} |
---|---|---|

U ≈ − Stan α _{ c} | W ≈ − Stan κ _{ c}sec α _{ c} | S = Rcos α _{ c}cos κ _{ c} |

U ≈ − Stan β _{ c}sec η _{ c} | W ≈ − Stan η _{ c}sec α _{ c} | S = Rcos β _{ c}cos η _{ c} |

Cartesian ( x, y)
( Figure 2A) | Azimuth longitude,
elevation longitude
( α, η) ( Figure 2B) | Azimuth longitude,
elevation latitude:
( α, κ) (Fick; Figure 2C) | Azimuth latitude,
elevation longitude
( β, η) (Helmholtz; Figure 2D) | Azimuth latitude,
elevation latitude:
( β, κ) ( Figure 2E) | |
---|---|---|---|---|---|

( x, y) | x = tan α y = tan η | x = tan α y = tan κ sec α | x = tan β sec η y = tan η | x = sin β / √ ( cos 2 κ sin 2 β ) y = sin κ / √ ( cos 2 β sin 2 κ ) | |

( α, η) | α = arctan ( x ) η = arctan y | α = α η = arctan ( tan κ cos α ) | α = arctan ( tan β · sec η ) η = η | α = arcsin ( sin β sec κ ) η = arcsin ( sin κ sec β ) | |

( α, κ) | α = arctan ( x ) κ = arctan ( y / x 2 + 1 ) | α = α κ = arctan ( tan η cos α ) | α = arcsin ( sin β cos κ ) κ = κ | α = arcsin ( sin β cos κ ) κ = κ | |

( β, η) | β = arctan ( x / y 2 + 1 ) η = arctan ( y ) | β = arctan ( tan α cos η ) η = η | β = arcsin ( sin α cos κ ) η = arctan ( tan κ sec α ) | β = β η = arcsin ( sin κ cos β ) | |

( β, κ) | β = arctan ( x / √ ( 1 + y 2 ) ) κ = arctan ( y / √ ( 1 + x 2 ) ) | β = arctan ( tan ( α ) cos ( η ) ) κ = arctan ( tan ( η ) cos ( α ) ) | β = arcsin ( cos κ sin α ) κ = κ | β = β κ = arcsin ( cos β sin η ) |

**m**

_{c k}and

**m**

_{ δk}defined in Equation A6. First, the inner product of any difference vector

**m**

_{ δk}with the corresponding cyclopean vector

**m**

_{c k}is identically zero:

*M*

_{c}and

*M*

_{ δ}in terms of the 6 Helmholtz gaze parameters for the two eyes:

*H*

_{L},

*V*

_{L},

*T*

_{L},

*H*

_{R},

*V*

_{R},

*T*

_{R}. We can then use trigonometric identities to re-express these in terms of the cyclopean (half-sum) and vergence (half-difference) equivalents:

*H*

_{ c},

*V*

_{ c},

*T*

_{ c},

*H*

_{ δ},

*V*

_{ δ},

*T*

_{ δ}. Needless to say, this yields extremely complicated expressions. However, we now introduce the first critical approximation of this paper. We assume that differences in eye posture are small. We therefore work to first order in the horizontal vergence

*H*

_{ δ}, the vertical vergence half-error

*V*

_{ δ}, and the half-cyclovergence

*T*

_{ δ}, i.e., we replace terms like cos

*H*

_{ δ}with 1, and we neglect terms in sin

^{2}

*H*

_{ δ}, sin

*H*

_{ δ}·sin

*V*

_{ δ}, and so on. Under these approximations, the 3

**m**

_{c}and the 3

**m**

_{ δ}are approximately orthonormal, i.e.,

**m**

_{c}and an

**m**

_{ δ}vector:

*V*

_{ δ}= 0) and there is no torsion (

*T*

_{ c}=

*T*

_{ δ}= 0), then the only non-zero inner product is

**m**

_{ δ1}.

**m**

_{c3}≈ −

*H*

_{ δ}.

**m**

_{ c1}.

**m**

_{ c2}.

**m**

_{ c3}.

*X*-axis. These are the entries in the top row of the cyclopean rotation matrix, which under the above approximation are

*T*

_{ c}

^{cos}= cos

*T*

_{ c}, and so on.

