Miles Hansard

Radu Horaud

Cyclo-rotation Models for Eyes and Cameras

Keywords: Biological control systems, visual system, robot kinematics

The human visual system obeys Listing's law, which means that the cyclo-rotation of the eye (around the line of sight) can be predicted from the direction of the fixation point. It is shown here that Listing's law can be conveniently formulated in terms of rotation matrices. The function that defines the observed cyclo-rotation is derived in this representation. Two polynomial approximations of the function are developed, and the accuracy of each model is evaluated by numerical integration over a range of gaze directions. The error of the most simple approximation, for typical eye movements, is less than half a degree. It is shown that, given a set of calibrated images, the effect of Listing's law can be simulated in a way that is physically consistent with the original camera. This is important for robotic models of human vision, which typically do not reproduce the mechanics of the oculomotor system.

I. INTRODUCTION

This paper explores a kinematic property of human eyemovements, using geometric and numerical methods. The objective is to model the rotation of the human eye in terms of the standard camera model from computer vision [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. This work is motivated by the need to use real image-data in computational models of human vision. It will be shown, in particular, that the images obtained from a standard robotic camera-mounting can be made compatible with the observed orientations of the human eye. This means that subsequent geometric analysis of the images will be consistent with the behaviour of the oculomotor system. The results described here provide a foundation for the further development of both monocular and binocular models of biological vision [START_REF] Koenderink | Geometry of Binocular Vision and a Model for Stereopsis[END_REF]- [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF].

A. Visual Orientation

There are several types of human eye-movement, including those that are used to stabilize the retinal image during motion of the head, and those that are specific to binocular vision [START_REF] Carpenter | Movements of the Eyes[END_REF]. This work, however, is chiefly concerned with saccadic eye movements, which are used to fixate visual targets in the scene.

The fixation of a target-point defines the direction but not the complete orientation of the eye. This is because the orientation includes the 'cyclo-rotation', around the ray that joins the optical centre 1 to the target, as well as the direction. In geometric terms, the direction can be specified by a suitable choice of two spherical angles, such as elevation and azimuth. The orientation involves a third angle, which accounts for the cyclo-rotation, or 'torsion' around the line of sight. Listing's law, as will be explained, describes the observed relationship between visual direction and orientation.

The complete orientation of the eye can be consistently determined from the gaze direction, as follows. Suppose that the visual target is represented, with respect to a reference direction, by elevation and azimuth angles α and β. Then the cyclo-rotation γ can be treated as a function γ(α, β). This is the principle of Donders' law [START_REF] Carpenter | Movements of the Eyes[END_REF], [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF], which states that the actual torsion of the eye is determined by the gaze direction, so that the final orientation is fully determined by the visual target. Note however, that Donders' law does not actually define the function γ(α, β).

B. Ocular Kinematics

Donders' law asserts that torsion is consistently determined by the oculomotor system, but does not make any further predictions. A more specific model can be formulated by noting that, given an initial orientation, the subsequent angles and axes of rotation can be used to predict the final torsion.

Hence the actual cyclo-rotation function γ(α, β) can be derived from a geometric model of visual orientation. This is the principle of Listing's law, which quantifies Donders' law in the following way: There exists a unique reference orientation, such that any other observed orientation of the eye can be obtained by a single rotation, around an axis perpendicular to the reference direction [START_REF] Carpenter | Movements of the Eyes[END_REF], [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF]. It follows that, although the axis of rotation depends on the target direction, it must lie in Listing's plane, which is itself perpendicular to the reference direction. The unique reference orientation of the eye is called primary position. The reference direction, which is determined by experiment, is approximately straightahead. It follows that Listing's plane is approximately parallel to the face [START_REF] Tweed | Geometric Relations of Eye Position and Velocity Vectors during Saccades[END_REF]. Note that Listing's law does not determine the rotational movement of the eye; rather, it states that the observed torsion is compatible with a particular choice of rotation.

Listing's law is applicable if the head is upright and static, and if the fixation point is distant. The torsion, in these conditions, can be predicted with an accuracy of around one degree [START_REF] Carpenter | Movements of the Eyes[END_REF], [START_REF] Tweed | Geometric Relations of Eye Position and Velocity Vectors during Saccades[END_REF]. Listing's law can be extended to describe the case in which the eye moves from a general (i.e. non-primary) position. The rotation axes remain co-planar in this case, but the plane is no longer orthogonal to the initial direction [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF], [START_REF] Tweed | Computing Three-Dimensional Eye Position Quaternions and Eye Velocity from Search Coil Signals[END_REF]. Most of the relevant experimental literature is concerned with primate vision, although support for Listing's law has also been found in other species, including Chameleons [START_REF] Sandor | Chameleon Eye-Position obeys Listing's Law[END_REF]. Donders' law can be justified with respect to the kinematics of the eye: Cyclo-rotation is not a component of visual direction, and so the oculomotor control problem can be simplified by removing this degree of freedom. Experimental evidence suggests that the human eye is not mechanically constrained to behave in this way. For example, irregular torsion is observed in eye movements that occur during sleep [START_REF] Zhu | Binocular Eye Movements Not Coordinated during REM Sleep[END_REF]. The particular form of Listing's law has been justified with respect to both 'motor' and 'visual' criteria. For example, it has been shown that the law is related to the minimization of muscular effort, of total cyclo-rotation, and of binocular disparity [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF], [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF], [START_REF] Hepp | Theoretical Explantions of Listing's Law and their Implications for Binocular Vision[END_REF].

It is useful, as explained above, to think of Donders' law as a rule γ(α, β) that associates a cyclo-rotation angle γ with each visual direction (α, β), such that the complete orientation of the eye is given by the three angles α, β and γ(α, β). In order to derive the form of this function from Listing's law, it is first necessary to define the parameterization of visual direction. Here the angles α and β will be assigned to the elevation and azimuth of the target, respectively. This is the 'Helmholtz' coordinate system [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF], [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF], [START_REF] Westheimer | Kinematics of the Eye[END_REF]- [START_REF] Haslwanter | Mathematics of Three-Dimensional Eye-Rotations[END_REF], in which the visual direction swings in a plane containing the inter-ocular axis; the plane itself rotates around the inter-ocular axis, as shown in figure 1.

This azimuth-elevation scheme is the natural choice, for four related reasons. Firstly, it means that torsion can be measured with respect to a reference plane, defined by the two optical centres and the fixation point, which is intrinsic to the viewing configuration. There is no need for an external definition of 'horizontal' and 'vertical'. Secondly, the elevation of any visual direction can be adjusted without affecting the azimuth. This is desirable with respect to the measurement of eye-movements, because the straight-ahead azimuth is easily defined (by making the visual direction orthogonal to the inter-ocular axis), whereas the straight-ahead elevation is more difficult to determine [START_REF] Carpenter | Movements of the Eyes[END_REF]. Thirdly, the geometry of binocular vision can be more readily described in Helmholtz coordinates, because each elevation plane contains corresponding epipolar lines in the left and right images [START_REF] Hansard | Cyclopean Geometry of Binocular Vision[END_REF]. Fourthly, the definition of γ(α, β) is simple and symmetric in the Helmholtz coordinate system (see section VI). The alternative definition in 'Fick' coordinates [START_REF] Carpenter | Movements of the Eyes[END_REF], [START_REF] Bolina | Kinematics of Eye Movement[END_REF], where α and β are longitude and latitude, is less concise.

C. Robotic Systems

The Helmholtz scheme is also the natural configuration for an active binocular robot-head [START_REF] Murray | Design of Stereo Heads[END_REF], [START_REF] Barreto | General Framework for Selecting World Coordinate Systems in Perspective and Catadioptric Imaging Applications[END_REF]. This is because the left and right pan-motors can be fixed parallel to each other, such that the visual axes are co-planar. It follows that, as the cameras converge, the axes will (ideally) intersect in space. Hence binocular fixation can be mechanically approximated. This type of robot does not obey Listing's law, as will be shown in section V. However, in order to simulate human vision with such a system, the method of section IX can be used to appropriately cyclo-rotate the original images. The present work, for the purpose of simulating human vision, provides an alternative to the mechanical implementation of Listing's law [START_REF] Jenkin | Active Stereo Vision and Cyclotorsion[END_REF], [START_REF] Cannata | Models for the Design of Bioinspired Robot Eyes[END_REF]. Robot-heads of the latter type are, in comparison with the Helmholtz configuration, more difficult to construct and control [START_REF] Murray | Design of Stereo Heads[END_REF].

D. Gaze Tracking

Listing's law is also relevant to the design of gaze-tracking systems [START_REF] Robinson | The Oculomotor Control System: A Review[END_REF], [START_REF] Young | Survey of Eye Movement Recording Methods[END_REF]. For example, greater accuracy can be achieved by accounting for the small angular difference between the line-of-sight (defined in relation to the fovea) and the optical axis of the eye [START_REF] Carpenter | Movements of the Eyes[END_REF], [START_REF] Park | A Real-Time Gaze Position Estimation Method Based on a 3-D Eye-Model[END_REF]. If the latter can be estimated, then the plane containing the two rays can be obtained from Listing's law. The line-of-sight is at a fixed angular offset, in this plane, from the optical axis [START_REF] Villanueva | A Novel Gaze Estimation System with One Calibration Point[END_REF]. Listing's law can also be used to relate the direction of gaze to the projection of the iris, in a calibrated video of the eye [START_REF] Haslwanter | A Theoretical Analysis of Three-Dimensional Eye Position Measurement Using Polar Cross-Correlation[END_REF].

