0042-6989/91 $3.00 + 0.00 Pergamon Press pIc
Vision Res. Vol. 31, No. II, pp. 2029-2032, 1991 Printed in Great Britain
SHORT COMMUNICATION
NATURAL COORDINATES FOR SPECIFICATION OF EYE MOVEMENTS R. A. CLEMENT Defence Research Agency, Military Division, Fort Halstead, Sevenoaks, Kent TNl4 7BP, England
(Received 9 November 1990; in revised form 21 February 1991) Abstract-Tweed and Vilis (Journal of Neurophysiology, 58,832-849, 1987) have argued that quaternion algebra provides the most appropriate description of the rotations of the eye, and have derived a three-dimensional model of gaze control based on quaternion operations. Euler angles give a simpler description of the rotations of the eye, and can also be used to formulate an alternative version of the three-dimensional gaze control model. Comparison of the two versions of the model highlights the distinction between the functional predictions of the model, and the predictions which depend only on the choice of mathematical descriptions. Quaternions
Euler angles
Listing's law
Saccadic eye movements
INTRODUCTION
Although the human eye can rotate in its socket with three degrees of freedom, fixation eye movements usually involve movements with two degrees of freedom (Helmholtz, 1867). It is therefore natural to use two coordinates to describe the kinematics of the eye, one for each degree of freedom. However, it has recently been argued that the rotations of the eye are best described by quaternions, which have 4 components (Tweed & Vilis, 1987, 1990; Tweed, Cadera & Vilis, 1990). Their arguments arose from consideration of the three-dimensional properties of rotations. In one dimension, the behaviour of the saccadic eye movement system can be described by a local feedback model (van Gisbergen, Robinson & Gielen, 1981). According to this model, the eye position is subtracted from the target position to provide an error signal. The error signal is converted to an eye velocity signal by the burst neurons, and the eye velocity signal is integrated to provide the position signal. In three dimensions, the position signal can be represented by an angular position vector a, which specifies the rotation of the eye from the primary position, and the movement signal can be represented by an angular velocity vector w. Tweed and Vilis (1987) pointed out that integration of w, does not give a because of the
non-commutatlVlty of rotations. They argued for a quaternion-based model, which correctly describes the three-dimensional properties of rotations. Despite such arguments, the fact remains that no more than two coordinates should be needed to describe the behaviour of a system which only moves with two degrees of freedom. In this paper it is shown that the kinematics of the saccadic eye movement system can be fully described by operations on two coordinates: the meridional angle and the angle of eccentricity of the line of fixation. THEORY
(aJ Description of eye movements
An appropriate system of coordinates for describing the rotation of the eye around its centre of rotation 0 is provided by the specifying a line through 0 and a rotation of the eye about that line. This system requires three angles, referred to as Euler angles (Rutherford, 1951). The first two angles specify the orientation of the line and the third specifies the rotation of the eye about that line. When applied to the eye, it is natural to specify the orientation of the line of fixation and the rotation of the eye about that line. The angles are defined with respect to two sets of Cartesian base vectors, one of which e" e2,
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e3 is fixed in space and the other of which el" e2" e3 ' is fixed with respect to the eye. The two
systems of base vectors have their origins at the centre of rotation of the eye and are coincident in the primary position. Let the el and el' base vectors point to the right along the line joining the centres of rotation of the two eyes, let the e2 and e/ base vectors point vertically upwards and let the e3 and e3' base vectors point in the direction of the line of fixation. The first two angles (4J, 0) specify the e3 ' direction with respect to the fixed set of base vectors, and the third angle (l/J) specifies the rotation of the eye about the e3 ' direction. The angle 4J is given by the clockwise rotation about e3 needed to rotate the plane spanned by the e2' e3 base vectors into the plane spanned by the e3 and e3' base vectors; 4J specifies the meridional angle of the line of fixation. The angle 0 is determined by the clockwise rotation about the newel direction which aligns e3 with e/; 0 specifies the eccentricity of the line of fixation. Finally, the angle l/J specifies the clockwise rotation about the e3 ' direction. The constraint implied by Listing's law is that l/J = - 4J (Helmholtz, 1867). An important property of the Euler angles is that the angular velocity of a rotating object W
can be expressed directly in terms of the Euler angles and their derivatives. With the systems of base vectors used here: = ~ sin l/J
+ (j cos l/J W2 = - ~ sin 0 cos l/J + (j sin l/J W3 = - (~ cos 0 + ~). WI
When Listing's law is obeyed, the equations take the simpler form: WI
= - ~ sin 4J sin 0
W2
= -
W3
= ~(l- cos 0).