**P**= (

*X, Y, Z*) in head-centered coordinates. Then, the object projects onto the left retina at a point given by (

*x*

_{L},

*y*

_{L}) in Cartesian coordinates, where ( Equation A9)

**N**

_{ L}is the vector from the origin to the nodal point of the left eye, and

**m**

_{L k}is the

*k*th column of the left eye's rotation matrix

*M*

_{L}. For the left eye, we have

**N**

_{ L}=

*i*

*X*-axis and

*i*is half the interocular distance, while for the right eye,

**N**

_{ R}= −

*i*

**m**

_{L}and

**m**

_{R}, in terms of the half-sum and half-difference between the two eyes:

*i*and

**m**

_{ δ}reversed:

*i*. The other arises from the fact that the eyes may point in different directions and contributes terms in

**m**

_{ δ}. We shall see these two sources emerging in all our future expressions for binocular disparity.

*i*and

**m**

_{ δ}, are small. We carry out a Taylor expansion in which we retain only first-order terms of these quantities. To do this, it is helpful to introduce dummy quantities

**s**and

*j,*where

**m**

_{ δj}=

*ɛ*

**s**

_{ j}and

*i*=

*ɛj,*and the variable

*ɛ*is assumed to be so small that we can ignore terms in

*ɛ*

^{2}:

*x*

_{L}under the small-eye-difference approximation:

*x*

_{R}is the same but with the signs of

*i*and

**m**

_{ δ}reversed. The expressions for

*y*are the same except with subscripts 1 replaced with 2. We can therefore derive the following expressions for the cyclopean position of the image:

**m**

_{ cj}.

**P**= (

*X, Y, Z*) in head-centered coordinates, we move to the gaze-centered coordinate system (

*U, W, S*) in which an object's position is specified relative to the cyclopean gaze direction ( Equation A7):

**m**

_{ δj}with the corresponding cyclopean vector

**m**

_{c j}is identically zero ( Equation B1). Thus, the term

**m**

_{ δ3}.

**P**is independent of the object's distance measured along the cyclopean gaze direction,

*S*:

**m**vectors ( Equations B1– B3), we obtain

*δ*quantities, we obtain the following expressions for an object's horizontal and vertical disparities in Cartesian planar coordinates, expressed as a function of its spatial location in gaze-centered coordinates:

*T*

_{ c}

^{cos}= cos

*T*

_{ c}, and so on.

*U, W, S*). However, this is not very useful, since the brain has no direct access to this. It is more useful to express disparities in terms of (

*x*

_{ c},

*y*

_{ c}), the position on the cyclopean retina or equivalently the visual direction currently under consideration, together with the distance to the object along the cyclopean gaze,

*S*. The brain has direct access to the retinal position (

*x*

_{ c},

*y*

_{ c}), leaving distance

*S*as the sole unknown, to be deduced from the disparity. Then we obtain the following expressions for an object's horizontal and vertical disparities in Cartesian planar coordinates, expressed as a function of its retinal location in Cartesian planar coordinates:

*x*

_{L}given in Equation B12, we have

*α*

_{R}is the same but with the signs of

*i*and

**m**

_{ δ}swapped. Thus, we obtain

**m**, we obtain, in terms of an object's spatial location in gaze-centered coordinates:

*κ*is related to azimuth longitude

*α*and elevation longitude

*η*as

*η*

_{c}in Equation B29 with

*κ*

_{ c}:

*η*

_{Δ}as a function of retinal location in an azimuth-latitude/elevation-longitude coordinate system (

*β, η*). Using tan

*β*= tan

*α*cos

*η,*Equation B29 becomes

*x*

_{L}and

*y*

_{L}given in Equation B12, we have

*M*

_{c},

**m**

_{ δ}, and

**m**

_{ c}, we obtain

*α*

_{ c},

*κ*

_{ c}), we obtain

*T*

_{Δ}, is small. They do not assume anything about the overall cycloversion,

*T*

_{ c}. Cycloversion rotates the eyes in the head, mixing up vertical and horizontal disparities. This can occur when the head tilts over, so that the interocular axis is no longer horizontal with respect to gravity. In the extreme case of

*T*

_{ c}= 90°, the vertical and horizontal directions have actually swapped over (

*y*→

*x*and

*x*→ −

*y*). One can verify from the above results that the expressions for vertical and horizontal disparities also swap over (i.e.,

*x*

_{Δ}with

*T*

_{ c}= 90° is the same as

*y*

_{Δ}with

*T*

_{c}= 0, after replacing

*y*with

*x*and

*x*with −

*y*), a quick “sanity check” on the results.