E. Geometric Models

The mathematical expression of Listing's law depends on the representation of the relevant eye-movements. The threedimensional rotation group can be parameterized in several different ways [START_REF] Stuelpnagel | On the Parametrization of the Three-Dimensional Rotation Group[END_REF]. The quaternion [START_REF] Tweed | Computing Three-Dimensional Eye Position Quaternions and Eye Velocity from Search Coil Signals[END_REF], [START_REF] Westheimer | Kinematics of the Eye[END_REF], [START_REF] Haslwanter | Mathematics of Three-Dimensional Eye-Rotations[END_REF], rotation vector [START_REF] Haustein | Considerations on Listing's Law and the Primary Position by Means of a Matrix Description of Eye Position Control[END_REF], [START_REF] Hepp | On Listing's Law[END_REF] and geometric algebra [START_REF] Hestenes | Invariant Body Kinematics: I. Saccadic and Compensatory Eye Movements[END_REF] parameterizations, which are closely related, are particularly well-suited to the modelling of ocular kinematics. It is also possible to represent a rotation by a pair of reflections, leading to a more geometric interpretation of Listing's law [START_REF] Judge | Reflection Makes Sense of Rotation of the Eyes[END_REF]. The present work emphasizes the computational aspects of Listing's law, which is formulated here in terms of rotation matrices. The matrix representation has the advantages of being both mathematically familiar, and computationally convenient. Furthermore, it can immediately be combined with standard projectionmodels from the computer vision literature. This is useful for geometric analysis of the retinal image, in relation to Listing's law, as described in section IX.

F. Novel Contributions

This work describes new results in the representation, approximation and simulation of Listing's law. Primarily, a new derivation of the torsion function γ(α, β) is made in sections IV-VI. There are two objectives here. Firstly, Listing's law will be expressed in terms of rotation matrices, so that it can readily be incorporated into the standard camera model of computer vision [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. Secondly, these rotation matrices will be suitable for the analysis of binocular vision. This means that the results obtained here provide a foundation for further analyses of binocular kinematics [START_REF] Koenderink | Geometry of Binocular Vision and a Model for Stereopsis[END_REF].

Two approximations (one of which has been used previously, e.g. [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF]) of the Listing cyclo-rotation are made in section VII. The advantage of these approximations is that they dispense with the trigonometric functions in the exact formula. This makes it easier to incorporate cyclo-rotation into theoretical models of oculomotor control [START_REF] Van Den | Kinematics of Eye Movement Control[END_REF], [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF]. The results described above are visualized by stereographic projection. The objective of the visualization is to understand the pattern of cyclo-rotation across the visual field.

A procedure for the numerical integration of ocular torsion is introduced in section VIII, where it is used to estimate the accuracy of the approximate cyclo-rotation models described above. This numerical evaluation is complementary to the analytic approach that has been taken elsewhere [START_REF] Hepp | Theoretical Explantions of Listing's Law and their Implications for Binocular Vision[END_REF]. The objective is establish the range of visual directions over which the approximate models can be used. This evaluation also justifies the approximations that have been used in previous work (e.g. [START_REF] Van Den | Kinematics of Eye Movement Control[END_REF], [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF]).

Finally, in section IX, an algorithm for the simulation of Listing's law is presented. The proper way to simulate ocular torsion, given a set of calibrated images, has not been addressed previously. Here the objective is to understand how the cyclo-rotation relates to the extrinsic and intrinsic parameters of the camera [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF].

G. Organization of the Paper

The body of the present work is organized as follows. section II introduces the necessary notation, and defines the primary position of the eye. A useful visualization procedure, based on stereographic projection, is described in section III. Matrix representations of Listing and Helmholtz orientation are developed in sections IV and V, respectively. The Listing cyclo-rotation is derived in section VI. Approximations of the cyclo-rotation are developed in section VII. The accuracy of each approximation is evaluated in section VIII. It is shown in section IX that, given a set of images, the preceding results can be used to simulate the effects of ocular torsion. section X contains the conclusion of the paper.

II. SCENE COORDINATES

Each scene-point p = (x, y, z) ⊤ is represented in a headfixed coordinate system, as illustrated in figure 1. The origin is located at the rotational centre of the left eye, e = (0, 0, 0) ⊤ , which is assumed to coincide with the optical centre [START_REF] Carpenter | Movements of the Eyes[END_REF]. The axes of the coordinate system are {x , y , z }, with x and y parallel to the coronal ('face') plane. The vector x points rightwards along the inter-ocular axis (from the subject's point of view), while y points downwards. The coordinate system is right-handed, and so it follows that z is perpendicular to the coronal plane, and points out into the scene. It will be assumed that, when the eye is in primary position, z is aligned with the visual axis. The plane that is perpendicular to x intersects the eyeball in a great circle which, in primary position, defines the vertical meridian of the eye. The plane that is perpendicular to the y axis defines, in primary position, the horizontal meridian of the eye. It will be assumed that these meridians are fixed on the eye (not in the head).

The target point is p, which lies in a visual direction represented by the unit-vector

v = p d, (1)
where d = |p| is the Euclidean distance to p. The target-point is fixated by a rotation of the primary line of sight, z , onto the target direction, v . This will be expressed as

Rz = v (2)
where R is a 3 × 3 rotation matrix. If the target point lies in either the horizontal plane {x , z }, or in the vertical plane {y , z }, then the eye is in a secondary position after the rotation. The fixation of a generic scene point leaves the eye in a tertiary position [START_REF] Carpenter | Movements of the Eyes[END_REF]. The two eyes fixate a point p in the scene, which is shown here from the back-left. The rotation of one eye (here the left), with optical centre e, will be analyzed with respect to the coordinate system {x , y , z }. The vectors x and z correspond to the inter-ocular and straight-ahead directions, respectively. The vector v corresponds to the visual direction of p from the left optical centre e. Note that the left and right visual directions lie in a common epipolar plane.

It will be assumed, in the following sections, that the eye begins in primary position z , and rotates to fixate the target point p. If the eye begins in a non-primary position, then it can be shown that Listing's plane is rotated in space, by half the angle of the initial direction [START_REF] Tweed | Geometric Relations of Eye Position and Velocity Vectors during Saccades[END_REF], [START_REF] Judge | Reflection Makes Sense of Rotation of the Eyes[END_REF]. This case will not be analyzed here.

Several different coordinate conventions have been used in the literature (e.g. [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF], [START_REF] Tweed | Computing Three-Dimensional Eye Position Quaternions and Eye Velocity from Search Coil Signals[END_REF], [START_REF] Westheimer | Kinematics of the Eye[END_REF]). The system described above was chosen for two reasons. Firstly it is consistent with the computer-vision literature, in which z is usually the optical axis. Secondly, it is convenient in the binocular case, which emphasizes the family of planes containing the inter-ocular axis. Here the convention of x pointing rightwards is kept for these planes.

III. STEREOGRAPHIC PROJECTION

The orientation of the eye is visualized, in the present work, by stereographic projection [START_REF] Carpenter | Movements of the Eyes[END_REF], [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF]. This procedure, which is illustrated in figure 2, is defined as follows. The eyeball is represented by the unit sphere S, located at e. The stereographic centre of projection s 0 = (0, 0, -1) ⊤ is fixed in the head, at the back of of the eyeball. A projection plane T is fixed in space, perpendicular to the z axis. Now consider a landmark on the eyeball, with coordinates (v x , v y , v z ) ⊤ ; this point, together with s 0 , defines a ray. The intersection of the ray with the plane T defines the stereographic projection (ξ, η) of the landmark. For example, if T passes through the centre of S, then the projection is

(ξ, η) = (v x , v y ) 1 + v z . (3)
Any point, other than s 0 , can be mapped to the plane in this way. In particular, the entire forward-hemisphere of visual directions v : |v | = 1, v z ≥ 0 is mapped to the unit disc, centred at (0, 0) ⊤ in T . The orientation of the eye will be visualized by projecting a notional cross, which is marked on the cornea of the eye, and aligned with the pair planes that contain the horizontal and vertical retinal meridians. If the eye rotates to fixate a point p, then the cross will take coordinates (v x , v y , v z ) ⊤ , which can be projected to (ξ, η) via equation [START_REF] Van Den | Kinematics of Eye Movement Control[END_REF]. The procedure is illustrated in fig. 2 where, for clarity, the plane T is shown in front of the eyeball (the principle is the same). The stereographic plane T is perpendicular to the head-fixed z -axis. The eye is fixating a scene-point p; the corresponding vector, v , points into the page. A reference cross, which is fixed to the cornea, is projected onto T . The centre of projection, s 0 , is fixed in the head, at the back of the eye.