+ (j cos 4J (~ cos 4J sin 0 + (j sin 4J) (2)
(b) Model of gaze control A block diagram of the quaternion model is shown in Fig. lea) and the corresponding diagram for the Euler angles version of the model is shown in Fig. l(b). In the quaterion model, the velocity signal is multiplied by a position feedback signal to give the derivative of the position signal, which can then be integrated. In the Euler angle version ofthe model, the derivatives of the orientation angles are obtained from the equations (2) which specify the angular velocity vector. If the equations for two
(0 )
E
/
1 I(Q
I---~
n )----+1 J
q
(b)
E
/
(I)
J
~/9
Fig. I. Block diagrams of alternative versions of a three-dimensional model of saccadic gaze control. In both versions, the target eye position is denoted by an asterisk. The overall structure of the model involves passing an error signal E to the burst neurons, to produce a velocity signal, which is combined with a position feedback signal and subsequently integrated. (a) The quaternion model, redrawn from Tweed and Vilis (1987). (b) The Euler angle version of the model, where IjJ denotes the angle of meridian and (J denotes the angle of eccentricity. The vector t is normal to the plane containing all the possible axes of rotation associated with an eye position (1jJ, (J), and A -I denotes multiplication by a matrix inverse.
Short Communication
components of the velocity vector are written in matrix form, then the derivatives of the orientation angles can be obtained by applying the inverse matrix to ¢ and e. Using W2 and W3' one has:
so (3)
The error signal E specifies the axis of rotation of the eye, and the size of the rotation. Helmholtz (1867) proved that, if the eye moves according to Listing's law, then the axes of rotation associated with all possible movements from a given direction lie in a plane. Furthermore, the normal to this plane bisects the angle between the actual direction of the line of fixation and its primary direction. Hence, if the direction of the line of fixation is specified by the Euler angles ¢ and e, then a unit vector normal to the plane of the axes of rotation is given by: t 1 = sin(¢) sin(e /2)
e
t2 = cos(¢) sin( /2) t3 = cos(e /2).
(4)
If the eye rotates about a fixed axis from its current position (¢, e) to a target position (¢*, e*) in accordance with Listing's law, then the fixed axis must lie in both of the planes of the axes of rotations associated with the two positions of the eye. Hence the axis is given by the intersection of these two planes. Let the unit vector normal to the plane of the axes of rotation of the current eye position be denoted by t, and the corresponding vector for the ~arget position be denoted by t *, then a vector in t~e direction of the intersection of the two planes IS given by the vector product of t * and t: E=t*rd.
(5)
By expanding out the expressions for E in both versions of the model into coordinates it can be shown that the size of the vector component of E is the same in both cases. However, equation (5) gives a particularly simple expression for the size. Since t * and t unit vectors, the size of their vector product is equal to the sine of the angle between them.