Horizontal disparity | Most general expressions |
---|---|

In planar Cartesian retinal coordinates as a function of spatial position in gaze-centered coordinates | x Δ ≈ I S ( U S H c sin − H c cos T c cos ) + [ ( U 2 S 2 + 1 ) T c cos − U W S 2 T c sin ] H Δ − [ ( U 2 S 2 + 1 ) H c cos T c sin + U W S 2 H c cos T c cos + W S H c sin ] V Δ − W S T Δ |

In planar Cartesian retinal coordinates as a function of retinal location in planar Cartesian coordinates | x Δ ≈ − ( x c H c sin + H c cos T c cos ) I S + [ ( x c 2 + 1 ) T c cos − x c y c T c sin ] H Δ + [ y c H c sin − ( x c 2 + 1 ) H c cos T c sin − x c y c H c cos T c cos ] V Δ + y c T Δ |

In azimuth longitude, as a function of spatial location in gaze-centered coordinates | α Δ ≈ 1 S 2 + U 2 { [ U H c sin − S H c cos T c cos ] I + [ ( S 2 + U 2 ) T c cos − U W T c sin ] H Δ − [ ( S 2 + U 2 ) H c cos T c sin + U W H c cos T c cos + W S H c sin ] V Δ − W S T Δ } |

In azimuth longitude, as a function of retinal location in azimuth-longitude/ elevation-longitude coordinates | α Δ ≈ − I S cos α c ( H c cos T c cos cos α c + H c sin sin α c ) + [ T c cos − sin α c cos α c tan η c T c sin ] H Δ − [ H c cos T c sin + sin α c cos α c tan η c H c cos T c cos − cos 2 α c tan η c H c sin ] V Δ + cos 2 α c tan η c T Δ |

In azimuth longitude, as a function of retinal location in azimuth-longitude/ elevation-latitude coordinates | α Δ ≈ − I S cos α c ( H c cos T c cos cos α c + H c sin sin α c ) + ( T c cos − sin α c tan κ c T c sin ) H Δ − ( H c cos T c sin + T c cos H c cos sin α c tan κ c − H c sin cos α c tan κ c ) V Δ + ( cos α c tan κ c ) T Δ α Δ ≈ − I R sec κ c ( H c cos T c cos cos α c + H c sin sin α c ) + ( T c cos − sin α c tan κ c T c sin ) H Δ − ( H c cos T c sin + T c cos H c cos sin α c tan κ c − H c sin cos α c tan κ c ) V Δ + ( cos α c tan κ c ) T Δ |

In azimuth latitude, as a function of spatial location in gaze-centered coordinates | β Δ = − I ( T c sin H c cos U W − H c sin U S + T c cos H c cos ( W 2 + S 2 ) ) ( U 2 + W 2 + S 2 ) W 2 + S 2 − ( − S T c cos H Δ + S H c cos T c sin V Δ + W H c sin V Δ + W T Δ ) W 2 + S 2 |

In azimuth latitude, as a function of retinal location in azimuth-latitude/ elevation-longitude coordinates | β Δ = − I R ( T c sin H c cos sin β c sin η c + H c sin sin β c cos η c + T c cos H c cos cos β c ) + ( T c cos cos η c ) H Δ + cos η c ( tan η c H c sin − H c cos T c sin ) V Δ + ( sin η c ) T Δ |

Vertical disparity | Most general expressions |
---|---|

In planar Cartesian retinal coordinates as a function of spatial position in gaze-centered coordinates | y Δ ≈ I S ( W S H c sin + T c sin H c cos ) + [ U W S 2 T c cos − ( W 2 S 2 + 1 ) T c sin ] H Δ + [ U S H c sin − ( W 2 S 2 + 1 ) H c cos T c cos − U W S 2 H c cos T c sin ] V Δ + U S T Δ |

In planar Cartesian retinal coordinates as a function of retinal location in planar Cartesian coordinates | y Δ ≈ − ( y c H c sin − T c sin H c cos ) I S + [ x c y c T c cos − ( y c 2 + 1 ) T c sin ] H Δ − [ x c H c sin + ( y c 2 + 1 ) H c cos T c cos + x c y c H c cos T c sin ] V Δ − x c T Δ |

In elevation longitude, as a function of spatial location in gaze-centered coordinates | η Δ ≈ 1 W 2 + S 2 { [ W H c sin + S T c sin H c cos ] I + [ − ( W 2 + S 2 ) T c sin + U W T c cos ] H Δ − [ ( W 2 + S 2 ) H c cos T c cos + U W H c cos T c sin − U S H c sin ] V Δ + U S T Δ } |