The stereographic projection preserves several properties of the underlying rotation group [START_REF] Stuelpnagel | On the Parametrization of the Three-Dimensional Rotation Group[END_REF]. This means that the procedure gives an effective visualization of ocular orientation, as can be seen in figures 4 and 5. For example, consider the 'spokes' of parallel crosses that appear in figure 5a; these correspond to geodesic paths through the space of orientations. The eye moves directly from the central point, with no additional rotation of the cross.

It should be emphasized that, in the present work, stereographic projection is used only for visualization. It is not used to describe the retinal projection of the scene. Indeed, the stereographic and optical centres of projection, s 0 and e respectively, are quite distinct (see fig. 2).

IV. LISTING ORIENTATION

Listing's law states that the actual orientation of the eye, as p is fixated, is consistent with a rotation of the primary line of sight z = (0, 0, 1) ⊤ around a perpendicular axis w . It follows that, for any target p, the corresponding axis w is in the plane {x , y }; this, in the present context, is Listing's plane. The axis is defined as

w = z × v (4) = sin(φ) sin θ, cos θ, 0 ⊤ ( 5
)
where φ is the angle of rotation (from z to v ), and θ is the direction of the axis in the {x , y } plane, as shown in figure 3. The angle θ is in the range 0 ≤ θ < 2π; in particular, if θ = 0, then w is parallel to y , and the rotated line of sight lies in the {x , z } plane. The angle φ can be unambiguously recovered from

sin φ = √ w • w , (6)
x y z p e v θ φ Fig. 3. Listing Coordinates (c.f. fig. 6). The optical centre of the eye is located at point e = (0, 0, 0) ⊤ . The initial gaze direction, with the eye in 'primary position', is along the z axis. In order to fixate the point p, a rotation through angle φ is required. The axis w of the rotation is perpendicular to the {z , v } plane, and in the {x , y } plane. The axis is inclined from the vertical by an angle θ.

because of the physical constraint 0 ≤ φ ≤ π/2.

The axis w and angle φ define the Listing rotation, which will be represented by a 3×3 matrix R L . Rodrigues' equation [START_REF] Stuelpnagel | On the Parametrization of the Three-Dimensional Rotation Group[END_REF] is used to compute R L from w and φ; this results in

R L = x L , y L , z L =   1 -λ cos 2 θ λ cos θ sin θ cos θ sin φ λ cos θ sin θ 1 -λ sin 2 θ -sin θ sin φ -cos θ sin φ sin θ sin φ cos φ   (7)
where x L , y L and z L are the columns of the matrix, and the versine

λ = 1 -z • v (8) = 1 -cos φ (9)
has been introduced for notational convenience. The vector z L is the rotated line of sight, R L z , while {x L , y L } are the rotated retinal axes. Equation ( 7) is a mathematical expression of Listing's law (cf. sec. I-B).

The stereographic coordinates of the Listing rotation are shown in figure 4a, where the concentric circles are parameterized by θ, and the radial spokes are parameterized by φ. The ocular orientations that result from Listing's law are illustrated in figure 5a. Note that the crosses, each of which encodes the ocular torsion, are parallel in this representation.

V. HELMHOLTZ ORIENTATION

It was shown, in the preceding section, that the target visual direction v can be reached by a rotation R L of z , around a variable axis w . In order to quantify the torsion that is associated with this rotation, it will be necessary to establish a reference-system of rotations. These will be defined with respect to the fixed axes {x , y , z }. It is, as described in the introduction, convenient to use the Helmholtz rotation, 4a and4b show the stereographic projection of the Listing and Helmholtz coordinate systems, respectively. The track from the origin shows the path of the visual axis as the eye turns to fixate in the direction θ = 45 • , φ = 65 • . Note that this involves one rotation in 4a, and two rotations in 4b. A spacing of 10 • is used for each family of coordinate lines. rotation matrix

R H = AB (
A =   1 0 0 0 cos α -sin α 0 sin α cos α   ( 11
)
is associated with the elevation angle α of the target, and the rotation matrix

B =   cos β 0 sin β 0 1 0 -sin β 0 cos β   ( 12
)
is associated with the azimuth angle β of the target. The azimuth and elevation are defined as where d is the distance to the target, as in [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. Note that points with positive elevation are above the optical centre (y < 0), while points with positive azimuth are to the right of the optical centre (x > 0). The angular ranges are -π/2 ≤ α ≤ π/2 and -π/2 ≤ β ≤ π/2. The Helmholtz rotation is illustrated in figure 6.

tan α = -v y v z ( 13
)
sin β = v x d (14)
The explicit form of the Helmholtz matrix R H is obtained by substituting [START_REF] Hepp | Theoretical Explantions of Listing's Law and their Implications for Binocular Vision[END_REF] and ( 12) into [START_REF] Zhu | Binocular Eye Movements Not Coordinated during REM Sleep[END_REF], and performing the multiplication; this results in

R H = x H , y H , z H =   cos β 0 sin β sin α sin β cos α -sin α cos β -cos α sin β sin α cos α cos β   (15)
Note that, by analogy with [START_REF] Tweed | Geometric Relations of Eye Position and Velocity Vectors during Saccades[END_REF], z H is the rotated line of sight, 3). The eye turns from the initial direction z , in order to fixate the point p, as in figure 3. This motion is represented as follows: The eye rotates through angle β around the axis y , after which the {x , z } plane is rotated by angle α around axis x . In order to match the final Listing orientation, an initial cyclo-rotation around z , of angle γ, would be required. R H z , while {x H , y H } are the rotated retinal axes.

The stereographic coordinates of the Helmholtz rotation are shown in figure 4b, where the bottom-to-top curves are parameterized by α, and the left-to-right curves are parameterized by β. Furthermore, note that each side-to-side curve can be identified with the intersection of an epipolar plane with the viewing sphere. The orientations that result from the Helmholtz rotation scheme are illustrated in figure 5b. It is important to see that the crosses, by comparison with figure 5a, are no longer parallel. Rather, the 'horizontal' axis of each cross lies in the corresponding epipolar plane.

VI. OCULAR TORSION

It has been shown in sections IV and V that the eye can be directed to a target p by a Listing rotation R L , or by a Helmholtz rotation R H ; hence

z L = z H = v x , v y , v z ⊤ , ( 16
)
where v is the visual direction of p, as in [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. The Listing rotation accounts for the observed orientation of the eye, as well as for the direction. Hence, if Listing's law is obeyed, then the vectors x L and y L are aligned with the horizontal and vertical meridians of the retina. However, the corresponding Helmholtz axes x H and y H are, in general, cyclo-rotated around the common direction v . The discrepancy can be resolved by introducing a third rotation matrix, which represents the torsion around the primary direction z ;

C =   cos γ sin γ 0 -sin γ cos γ 0 0 0 1   . (17)
Note that this cyclo-rotation is anti-clockwise from the subject's point of view (i.e. looking out along the increasing z axis). For example, if γ = π/2, then Cy = x . This is consistent with the sense of θ, which can be seen by comparison of figures 3 and 6.

The Listing rotation R L can now be represented in the Helmholtz system. The appropriate composition of ( 10) and [START_REF] Murray | Design of Stereo Heads[END_REF] gives the equation

R L = ABC . ( 18
)
The torsion angle γ in (17) that solves equation ( 18) must now be obtained in terms of the Euler angles α and β in ( 11) and ( 12). This will be done in two steps; firstly the Listing meridian y L will be expressed in terms of α and β.

Secondly, the inclination of this meridian will be computed in the Helmholtz axes x H and y H . It is straightforward to express y L in terms of the components of z L , by inspection of the second and third columns of the rotation matrix [START_REF] Tweed | Geometric Relations of Eye Position and Velocity Vectors during Saccades[END_REF]. The result, written in the notation of ( 16), is

y L = -v x v y 1 + v z , 1 + v z -v 2 y 1 + v z , -v y ⊤ ( 19
)
where the fact that 19) is the crux of the derivation because, using the equality ( 16), the components of v can be substituted from z H , which is defined in the other coordinate system [START_REF] Hansard | Cyclopean Geometry of Binocular Vision[END_REF]. Now that both y L and y H are known in terms of (α, β), the cosine of the torsion angle is easily obtained;

sin 2 φ = 1 -cos 2 φ = (1 + v z )(1 -v z ) has been used. Equation (
cos γ = y L • y H = cos α + cos 2 α cos β + sin 2 α cos β 1 + v z = cos α + cos β 1 + cos α cos β ( 20
)
where the identity cos 2 α + sin 2 α = 1 has been used. This result is sufficient to define the torsion angle, as |γ| < π/2. However, it is also possible to compute the sine of the angle, as the dot product of y L with the reference axis x H . This is, by a derivation analogous to [START_REF] Cannata | Models for the Design of Bioinspired Robot Eyes[END_REF],

sin γ = y L • x H = sin α sin β 1 + cos α cos β . ( 21
)
The cosine [START_REF] Cannata | Models for the Design of Bioinspired Robot Eyes[END_REF] and sine equations ( 21) can be combined, to give the final result:

tan γ = sin α sin β cos α + cos β . ( 22
)
This equation can be put into a useful half-angle form, via the trigonometric identity tan(µ/2) = tan µ sin µ (tan µ+sin µ).