2031 DISCUSSION
The Euler angle based model of the saccadic system demonstrates that only two independent coordinates are needed to describe the behaviour of the system. Euler angles are suited to describing the behaviour of any system which obeys Listing's law, because the constraint implied by Listing's law takes the simple form t/J = - ¢. This feature of Euler angles has previously been exploited in analysis of the functional significance of Listing's law. Thus, Helmholtz (1897) argued that whenever the object which is being fixated is subsequently indirectly fixated, so that it is imaged on a particular retinal location, then the pattern of displacement of the individual points of t~e image should always be the same. If a defimte pattern of image point displacements is associated with indirect fixation of an object on a given retinal position, then the nervous system can learn to interpret the displacements as due to movements of the eye and not the object. Subsequently Lamb (1919) applied the calculus of variations to the torsional component of angular velocity specified by equation (1), to show that the patterns of image point displacements are most similar when t/J = - ¢, thus supporting Helmholtz's explanation. Tweed and Vilis (1987) pointed out that alternative forms of their model could be constructed, but dismissed them as lacking in simplicity, compared to the quaternion model. Certainly, the insight that an angular position feedback signal is required, if the position signal is to be recovered by integration of the velocity signal, and the prediction that the eyes move around fixed axes during saccades emerge most clearly in the quaterion model. However, the Euler angle version of their model does help in distinguishing between the necessary features of their model, and those which are contingent on the use of quaternions. The angular position feedback signal does .not .have to be combined with the angular velocity signal by quaternion multiplication, and the er~or signal does not have to be formed by quatermon division.
REFERENCES van Gisbergen, 1., Robinson, D. A. & Gielen, S. A. (1981). A quantitative analysis of generation of saccadic eye movements by burst neurons. Journal ofNeurophysiology, 45,417-442.
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von Helmholtz, H. (1867). Handbuch der Physiologischen Optik (1st edn, Vol. 3). Hamburg: Voss. Third edition translated into English by Southall J. P. C. (1925) as Treatise on physiological optics. Rochester, N.Y.: Optical Society of America. Lamb, H. (1919). The kinematics of the eye. Philosophical Magazine, 38, 685-695. Rutherford, D. E. (1951). Classical mechanics. Edinburgh: Oliver It Boyd.
Tweed, D. & Vilis, T. (1987). Implications of rotational kinematics for the oculomotor system in three dimensions. Journal of Neurophysiology, 58, 832-849. Tweed, S. & Vilis, T. (1990). Geometric relations of eye position and velocity vectors during saccades. Vision Research, 30, 111-127. Tweed, D., Cadera, W. It Vilis, T. (1990). Computing three-dimensional eye position quaternions and eye velocity from search coil signals. Vision Research, 30, 97-110.
Vision Res. Vol. 31, No. II, pp. 2029-2032, 1991 Printed in Great Britain
SHORT COMMUNICATION
NATURAL COORDINATES FOR SPECIFICATION OF EYE MOVEMENTS R. A. CLEMENT Defence Research Agency, Military Division, Fort Halstead, Sevenoaks, Kent TNl4 7BP, England
(Received 9 November 1990; in revised form 21 February 1991) Abstract-Tweed and Vilis (Journal of Neurophysiology, 58,832-849, 1987) have argued that quaternion algebra provides the most appropriate description of the rotations of the eye, and have derived a three-dimensional model of gaze control based on quaternion operations. Euler angles give a simpler description of the rotations of the eye, and can also be used to formulate an alternative version of the three-dimensional gaze control model. Comparison of the two versions of the model highlights the distinction between the functional predictions of the model, and the predictions which depend only on the choice of mathematical descriptions. Quaternions
Euler angles
Listing's law
Saccadic eye movements
INTRODUCTION
Although the human eye can rotate in its socket with three degrees of freedom, fixation eye movements usually involve movements with two degrees of freedom (Helmholtz, 1867). It is therefore natural to use two coordinates to describe the kinematics of the eye, one for each degree of freedom. However, it has recently been argued that the rotations of the eye are best described by quaternions, which have 4 components (Tweed & Vilis, 1987, 1990; Tweed, Cadera & Vilis, 1990). Their arguments arose from consideration of the three-dimensional properties of rotations. In one dimension, the behaviour of the saccadic eye movement system can be described by a local feedback model (van Gisbergen, Robinson & Gielen, 1981). According to this model, the eye position is subtracted from the target position to provide an error signal. The error signal is converted to an eye velocity signal by the burst neurons, and the eye velocity signal is integrated to provide the position signal. In three dimensions, the position signal can be represented by an angular position vector a, which specifies the rotation of the eye from the primary position, and the movement signal can be represented by an angular velocity vector w. Tweed and Vilis (1987) pointed out that integration of w, does not give a because of the