In elevation longitude, as a function of retinal location in azimuth-longitude/ elevation-longitude coordinates | η Δ ≈ cos 2 η c [ T c sin H c cos − H c sin tan η c ] I S + [ tan α c sin η c cos η c T c cos − T c sin ] H Δ − [ H c cos T c cos + tan α c sin η c cos η c H c cos T c sin + tan α c cos 2 η c H c sin ] V Δ − tan α c cos 2 η c T Δ |

In elevation longitude, as a function of retinal location in azimuth-latitude/ elevation-longitude coordinates | η Δ ≈ cos 2 η c [ T c sin H c cos − H c sin tan η c ] I S + [ tan β c sin η c T c cos − T c sin ] H Δ − [ H c cos T c cos + tan β c sin η c H c cos T c sin + tan β c cos η c H c sin ] V Δ − tan β c cos η c T Δ η Δ ≈ I R cos η c cos β c [ T c sin H c cos − H c sin tan η c ] + [ tan β c sin η c T c cos − T c sin ] H Δ − [ H c cos T c cos + tan β c sin η c H c cos T c sin + tan β c cos η c H c sin ] V Δ − tan β c cos η c T Δ |

In elevation latitude, as a function of spatial location in gaze-centered coordinates | κ Δ = I ( H c cos T c cos U W + H c sin W S + T c sin H c cos ( U 2 + S 2 ) ) ( U 2 + W 2 + S 2 ) U 2 + S 2 − ( S T c sin H Δ + S H c cos T c cos V Δ − U H c sin V Δ − U T Δ ) U 2 + S 2 |

In elevation latitude, as a function of retinal location in azimuth-longitude/ elevation-latitude coordinates | κ Δ = I S cos α c cos κ c ( T c cos H c cos sin α c sin κ c − H c sin cos α c sin κ c + T c sin H c cos cos κ c ) − T c sin H Δ cos α c − ( H c cos T c cos cos α c + sin α c H c sin ) V Δ − sin α c T Δ κ Δ = I R ( T c cos H c cos sin α c sin κ c − H c sin cos α c sin κ c + T c sin H c cos cos κ c ) − T c sin H Δ cos α c − ( H c cos T c cos cos α c + sin α c H c sin ) V Δ − sin α c T Δ |

Horizontal disparity | With zero overall cycloversion, T _{c} = 0 |
---|---|

In planar Cartesian retinal coordinates as a function of spatial position in gaze-centered coordinates | x Δ ≈ I S ( U S H c sin − H c cos ) + ( U 2 S 2 + 1 ) H Δ − [ U W S 2 H c cos + W S H c sin ] V Δ − W S T Δ |

In planar Cartesian retinal coordinates as a function of retinal location in planar Cartesian coordinates | x Δ ≈ − ( x c H c sin + H c cos ) I S + ( x c 2 + 1 ) H Δ + ( y c H c sin − x c y c H c cos ) V Δ + y c T Δ |

In azimuth longitude, as a function of spatial location in gaze-centered coordinates | α Δ ≈ 1 S 2 + U 2 { [ U H c sin − S H c cos ] I + ( S 2 + U 2 ) H Δ − [ U W H c cos + W S H c sin ] V Δ − W S T Δ } |

In azimuth longitude, as a function of retinal location in azimuth-longitude/ elevation-longitude coordinates | α Δ ≈ − I S cos α c cos ( H c − α c ) + H Δ + V Δ cos α c tan η c sin ( H c − α c ) + T Δ cos 2 α c tan η c |

In azimuth longitude, as a function of retinal location in azimuth-longitude/ elevation-latitude coordinates | α Δ ≈ − I S cos α c cos ( H c − α c ) + H Δ + V Δ tan κ c sin ( H c − α c ) + T Δ cos α c tan κ c α Δ ≈ − I R sec κ c cos ( H c − α c ) + H Δ + V Δ tan κ c sin ( H c − α c ) + ( cos α c tan κ c ) T Δ |

In azimuth latitude, as a function of spatial location in gaze-centered coordinates | β Δ = − I ( − H c sin U S + H c cos ( W 2 + S 2 ) ) ( U 2 + W 2 + S 2 ) W 2 + S 2 − ( − S H Δ + W H c sin V Δ + W T Δ ) W 2 + S 2 |

In azimuth latitude, as a function of retinal location in azimuth-latitude/ elevation-longitude coordinates | β Δ = − I R ( H c sin sin β c cos η c + H c cos cos β c ) + H Δ T c cos cos η c + V Δ H c sin cos η c tan η c + T Δ sin η c |