With reference to ( 21) and ( 22), this gives

tan γ 2 = tan α 2 tan β 2 . ( 23
)
The results ( 22) and ( 23) were originally obtained by Helmholtz [START_REF] Helmholtz | Treatise on Physiological Optics[END_REF], although the present derivation is different. The meaning of the torsion formula ( 22) is illustrated in figure 7a. Here the location of each cross is determined by (α, β), as in figure 5b. However, in each direction (α, β), the eye has been cyclo-rotated by γ(α, β), in accordance with [START_REF] Young | Survey of Eye Movement Recording Methods[END_REF]. The result is that the projected crosses are made mutually parallel, as in the Listing figure 5a. Hence it can be seen that Listing's law has been simulated in the Helmholtz coordinate system.

Equation [START_REF] Park | A Real-Time Gaze Position Estimation Method Based on a 3-D Eye-Model[END_REF] shows that the torsion γ will be zero if either α or β is zero. These 'secondary positions' of the eye correspond to the parallel crosses on the horizontal and vertical meridians of figure 5b. If α and β are both non-zero, then there are two qualitative cases. If α and β have the same sign, then γ is positive, and the resulting cyclo-rotation is anti-clockwise from the subject's point of view [START_REF] Murray | Design of Stereo Heads[END_REF]. These rotations map each cross in the upper-right and lower-left quadrants of figure 5b onto the corresponding cross in figure 7a. If α and β have different signs, then γ is negative, and the rotation is clockwise. These rotations map each cross in the upper-left and lowerright quadrants of figure 5b onto the corresponding cross in figure 7a.

Consider, for example, the fixation of a distant scene-point in the upper mid-sagittal plane (i.e. 'ahead' and 'up'), as in figure 1. This means that the elevation is α > 0, while the left and right azimuths are β ℓ > 0 and β r < 0, respectively. Then it follows from ( 23) that, for each eye obeying Listing's law, the nasal half of the horizontal retinal meridian will turn up out of the Helmholtz elevation plane (as in e.g. [START_REF] Somani | Visual Test of Listing's Law during Vergence[END_REF], [START_REF] Hooge | Visually Evoked Cyclovergence and Extended Listing's Law[END_REF]). Furthermore, a rotation [START_REF] Murray | Design of Stereo Heads[END_REF] of the elevation plane by γ ℓ > 0 around the left visual axis would align it with the horizontal meridian of the left retina. Likewise, a rotation [START_REF] Murray | Design of Stereo Heads[END_REF] of the elevation plane by γ r < 0 around the right visual axis would align it with the horizontal meridian of the right retina. 2The preceding example of 'ahead' and 'up' fixation can also be described in relation to the vertical retinal meridians. In this case, the upper-halves of these meridians turn inwards (nasally) with respect to the head-fixed vertical direction. 3This is commonly referred to as 'intorsion' of the vertical meridians [START_REF] Carpenter | Movements of the Eyes[END_REF].

It should be noted that equation [START_REF] Barreto | General Framework for Selecting World Coordinate Systems in Perspective and Catadioptric Imaging Applications[END_REF] expresses the Listing rotation R L = ABC in relation to head-fixed axes x , y and z . It is straightforward to transform this to an eye-fixed representation R L = C ′ B ′ A ′ . The matrices C ′ , B ′ and A ′ can be obtained from Rodrigues' formula, with axes B ′ A ′ z , A ′ y and x , respectively.

VII. APPROXIMATE MODELS

The Helmholtz torsion equation ( 22) is valid over the hemisphere of gaze angles α, β ∈ [-90 • , 90 • ]. The maximum range of human eye movements is smaller than this, and the typical range is much smaller; an average saccade magnitude of 15 • has been reported [START_REF] Bahill | Most Naturally Occurring Human Saccades have Magnitude of 15 degrees or less[END_REF]. This suggests that a simplified form of the torsion function ( 22) might be valid in practice. Moreover, there are three particular reasons to consider approximate torsion functions; firstly, a better understanding of the exact torsion function can be obtained. Secondly, the ease with which the visual system could represent the function γ(α, β) is established. Thirdly, the approximations can be used to simplify the kinematics of the oculomotor system [START_REF] Van Den | Kinematics of Eye Movement Control[END_REF], [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF].

The Helmholtz torsion function [START_REF] Young | Survey of Eye Movement Recording Methods[END_REF], as noted in section VI, can be expressed in the half-angle form [START_REF] Park | A Real-Time Gaze Position Estimation Method Based on a 3-D Eye-Model[END_REF], with tan(γ/2) on the left-hand side. Recall the truncated Taylor series tan(σ) = σ + σ 3 /3 + O(σ 5 ). The half-angle version of this is therefore

tan σ 2 = σ 2 + σ 3 3 × 2 3 + O(σ 5 ).
The corresponding series for tan(α/2) and tan(β/2) are combined, according to [START_REF] Park | A Real-Time Gaze Position Estimation Method Based on a 3-D Eye-Model[END_REF]. The following half-angle approximation is obtained, having discarded terms of total degree six (i.e. 3 + 3, 1 + 5, 5 + 1) and higher:

tan γ 2 ≈ αβ 4 + α 3 β 48 + αβ 3 48 . (24)
Note that terms of total degree three and five are absent from the expansion of ( 23), owing to the saddle-like symmetries of the function.

The inverse-tangent can be approximated by the formula tan -1 (στ ) = στ +O(σ 3 τ 3 ), where the total truncation degree matches that of [START_REF] Villanueva | A Novel Gaze Estimation System with One Calibration Point[END_REF]. It follows that the second and fourth degree approximations of γ(α, β) are

γ 2 (α, β) = αβ 2 (25)
γ 4 (α, β) = αβ 2 + α 3 β 24 + αβ 3 24 . (26)
The first of these ( 25) has appeared elsewhere (e.g. [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF]). It can be seen that the approximations are much simpler than the original function [START_REF] Young | Survey of Eye Movement Recording Methods[END_REF]; in particular, neither trigonometric nor inverse-trigonometric functions are involved.

The accuracy of the second-degree approximation ( 25) is visualized in figure 7b. If the approximation were perfect, then this plot would be identical to that in figure 7a. It can be seen that, inside the second (60 • ) circle, the two plots are visually indistinguishable. Differences in cyclo-rotation can, however, be clearly seen along the outer (90 • ) circle. The corresponding plot of the fourth-order (26) approximation is visually identical to that in figure 7a. A more formal evaluation will be made in the following section.

VIII. NUMERICAL EVALUATION

The exact and approximate torsion functions will now be compared, over a range of visual directions. This will be done, intuitively, by moving the eye from primary position to a new direction, and evaluating the resulting torsion. This measurement will be averaged over a continuous range of visual directions. The range of directions will be defined by putting an upper limit on the eccentricity r of a visual target. Hence it is natural to choose a set of Listing directions, with 0 ≤ φ ≤ r, which can be converted to Helmholtz coordinates, for use in [START_REF] Haslwanter | A Theoretical Analysis of Three-Dimensional Eye Position Measurement Using Polar Cross-Correlation[END_REF] and [START_REF] Stuelpnagel | On the Parametrization of the Three-Dimensional Rotation Group[END_REF]. The conversion is obtained by comparing the third columns of the matrices [START_REF] Tweed | Geometric Relations of Eye Position and Velocity Vectors during Saccades[END_REF] and [START_REF] Hansard | Cyclopean Geometry of Binocular Vision[END_REF]. If, as before, the eccentricity of the target is in the range Listing orientations (7a), plotted in (α, β) coordinates. Each orientation has been subject to a cyclo-rotation γ(α, β), resulting in mutually parallel stereographic crosses, as in fig. 5a. The approximate cyclo-rotation γ 2 (α, β), defined in equation 25, has been used in (7b). The approximation is good for eccentricities of less than 60 • (i.e. inside the second grey circle), but some tilting of the crosses can be seen in the periphery. See fig. 8 for a visualization of the difference between (7a) and (7b).

0 ≤ φ ≤ π/2, then α(θ, φ) = tan -1 sin θ sin φ cos φ (27)
β(θ, φ) = sin -1 (cos θ sin φ). (28)
It is now straightforward to define a function δ(θ, φ), being the torsion value γ, measured in direction (θ, φ); δ(θ, φ) = γ α(θ, φ), β(θ, φ) .

(

) 29
The functions δ 2 (θ, φ) and δ 4 (θ, φ) are similarly defined from the approximations γ 2 and γ 4 respectively [START_REF] Haslwanter | A Theoretical Analysis of Three-Dimensional Eye Position Measurement Using Polar Cross-Correlation[END_REF][START_REF] Stuelpnagel | On the Parametrization of the Three-Dimensional Rotation Group[END_REF]. The torsion error ǫ k (θ, φ), will also be defined in Listing coordinates. This function measures the difference between the approximate and actual values, δ k and δ respectively;

ǫ k (θ, φ) = δ k (θ, φ) -δ(θ, φ). ( 30
)
The accuracy of the k-th order approximation can be evaluated by integrating a suitable function of ǫ k over a region of visual directions (θ, φ). Such integrals could studied analytically, but a numerical approach will be preferred here. This choice allows the integrand, and the region of integration, to be chosen with more freedom. In particular, the absolute value of the error can be integrated.