non-commutatlVlty of rotations. They argued for a quaternion-based model, which correctly describes the three-dimensional properties of rotations. Despite such arguments, the fact remains that no more than two coordinates should be needed to describe the behaviour of a system which only moves with two degrees of freedom. In this paper it is shown that the kinematics of the saccadic eye movement system can be fully described by operations on two coordinates: the meridional angle and the angle of eccentricity of the line of fixation. THEORY
(aJ Description of eye movements
An appropriate system of coordinates for describing the rotation of the eye around its centre of rotation 0 is provided by the specifying a line through 0 and a rotation of the eye about that line. This system requires three angles, referred to as Euler angles (Rutherford, 1951). The first two angles specify the orientation of the line and the third specifies the rotation of the eye about that line. When applied to the eye, it is natural to specify the orientation of the line of fixation and the rotation of the eye about that line. The angles are defined with respect to two sets of Cartesian base vectors, one of which e" e2,
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Short Communication
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e3 is fixed in space and the other of which el" e2" e3 ' is fixed with respect to the eye. The two
systems of base vectors have their origins at the centre of rotation of the eye and are coincident in the primary position. Let the el and el' base vectors point to the right along the line joining the centres of rotation of the two eyes, let the e2 and e/ base vectors point vertically upwards and let the e3 and e3' base vectors point in the direction of the line of fixation. The first two angles (4J, 0) specify the e3 ' direction with respect to the fixed set of base vectors, and the third angle (l/J) specifies the rotation of the eye about the e3 ' direction. The angle 4J is given by the clockwise rotation about e3 needed to rotate the plane spanned by the e2' e3 base vectors into the plane spanned by the e3 and e3' base vectors; 4J specifies the meridional angle of the line of fixation. The angle 0 is determined by the clockwise rotation about the newel direction which aligns e3 with e/; 0 specifies the eccentricity of the line of fixation. Finally, the angle l/J specifies the clockwise rotation about the e3 ' direction. The constraint implied by Listing's law is that l/J = - 4J (Helmholtz, 1867). An important property of the Euler angles is that the angular velocity of a rotating object W
can be expressed directly in terms of the Euler angles and their derivatives. With the systems of base vectors used here: = ~ sin l/J
+ (j cos l/J W2 = - ~ sin 0 cos l/J + (j sin l/J W3 = - (~ cos 0 + ~). WI
When Listing's law is obeyed, the equations take the simpler form: WI
= - ~ sin 4J sin 0
W2
= -
W3
= ~(l- cos 0).
+ (j cos 4J (~ cos 4J sin 0 + (j sin 4J) (2)
(b) Model of gaze control A block diagram of the quaternion model is shown in Fig. lea) and the corresponding diagram for the Euler angles version of the model is shown in Fig. l(b). In the quaterion model, the velocity signal is multiplied by a position feedback signal to give the derivative of the position signal, which can then be integrated. In the Euler angle version ofthe model, the derivatives of the orientation angles are obtained from the equations (2) which specify the angular velocity vector. If the equations for two
(0 )
E
/
1 I(Q
I---~
n )----+1 J
q
(b)
E
/
(I)
J
~/9
Fig. I. Block diagrams of alternative versions of a three-dimensional model of saccadic gaze control. In both versions, the target eye position is denoted by an asterisk. The overall structure of the model involves passing an error signal E to the burst neurons, to produce a velocity signal, which is combined with a position feedback signal and subsequently integrated. (a) The quaternion model, redrawn from Tweed and Vilis (1987). (b) The Euler angle version of the model, where IjJ denotes the angle of meridian and (J denotes the angle of eccentricity. The vector t is normal to the plane containing all the possible axes of rotation associated with an eye position (1jJ, (J), and A -I denotes multiplication by a matrix inverse.