Vertical disparity | With zero overall cycloversion, T _{c} = 0 |
---|---|

y Δ ≈ I W S 2 H c sin + H Δ U W S 2 + [ U S H c sin − ( W 2 S 2 + 1 ) H c cos ] V Δ + U S T Δ | |

y Δ ≈ − I S y c H c sin + H Δ x c y c − [ x c H c sin + ( y c 2 + 1 ) H c cos ] V Δ − x c T Δ | |

In elevation longitude, as a function of spatial location in gaze-centered coordinates | η Δ ≈ 1 W 2 + S 2 { [ W H c sin ] I + U W H Δ − [ ( W 2 + S 2 ) H c cos − U S H c sin ] V Δ + U S T Δ } |

In elevation longitude, as a function of retinal location in azimuth-longitude/elevation-longitude coordinates | η Δ ≈ − I S H c sin sin η c cos η c + H Δ tan α c sin η c cos η c − V Δ ( H c cos + tan α c cos 2 η c H c sin ) − tan α c cos 2 η c T Δ |

In elevation longitude, as a function of retinal location in azimuth-latitude/elevation-longitude coordinates | η Δ ≈ − I S H c sin sin η c cos η c + H Δ tan β c sin η c − V Δ ( H c cos + tan β c cos η c H c sin ) − tan β c cos η c T Δ η Δ ≈ − I R H c sin sin η c cos β c + H Δ tan β c sin η c − V Δ ( H c cos + tan β c cos η c H c sin ) − tan β c cos η c T Δ |

In elevation latitude, as a function of spatial location in gaze-centered coordinates | κ Δ = I ( H c cos U W + H c sin W S ) ( U 2 + W 2 + S 2 ) U 2 + S 2 − ( S H c cos V Δ − U H c sin V Δ − U T Δ ) U 2 + S 2 |

In elevation latitude, as a function of retinal location in azimuth-longitude/elevation-latitude coordinates | κ Δ = − I R sin κ c sin ( H c − α c ) − V Δ cos ( H c − α c ) − sin α c T Δ |

Horizontal disparity | For zero torsion and vertical vergence error |
---|---|

x Δ ≈ − ( x c H c sin + H c cos ) I S + ( x c 2 + 1 ) H Δ | |

x Δ ≈ I S 2 ( U H c sin − S H c cos ) + U 2 + S 2 S 2 H Δ | |

In azimuth longitude, as a function of spatial location in gaze-centered coordinates | α Δ ≈ I S 2 + U 2 ( U H c sin − S H c cos ) + H Δ |

In azimuth longitude, as a function of retinal location in azimuth-longitude/elevation-longitude coordinates | α Δ ≈ − I S cos α c cos ( α c − H c ) + H Δ |

In azimuth longitude, as a function of retinal location in azimuth-longitude/elevation-latitude coordinates | (same as above since α _{Δ} is then independent of retinal elevation) |

In azimuth latitude, as a function of spatial location in gaze-centered coordinates | β Δ = I ( H c sin U S − H c cos ( W 2 + S 2 ) ) ( U 2 + W 2 + S 2 ) W 2 + S 2 + S H Δ W 2 + S 2 |

In azimuth latitude, as a function of retinal location in azimuth-latitude/elevation-longitude coordinates | β Δ = − I S cos β c cos η c ( H c sin sin β c cos η c + H c cos cos β c ) + cos η c H Δ |

Vertical disparity | For zero torsion and vertical vergence error |
---|---|

y Δ ≈ y c ( − H c sin I S + x c H Δ ) | |

y Δ ≈ W S 2 ( I H c sin + U H Δ ) | |

In elevation longitude, as a function of spatial location in gaze-centered coordinates | η Δ ≈ W W 2 + S 2 ( I sin H c + U H Δ ) |

In elevation longitude, as a function of retinal location in azimuth-longitude/elevation-longitude coordinates | η Δ ≈ sin η c cos η c ( − I S sin H c + H Δ tan α c ) |

In elevation longitude, as a function of retinal location in azimuth-latitude/elevation-longitude coordinates | η Δ ≈ sin η c ( − I S sin H c cos η c + H Δ tan β c ) |

In elevation latitude, as a function of spatial location in gaze-centered coordinates | κ Δ = I W ( U cos H c + S sin H c ) ( U 2 + W 2 + S 2 ) U 2 + S 2 |

In elevation latitude, as a function of retinal location in azimuth-longitude/elevation-latitude coordinates | κ Δ = I S sin κ c cos κ c cos α c sin ( α c − H c ) κ Δ = I R sin κ c sin ( α c − H c ) |