The notation |g| r 0 represents the functional that returns the average absolute value of a function g(θ, φ), computed over a range of visual directions 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ r. This is defined as

|g| r 0 = 1 A(r) 2π 0 r 0 sin(φ) g(θ, φ) dφ dθ (31)
where sin(φ) is the scalar Jacobian. The normalization term A(r) is the area of the spherical cap, over which the integration is performed. This term is easily obtained from the formula

A(r) = 2π(1 -cos r). (32)
The integral [START_REF] Somani | Visual Test of Listing's Law during Vergence[END_REF] was evaluated by a standard numerical routine [START_REF] Genz | An Adaptive Algorithm for Numerical Integration over an N -Dimensional Rectangular Region[END_REF]. Table I shows the results of the evaluation. Each functional |g| r 0 was evaluated for all eye-movements up to eccentricity r = 15 • , 30 • , 45 • , 60 • , 75 • . Note that 0 ≤ φ ≤ 30 • represents a 'typical' range of human eye movements, while 0 ≤ φ ≤ 60 • extends to the maximum physical range. The upper limit can be reached, by humans, in the downward direction. The first row of the table gives the mean absolute torsion |δ| r 0 , evaluated in the Helmholtz system. These values, which do not seem to have been computed previously, are useful for gauging the significance of the approximation errors. The second and third rows of the table give the average errors ǫ 2 and ǫ 4 that are associated with torsion approximations γ 2 and γ 4 , respectively [START_REF] Haslwanter | A Theoretical Analysis of Three-Dimensional Eye Position Measurement Using Polar Cross-Correlation[END_REF][START_REF] Stuelpnagel | On the Parametrization of the Three-Dimensional Rotation Group[END_REF]. Hence the relative error of the approximations can be computed from |ǫ k | r 0 |δ| r 0 . The principal conclusion is that the second-order approximation γ 2 (α, β) = 1 2 αβ has an average error of 0.02 • for φ ≤ 30 • , and of 0.4 • for φ ≤ 60 • . As fractions of the average torsion |δ| r 0 , these numbers represent errors of 1.6% and 7.3%, respectively. Hence for typical eye movements (φ ≤ 30 • ) the second order approximation is adequate, given that eyemovements can typically be measured to a precision of around one degree [START_REF] Carpenter | Movements of the Eyes[END_REF]. The table also shows that, for all achievable eye movements (φ ≤ 60 • ), the fourth-order model γ 4 is adequate.

IX. SYNTHETIC TORSION

It will now be shown that the results obtained in sections IV-VII can be used to synthesize the effect of Listing's law on images that have been obtained from a pan-tilt camera mounting. Moreover, this will be done in a way that is physically consistent with the original camera. This means that the synthetic image will match the one that would have been obtained after the corresponding cyclo-rotation of the camera. It is important to ensure physical consistency, so that subsequent algorithms need not distinguish between the original and the synthetic images. This cannot, in general, be achieved simply by rotating the original images around their centre-points. The appropriate image transformation must be defined in relation to the projection matrix of the camera, as shown below. Note that it will usually be more convenient, and faster, to extract features (e.g. points or edges) in the original image, and then to transform the coordinates of the features. This avoids the need to resample or crop the pixel-data.

It will be assumed that the camera and ocular projections can be approximated by the usual pin-hole model [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. Furthermore, it will be assumed that the nodal points of the optics coincide at the fixed rotational centre e = (0, 0, 0) ⊤ of the camera, as in [START_REF] Koenderink | Geometry of Binocular Vision and a Model for Stereopsis[END_REF]. Each scene point q has coordinates (x, y, z) ⊤ , as described in section II. The perspective projection is obtained via the 3 × 3 matrix M , resulting in homogeneous 'eye-coordinates' q E = (x E , y E , z E ) ⊤ , with corresponding pixel-positions x E /z E , y E /z E ) ⊤ . This means, with reference to (2), that

q E = Mq (33) = SR ⊤ q (34)
where R ⊤ is the scene-to-eye rotation, and S performs an affine transformation of the image, to account for the 'intrinsic' parameters of the camera (as described below). Note that the rotation is transposed, for consistency with [START_REF] Koenderink | Geometry of Binocular Vision and a Model for Stereopsis[END_REF]. In particular, it follows from (1) and ( 2) that the projection of the fixated point p is Sz . For a perfect pan-tilt mounting, the rotation R ⊤ would have the form B ⊤ A ⊤ , based on definitions [START_REF] Hepp | Theoretical Explantions of Listing's Law and their Implications for Binocular Vision[END_REF] and [START_REF] Westheimer | Kinematics of the Eye[END_REF]. In practice, the camera matrix M (as in 33) can be estimated by standard methods, and decomposed as follows [START_REF] Hartley | Multiple View Geometry in Computer Vision[END_REF]. The matrices S and R ⊤ (as in 34) are obtained by RQ-factorization4 of M . The first factor can be written as

S =    s 11 s 12 x 0 0 s 22 y 0 0 0 1    . (35)
This matrix contains the pixel-coordinates (x 0 , y 0 ) ⊤ of the principal point (intersection of the optical axis with the image plane), two scale-factors (s 11 , s 22 ), and a parameter s 12 ≈ 0 which allows for a skew between the horizontal and vertical axes of the sensor. In more detail, s 11 = f g x and s 22 = f g y , where f is the focal-distance of the camera, and the scales (g x , g y ) determine the number of pixels per unit distance in the horizontal and vertical dimensions. These intrinsic parameters [START_REF] Curcio | Human Photoreceptor Topography[END_REF] should not be changed by a synthetic cyclo-rotation C ⊤ of the image. Note, however, that the naive procedure C ⊤ q E would imply a camera C ⊤ SR ⊤ , according to the projection model [START_REF] Genz | An Adaptive Algorithm for Numerical Integration over an N -Dimensional Rectangular Region[END_REF]. The matrices C ⊤ and S do not, in general, commute. It follows that S would not be recovered from the RQ-factorization of the implied camera (owing to the uniqueness of the factorization). For this reason, the cyclo-rotated points are properly defined as

q ′ E = SC ⊤ S -1 q E . (36)
This definition, with reference to [START_REF] Genz | An Adaptive Algorithm for Numerical Integration over an N -Dimensional Rectangular Region[END_REF], implies the existence of a synthetic camera matrix

M ′ = SC ⊤ S -1 M = SC ⊤ R ⊤
It is clear from [START_REF] Genz | An Adaptive Algorithm for Numerical Integration over an N -Dimensional Rectangular Region[END_REF] that the matrix M ′ has the RQfactorization S • C ⊤ B ⊤ A ⊤ , and is therefore consistent with the original set of intrinsic parameters [START_REF] Curcio | Human Photoreceptor Topography[END_REF]. In practice, it may be more convenient to work with 'normalized coordinates' S -1 q E , where the effects of the intrinsic parameters have been undone. In this case, the cyclo-rotated and normalized coordinates are simply

q ′′ E = C ⊤ S -1 q E ( 37
)
The product C ⊤ S -1 can be formed first, so that a single affine transformation is applied to the observed feature-coordinates q E . Note that this is a 2-D transformation, depending only on the intrinsic parameters [START_REF] Curcio | Human Photoreceptor Topography[END_REF] and the cyclo-rotation angle γ. In the special case that the upper-left 2×2 block of S is a multiple sI 2 of the identity matrix, the procedure (37) simply translates the principal point (x 0 , y 0 ) ⊤ to (0, 0) ⊤ , before scaling and rotating the feature-coordinates by s and γ, respectively. In general, however, s 11 = s 22 and s 12 = 0, meaning that the transformation (37) should be used.

If the cyclo-rotation angle γ is defined by [START_REF] Young | Survey of Eye Movement Recording Methods[END_REF], then the new coordinates q ′′ E (as well as the Q factor of the implied camera matrix) are subject to Listing's law, as well being physically consistent with the original camera. Other aspects of biological vision, such as the non-uniform distribution of photoreceptors on the retina [START_REF] Curcio | Human Photoreceptor Topography[END_REF], can be modelled by further transformations of the new coordinates. It is also possible to adapt the above model to a spherical projection, which may be more appropriate for the human eye; in this case the observed feature-coordinates are (x E /d, y E /d) ⊤ , where d = |q | is the distance to the point, as in (1).

X. DISCUSSION

It has been shown in sections II-VI that Listing's law can be formulated in terms of rotation matrices. This means that the usual computer vision camera-model [START_REF] Genz | An Adaptive Algorithm for Numerical Integration over an N -Dimensional Rectangular Region[END_REF] can easily be adapted to the human eye, as shown in section IX. Two polynomial approximations of the torsion function γ(α, β) were derived in section VII. A procedure for numerical integration over visual directions was introduced in section VIII. The average cyclo-rotation was computed, and the two approximations were validated. Finally, in section IX, it was shown that Listing's law can be imposed on a suitable set of calibrated images.

There is considerable interest in the relationship between Listing's law and other visual processes, such as stereopsis [START_REF] Koenderink | Geometry of Binocular Vision and a Model for Stereopsis[END_REF], [START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF]. For example; if the binocular fixation point is relatively close, then Listing's law must be modified [START_REF] Mok | Rotation of Listing's Plane during Vergence[END_REF], [START_REF] Van Rijn | Binocular Eye Orientation during Fixation: Listing's Law extended to include Eye Vergence[END_REF]. Future work will include an extension of the present analysis to the binocular case [START_REF] Hansard | Cyclopean Geometry of Binocular Vision[END_REF]. The results presented here, as described in the introduction, make it possible to evaluate such models with respect to real images.

The implications of section VII, with regard to the neural representation of eye-movements, are also interesting. It is clear that the cyclo-rotation angle is a slowly-varying function of visual direction, over the typical range of eye movements. Indeed, the results of section VIII show that the observed cyclo-rotation is effectively proportional the product of the eye's azimuth and elevation. This suggests that Listing's law could be be represented quite directly in the primate oculomotor system [START_REF] Van Den | Kinematics of Eye Movement Control[END_REF].

  Fig.1. The two eyes fixate a point p in the scene, which is shown here from the back-left. The rotation of one eye (here the left), with optical centre e, will be analyzed with respect to the coordinate system {x , y , z }. The vectors x and z correspond to the inter-ocular and straight-ahead directions, respectively. The vector v corresponds to the visual direction of p from the left optical centre e. Note that the left and right visual directions lie in a common epipolar plane.

Fig. 2 .

 2 Fig.2. Stereographic Projection. A schematic diagram of the procedure used to construct figures 4, 5, 7 and 8. The eyeball, which is centred on the point e, is seen from the back-left, as in fig.1. The stereographic plane T is perpendicular to the head-fixed z -axis. The eye is fixating a scene-point p; the corresponding vector, v , points into the page. A reference cross, which is fixed to the cornea, is projected onto T . The centre of projection, s 0 , is fixed in the head, at the back of the eye.

  [START_REF] Zhu | Binocular Eye Movements Not Coordinated during REM Sleep[END_REF] where A represents a rotation around x , and B represents a rotation around y . The two factors are defined as follows. The θ φ (a) Stereographic projection of the Listing coordinate system (θ, φ), cf. fig. 3. The primary direction is mapped to the centre of the disc; the boundary represents targets at φ = 90 • eccentricity. α β (b) Stereographic projection of the Helmholtz coordinate system (α, β), cf. fig. 6. The points on the far left and right represent the intersection of the inter-ocular axis with the eyeball.

Fig. 4 .

 4 Fig.[START_REF] Tweed | Visual-Motor Optimization in Binocular Control[END_REF]. Figures4a and 4bshow the stereographic projection of the Listing and Helmholtz coordinate systems, respectively. The track from the origin shows the path of the visual axis as the eye turns to fixate in the direction θ = 45 • , φ = 65 • . Note that this involves one rotation in 4a, and two rotations in 4b. A spacing of 10 • is used for each family of coordinate lines.

  (a) Listing Orientations, plotted in (θ, φ) coordinates, cf. figure 4a. (b) Helmholtz Orientations, plotted in (α, β) coordinates, cf. figure 4b.

Fig. 5 .

 5 Fig. 5. (a) Stereographic projection of Listing orientations, plotted in polar coordinates (θ, φ). Note that the retinal orientations are mutually parallel in this representation. Solid circles indicate eccentricities of 30 • , 60 • and 90 • . Some crosses are omitted in the central region, for clarity. (b) Helmholtz orientations, plotted in (α, β) coordinates. Note that the retinal orientations are not mutually parallel in this representation. The horizontal meridian of each possible orientation lies in an azimuthal plane.

Fig. 6 .

 6 Fig.6. Helmholtz Coordinates (c.f. fig.3). The eye turns from the initial direction z , in order to fixate the point p, as in figure3. This motion is represented as follows: The eye rotates through angle β around the axis y , after which the {x , z } plane is rotated by angle α around axis x . In order to match the final Listing orientation, an initial cyclo-rotation around z , of angle γ, would be required.

Fig. 7 .

 7 Fig. 7.Listing orientations (7a), plotted in (α, β) coordinates. Each orientation has been subject to a cyclo-rotation γ(α, β), resulting in mutually parallel stereographic crosses, as in fig.5a. The approximate cyclo-rotation γ 2 (α, β), defined in equation 25, has been used in (7b). The approximation is good for eccentricities of less than 60 • (i.e. inside the second grey circle), but some tilting of the crosses can be seen in the periphery. See fig.8for a visualization of the difference between (7a) and (7b).

Fig. 8 .

 8 Fig.8.Visualization of the difference between the actual torsion (7a) and the γ 2 approximation (7b), plotted in (α, β) coordinates. Larger discs represent worse approximations, as can be confirmed by comparing figs. (7a) and (7b). Note that the large errors occur in directions that have high absolute elevation |α| and high absolute azimuth |β|. Errors of less than 0.5 • are not shown; otherwise the disc radii are proportional to ˛γ2 (α, β)γ(α, β) ˛. The horizontal and vertical meridians shown in grey, for reference.

TABLE I ABSOLUTE

 I VALUES OF THE ACTUAL TORSION δ(θ, φ), AND APPROXIMATION ERROR ǫ k (θ, φ), AVERAGED OVER INCREASINGLY LARGE REGIONS OF THE VIEWING SPHERE.

		r = 15 •	r = 30 •	r = 45 •	r = 60 •	r = 75 •
	|δ| r 0	0.314 •	1.280 •	2.978 •	5.596 •	9.630 •
	|ǫ 2 | r 0	0.001 •	0.020 •	0.112 •	0.407 •	1.271 •
	|ǫ 4 | r 0	1.45 • ×10 -5	0.001 •	0.011 •	0.075 •	0.396 •

For comparison with the literature, these are the 'Upper-nasal; Out' cases in Westheimer's tables I and II[START_REF] Westheimer | Kinematics of the Eye[END_REF].

Westheimer's 'Upper-nasal; In' cases[START_REF] Westheimer | Kinematics of the Eye[END_REF].

The decomposition M = RQ has an upper-triangular factor R, and an orthogonal factor Q. The decomposition is unique if, as here, M has full-rank, and R is required to have positive elements on the diagonal.

This work is part of the Perception on Purpose project, supported by EU grant 027268. the eye has a fixed centre of rotation, and that this point coincides with the optical centre [5].

Miles Hansard is a post-doctoral researcher at INRIA Rhône-Alpes, France. He is interested in geometric and computational aspects of visual perception. His recent work is concerned with human and robotic stereopsis. He studied Experimental Psychology (BSc) and Computer Vision (MRes, PhD) at University College London.

Radu Horaud holds a position of director of research at INRIA Rhône-Alpes, France. His research interests cover computer vision, machine learning, multi-sensory fusion, and robotics. He is the author of over 100 scientific publications. He is an area editor of Computer Vision and Image Understanding (Elsevier), a member of the advisory board of The International Journal of Robotics Research (Sage), and a member of the editorial boards of The International Journal of Computer Vision (Kluwer), Image and Vision Computing, and Machine Vision and Applications (Springer). In 2001 he served as program co-chair of the IEEE Eighth International Conference on Computer Vision (ICCV'01).


A Harouri

F Bouamrane

D Bourrier

A Garnache

N Isac

G Beaudoin

K Pantzas

I Sagnes

Sustainable Development Goals* eventually involved in your research: Thermal management of VECSEL-GaAs for high-power single-mode emission

Keywords: Laser, VECSEL, Thermal management Disciplinary fields involved : Physics

Information and communication technologies have revolutionized our world and reshaped our industries. Vertical emission in external cavity lasers (VECSEL) combine simultaneously high-power and highly coherent tunable single-frequency laser emission. VECSELs are promising candidates for various applications such as transmitter, sensors.

In our study, we are interested in reducing the thermal impedance of our ½ VECSELs on GaAs substrate, emitting between 800 and 1100nm. Part of the power of the optical pump is converted into a significant heating of the component. This heating reduces the gain of the QWs and induces a shift of the emission wavelength towards longer wavelengths [1]. In the literature, many strategies to improve the heat dissipation for ½ VECSEL are proposed [2] [3]. In our case, our approach is to integrate a heat sink as close as possible to the Bragg mirror by dry etching and electrolytic metal growth. As a first step, we studied both the diameter of the aperture and the metal material to be used in order to have a good mechanical stability of the semiconductor structure while ensuring very good heat dissipation


La mémoire en morceaux. Une ethnographie de la patrimonialisation des minorités LGBTQI et de la lutte contre le sida.

Renaud Chantraine, postdoctorant en anthropologie (Sidaction Fellow, Aix-Marseille Université, SESSTIM).

Thèse de doctorant en anthropologie sociale et ethnologie, École des hautes études en sciences sociales, soutenue le 23 novembre 2023, sous la direction de Madame Véronique Moulinié.

Mots clés : Patrimoine, Mémoires, Musée, Transmission, Minorités LGBTQI, VIH/sida Entre 2012 et 2020, la manière de poser le problème des archives, des mémoires et de l'héritage culturel des communautés LGBTQI et de la lutte contre le sida a sensiblement évolué. Durant cette période, plusieurs personnes sont entrées en contact avec des institutions nationales afin de transférer, la plupart du temps définitivement, une partie des traces de leurs histoires ou de leurs luttes, individuelles ou collectives. En 2012, Alain Lecoultre, ancien éditeur de Masques, la revue des homosexualités (1979)(1980)(1981)(1982)(1983)(1984)(1985), donne ses archives à la Bibliothèque nationale de France (BnF). L'année suivante, Daniel Defert, un des fondateurs de l'association de lutte contre le sida AIDES, dépose ses archives personnelles aux Archives nationales. Le photoreporter Tom Craig donne une partie de ses images documentant les actions d'Act Up-Paris au Mucem. Toujours la même année, Jean-Marcel Michel donne aux Archives départementales des Bouches-du-Rhône un fonds d'archives privé lié à son parcours de militant homosexuel engagé dans la lutte contre le sida. En 2015, la bibliothèque homosexuelle d'André Baudry, fondateur du mouvement homophile Arcadie, est donnée à la Bibliothèque historique de la ville de Paris. J'ai enquêté sur plusieurs de ces dons ou de ces dépôts, en essayant de comprendre les stratégies, les attentes et les motivations des personnes qui les ont initiées. S'agit-il d'un désir de reconnaissance, d'une forme de revendication d'égalité via l'intégration au patrimoine et/ou au récit historique du corps de la nation ? S'agit-il du désir de laisser une trace dans le temps long, du souhait de tourner une page, ou plus pratiquement de se décharger d'un poids ou d'un encombrement de cartons ? Quels effets ces démarches produisent-elles, en retour, sur les institutions, mais aussi sur l'idée de communauté ?

Au niveau scientifique, ce travail m'a conduit à mobiliser différents champs de la recherche en sciences sociales comme la sociologie politique de la mémoire et des mobilisations [START_REF] Pollak | Mémoire, oubli, silence[END_REF]Broqua, 1998), l'anthropologie critique du patrimoine [START_REF] Hertz (ellen | Introduction : le "patrimoine" a-t-il fait son temps ?[END_REF][START_REF] Fabre | Le tournant patrimonial. Mutations contemporaines des métiers du patrimoine / sous la direction de Christian Hottin et Claudie Voisenat[END_REF][START_REF] Tornatore | Pour une anthropologie pragmatiste et plébéienne du patrimoine : un scénario contre-hégémonique[END_REF], l'historiographie [START_REF] Juhasz | Forgetting ACT UP[END_REF], la muséologie (Meijer-Van Mensch et [START_REF] Meijer-Van Mensch | AIDS Memorial Quilts: From Mourning and Activism to Heritage Objects[END_REF][START_REF] Nguyen (tuan | Co-existence and collaboration: Australian AIDS quilts in public museums and community collection[END_REF] ou encore l'archivistique [START_REF] Cook | Evidence, memory, identity, and community: four shifting archival paradigms[END_REF][START_REF] Cifor | Stains and Remains: Liveliness, Materiality, and the Archival Lives of Queer Bodies[END_REF]. Ce travail de terrain s'appuie sur l'exploration de différents fonds d'archives et de documentation, qu'ils soient publics ou privés, sur des éléments de mise en perspective internationale (Allemagne, Pays-Bas, États-Unis, Royaume-Uni, Nouvelle-Zélande, Australie) et sur la réalisation d'une trentaine d'entretiens compréhensifs avec des professionnels du patrimoine, des responsables politiques, des chercheurs comme des militants associatifs.

Le manuscrit est organisé en six chapitres. Les trois premiers portent globalement sur la patrimonialisation institutionnelle de la lutte contre le sida, les trois derniers cherchent à rendre compte d'autres façons de penser ou de faire les traces d'un passé ou d'un présent communautaire.

Le premier chapitre, « Déplier le Patchwork », débute par la description d'une cérémonie organisée au Mucem le 25 mai 2019. Plusieurs carrés de tissus cousus en souvenir de personnes mortes du sida -appelés patchwork -sont déployés dans l'espace du Fort Saint Jean. L'un des carrés, venant d'Amsterdam, est sur le point d'entrer dans les collections du musée. Quelle est la signification de cette cérémonie ? S'agit-il du premier déploiement organisé dans un musée ? Pour répondre à ces questions, je me plonge dans les transcriptions d'une série de débats organisés à partir de 1994 au Musée national des arts et traditions populaires -Centre d'ethnologie française (MNATP-CEF) autour de la mémoire et des nouveaux rituels de deuil ayant émergé en réponse à l'épidémie. Cette première phase correspond à l'élaboration d'un cadre anthropologique, façonné au contact et en relation avec des associations. Ce moment de problématisation (et les traces qui en restent) est précieux pour penser dans un temps long les processus de patrimonialisation de la lutte contre le sida. Concernant le carré néerlandais, le bloc 23 du Dutch AIDS Memorial Quilt, je me demande, en utilisant la méthode de la biographie d'objet, quelles étapes, matérielles et symboliques, cet objet voyageur a-t-il traversées pour (en) arriver là ? Quels rapports et contacts peuvent, ou non, être négociées par les communautés face aux règles édictées par le Code du patrimoine et ses gardiens, les conservateurs ? Comment d'autres institutions muséales, à l'étranger (Australie, Nouvelle-Zélande, Pays-Bas) se sont-elles positionnées pour accueillir, ou non, ces objets mémoriels ? Comment, enfin, mesurer les conséquences de cet acte de transfert de propriété ?

Le deuxième chapitre, « Une histoire de robes… », propose de suivre plusieurs robes ayant appartenu à des membres du mouvement des Soeurs de la Perpétuelle Indulgence, robes aujourd'hui conservées dans les réserves du Mucem. Une Soeur, présente à la cérémonie décrite au premier chapitre, demande à consulter ces objets. Les fiches qui les accompagnent ne disent rien de l'histoire des personnes qui les ont portées, comme si les robes étaient vides. Je pars à la recherche de ces informations, en commençant par revenir au moment -central dans le processus de patrimonialisation -de la vaste campagne d'acquisition sur « l'histoire et les mémoires du sida en France, en Europe et en Méditerranée » (2002)(2003)(2004)(2005). Comment fonctionne ce dispositif ? Comment deux anthropologues, Françoise Loux et Stéphane Abriol, ont-ils rencontré les Soeurs ? Comment le transfert de ces costumes au musée s'est-il passé ? Les relations nouées pendant la collecte sont-elles durables ou sont-elles vouées à se dissiper avec le temps ? Le troisième chapitre, « Oublier ou ne pas oublier Act Up », cherche à explorer la complexité d'une série de transferts d'archives et objets liés à Act Up-Paris aux institutions nationales, tout en proposant des pistes de contextualisation plus large sur les enjeux de mémoires et d'oublis. En 1997, l'association Act Up-Paris met en place en son sein un groupe destiné à centraliser en interne archives, objets et documentation. En 2013, l'association, placée en redressement judiciaire, est contrainte de transférer une large partie de cette mémoire patiemment rassemblée aux Archives nationales. Les papiers sont triés, tandis que les objets sont transférés au Mucem qui procède, à son tour, à un tri. Tout en interrogeant et en étudiant les différentes décisions internes et externes ayant conduit à cette (double) patrimonialisation, je réfléchis à ce que ces traces de l'action et du passé représentent, essayant de comprendre ce qu'elles sont et ce qu'elles ne sont pas. Renvoient-elles à une forme de mémoire officielle de l'association ? Comportent-elles des traces plus personnelles de la vie de ses membres, vivants ou disparus ? Qu'est-ce que les militants gardent pour eux ? À partir du quatrième chapitre, « Tout perdre, tout garder », il s'agit, en s'éloignant du terrain institutionnel, de comprendre comment et à partir de quelles impulsions, au sein des communautés LGBTQI, les traces de l'expérience du passé et du présent sont gardées et éventuellement partagées. Le cas de l'Académie Gay et Lesbienne, lieu de rassemblement d'archives et de documentation, installé dans un pavillon privé de la région parisienne, qui existe officiellement depuis une vingtaine d'années, est instructif. Écouter leurs fondateurs, Hoàng Phan Bigotte, réfugié politique d'origine vietnamienne arrivé en France en 1975, et son compagnon Thomas Leduc permet d'adopter un autre regard sur les archives. Leur pratique de collecte illustre celle d'un centre d'archives communautaire indépendant. Elle s'effectue sur les lieux de drague ou dans les espaces de militantisme (à la fois homosexuel et de lutte contre le sida). Leurs critères de sélection et de conservation diffèrent de ceux des institutions d'archives nationales.

Le cinquième chapitre, « Faire-Mémoire », a comme sujet l'association marseillaise Mémoire des sexualités. Un court texte d'introduction revient sur la fondation du projet autour de la figure du militant homosexuel Christian de Leusse au début des années 1980, sur ma rencontre avec celui-ci en 2013, et sur la problématisation, à partir de 2017, d'une situation et d'un enjeu de transmission intra-communautaire. Ce chapitre adopte lui aussi une forme spécifique : il s'écoute (un casque est donc à prévoir). Ce choix vient du fait que j'ai décidé de ne pas objectiver à partir d'une position classique d'autorité un projet militant et collectif dans lequel je suis impliqué. Il m'a semblé plus juste de produire en substitution une archive sonore, de donner à entendre des paroles qui racontent l'attachement aux strates de mémoires accumulées que le lieu contient. Ces paroles imaginent le passé comme elles imaginent le futur. Avec ce chapitre, je propose d'ajouter une archive supplémentaire au fonds de l'association. Ce geste est une double manière de restituer la forme de mon engagement spécifique.


Fabio Boudis
email: fabio.boudis.etu@univ-lille.fr.

Guillaume Clement

Amelie Bruandet

Antoine Lamer

Automated Generation of Individual and Population Clinical Pathways with the OMOP Common Data Model

Keywords: Pathway, Data visualization, Data reuse, OMOP, Sankey diagram

Introduction. Clinical pathways represents the sequence of interventions from which the patients benefit during their encounters with health care structures. There are several complex issues which make it difficult to represent these pathways (e.g. high numbers of patients, heterogeneity of variables). Methods. We developed a tool to automate the representation of clinical pathways, from an individual and population points of view, and based on the OMOP CDM. The tool implemented the Sankey diagram in three stages: (i) data extraction, (ii) generation of individual sequence of steps and (iii) aggregation of sequence to obtain the population-level diagram. We tested the tool with three surgery procedures : the total hip replacement, the coronary bypass and the transcatheter aortic valve implantation.

Results. The tool provided different ways of visualizing pathways depending on the question asked: a pathway before a surgery, the pathway of deceased patients or the complete pathway with different steps of interest. Discussion. We proposed a tool automating the representation of the clinical pathways, and reducing complexity of visualization. Representations are detailed from an individual and population points of view. It has been tested with three surgical procedures. The tool functionalities will be extended to cover a greater number of use cases.

Introduction

Clinical pathways represents the sequence of interventions from which the patients benefit during their encounters with health care structures [START_REF] Perer | Mining and exploring care pathways from electronic medical records with visual analytics[END_REF][START_REF] Martignene | Heimdall, a Computer Program for Electronic Health Records Data Visualization[END_REF]. They are used in two contexts: an individual-based context when the visualization targets a single patient to provide appropriate care, and a population-based context when it relates to a whole population and addresses research or decision-making objectives [START_REF] Martignene | Heimdall, a Computer Program for Electronic Health Records Data Visualization[END_REF]. There are several complex issues which make it difficult to represent these pathways. First, the high numbers of patients and the heterogeneity of variables increase the complexity of datasets and induce information overload. Second, pathway data contains temporal events that need to be aligned for population-based data analysis. Existing solutions have some limitations. They were often developed for representing the pathway of a selected pathology or a single patient, or they used data stored in a handmade data model, which complicates the use of the tool in other contexts or with other database formats [START_REF] Martignene | Heimdall, a Computer Program for Electronic Health Records Data Visualization[END_REF][START_REF] Goren-Bar | The Definition and Implementation of an Intelligent Tool for Visualization and Exploration of Time-oriented Clinical Data[END_REF][START_REF] Plaisant | LifeLines: Using Visualization to Enhance Navigation and Analysis of Patient Records[END_REF][START_REF] Bui | Visualizing Integrated Patient Records[END_REF][START_REF] Wongsuphasawat | Exploring Flow, Factors, and Outcomes of Temporal Event Sequences with the Outflow Visualization[END_REF].

Observational Health Data Sciences and Informatics (OHDSI) developed the Observational Medical Outcomes Partnership (OMOP) common data model (CDM). It allows to standardize the data structure and vocabulary of data, which are originally collected by different software and characterized by heterogeneous vocabularies, depending on the country [START_REF] Hripcsak | Observational Health Data Sciences and Informatics (OHDSI): Opportunities for Observational Researchers[END_REF]. Based on the CDM, it is possible to share tools, methods and results. The OHDSI consortium thus shares open access R packages, ETL tools and scripts for performing reproducible statistical analyses and evidence-based research [START_REF] Reps | Design and implementation of a standardized framework to generate and evaluate patient-level prediction models using observational healthcare data[END_REF].

The objective of this work is to propose a tool to automate the representation of the clinical pathways, from an individual and population points of view, based on the OMOP CDM.

Methods

Concepts

In a previous study, we identified and used the Sankey diagram to represent a sequence of steps [START_REF] Lamer | Exploring Patient Path Through Sankey Diagram: A Proof of Concept[END_REF]. The Sankey diagram is a flow diagram composed of nodes and links between nodes. In the hospital field, the nodes correspond to steps or events and are labeled according to unit care, drugs administrations, medical procedures, diagnoses. The links represent a set of patients going from one step to another. The width of nodes and links is proportional to the number of patients. Generally, a step that all patients go through is used to align the diagram.

Implementation

Data were extracted from the database of the Lille University Hospital and covering hospital stays, medical procedures and diagnoses, for year 2019. Local data was transformed into the OMOP CDM before processing.

In a first step, concepts corresponding to inclusion criteria were selected to filter the patients. Data were extracted from visit_detail, condition_occurrence, procedure_occurrence and concept_relationship tables.

The second step dealt with the generation of the individual pathways. For this, the temporal sequence of steps is reproduced, in integrating the inclusion event. Four lists of indices are developed : in starting from the first step of the sequence (list A), in starting from the last step of the sequence (list B), in starting from the inclusion event by moving to the endpoints (list C) and in starting from the endpoints to the inclusion event (list D). The steps of interest are selected according to their indices in the sequence of steps. An example of the lists of indices is presented in Figure 1. The steps labels may be simplified with mapping from concept_relationship table (e.g. all intensive care units are mapped to a unique intensive care unit label).

In the last step, individual pathways are gathered to obtain population pathways. The size of the nodes represents the population in the step. The flows between the nodes represent the size of the population moving from one step to another. The Sankey diagram is indexed on the inclusion event, and only previously selected steps are represented.

The tool was developed with R (version 4.0.3) and uses DatabaseConnector, SqlRender, d3js, plotly, tidyverse, and lubridate packages. Our R package for the generation of clinical pathways is available here : https://gitlab.com/Fboudis/patient_pathway.

Study cases

We choose 3 study cases in connection with the issues we routinely handle in our hospital : the total hip surgery (SC1), the coronary bypass (SC2) and the transcatheter aortic valve implantation (SC3). These 3 procedures benefit from different care pathways, and we wished to visualize the steps prior to surgery in SC1, to study the pathway of dead patients in SC2 and to visualize the whole hospital stay in SC3.

Results

We tested our package with 1013, 566, and 357 patients for SC1, SC2, and SC3 respectively. Due to label simplification, the number of distinct labels have been reduced from 31 to 11, from 37 to 12, and from 43 to 10 for SCI, SC2 and SC3 respectively. Due to simplification, the number of unique individual pathways have been decreased from 43 to 41, from 188 to 145, from 179 to 124 for SCI, SC2 and SC3 respectively. The Figure 2 represents the 3 Sankey diagrams for SC1 (A), SC2 (B), and SC3 (C).

In SC1, there were two main flows before surgery: (i) patients came from home and were admitted in conventional or weekly units, or (ii) patients were admitted in emergency units for unplanned admission.

In SC2, all patients went directly to intensive care units. The diagram showed that some of the deceased patients had returned to continuous care before their situation deteriorated and died, or got readmitted to intensive care before dying (orange flow, Figure 2 -B).

In SC3, the diagram describes the global pathway before and after the surgery. Patients went in several types of units, in equivalent proportions. It highlighted the heterogeneity of the care. Around a quarter of patients were transferred to another hospital at the end of the stay.

Discussion

In this paper, we described a tool to automate the representation of the clinical pathway. This tool is in open access on our git repository and supports databases in the OMOP CDM. By proposing the selection of steps through different index lists, and by simplifying the heterogeneous labels, our tool helps to reduce the complexity of the patient's journey. The tool has been tested with three different surgical operations, each time focusing on a different part of the hospital stay. The graphical representation at patient level allows the clinician to have an overview of the hospital stay of a selected patient. The representation at the population level offers the possibility to identify atypical care pathways, forecasting the resources to be allocated, and conducting research. In this study, only inpatient stays were considered and we do not take into account outpatient study cases. The index date was defined according to a procedure, and could take into account other events such as drug administration or diagnosis. Further evaluation of the tool is needed, in particular with clinicians and researchers to ensure the contribution of sequence simplification methods.

Figure 1 .

 1 Figure 1. Sequence of steps and inclusion event, and the related indices lists to identify a step.

Figure 2 .

 2 Figure 2. Sankey diagrams for : A -Total hip prosthesis, B -Coronary bypass, C -Transcatheter aortic valve implantation. CB: Coronary bypass; CC: Continuous care; CU: Conventional unit; EU: Emergency unit; IC: Intensive care; REA: Reanimation; WU: Weekly unit.

Conclusions

We proposed a tool to automate the representation of the clinical pathways, from an individual and population points of view, based on the OMOP CDM. It has been tested with three surgical procedures. The tool functionalities will be extended to cover a greater number of use cases.