Short Communication
components of the velocity vector are written in matrix form, then the derivatives of the orientation angles can be obtained by applying the inverse matrix to ¢ and e. Using W2 and W3' one has:
so (3)
The error signal E specifies the axis of rotation of the eye, and the size of the rotation. Helmholtz (1867) proved that, if the eye moves according to Listing's law, then the axes of rotation associated with all possible movements from a given direction lie in a plane. Furthermore, the normal to this plane bisects the angle between the actual direction of the line of fixation and its primary direction. Hence, if the direction of the line of fixation is specified by the Euler angles ¢ and e, then a unit vector normal to the plane of the axes of rotation is given by: t 1 = sin(¢) sin(e /2)
e
t2 = cos(¢) sin( /2) t3 = cos(e /2).
(4)
If the eye rotates about a fixed axis from its current position (¢, e) to a target position (¢*, e*) in accordance with Listing's law, then the fixed axis must lie in both of the planes of the axes of rotations associated with the two positions of the eye. Hence the axis is given by the intersection of these two planes. Let the unit vector normal to the plane of the axes of rotation of the current eye position be denoted by t, and the corresponding vector for the ~arget position be denoted by t *, then a vector in t~e direction of the intersection of the two planes IS given by the vector product of t * and t: E=t*rd.
(5)
By expanding out the expressions for E in both versions of the model into coordinates it can be shown that the size of the vector component of E is the same in both cases. However, equation (5) gives a particularly simple expression for the size. Since t * and t unit vectors, the size of their vector product is equal to the sine of the angle between them.
2031 DISCUSSION
The Euler angle based model of the saccadic system demonstrates that only two independent coordinates are needed to describe the behaviour of the system. Euler angles are suited to describing the behaviour of any system which obeys Listing's law, because the constraint implied by Listing's law takes the simple form t/J = - ¢. This feature of Euler angles has previously been exploited in analysis of the functional significance of Listing's law. Thus, Helmholtz (1897) argued that whenever the object which is being fixated is subsequently indirectly fixated, so that it is imaged on a particular retinal location, then the pattern of displacement of the individual points of t~e image should always be the same. If a defimte pattern of image point displacements is associated with indirect fixation of an object on a given retinal position, then the nervous system can learn to interpret the displacements as due to movements of the eye and not the object. Subsequently Lamb (1919) applied the calculus of variations to the torsional component of angular velocity specified by equation (1), to show that the patterns of image point displacements are most similar when t/J = - ¢, thus supporting Helmholtz's explanation. Tweed and Vilis (1987) pointed out that alternative forms of their model could be constructed, but dismissed them as lacking in simplicity, compared to the quaternion model. Certainly, the insight that an angular position feedback signal is required, if the position signal is to be recovered by integration of the velocity signal, and the prediction that the eyes move around fixed axes during saccades emerge most clearly in the quaterion model. However, the Euler angle version of their model does help in distinguishing between the necessary features of their model, and those which are contingent on the use of quaternions. The angular position feedback signal does .not .have to be combined with the angular velocity signal by quaternion multiplication, and the er~or signal does not have to be formed by quatermon division.
REFERENCES van Gisbergen, 1., Robinson, D. A. & Gielen, S. A. (1981). A quantitative analysis of generation of saccadic eye movements by burst neurons. Journal ofNeurophysiology, 45,417-442.
2032
Short Communication
von Helmholtz, H. (1867). Handbuch der Physiologischen Optik (1st edn, Vol. 3). Hamburg: Voss. Third edition translated into English by Southall J. P. C. (1925) as Treatise on physiological optics. Rochester, N.Y.: Optical Society of America. Lamb, H. (1919). The kinematics of the eye. Philosophical Magazine, 38, 685-695. Rutherford, D. E. (1951). Classical mechanics. Edinburgh: Oliver It Boyd.
Tweed, D. & Vilis, T. (1987). Implications of rotational kinematics for the oculomotor system in three dimensions. Journal of Neurophysiology, 58, 832-849. Tweed, S. & Vilis, T. (1990). Geometric relations of eye position and velocity vectors during saccades. Vision Research, 30, 111-127. Tweed, D., Cadera, W. It Vilis, T. (1990). Computing three-dimensional eye position quaternions and eye velocity from search coil signals. Vision Research, 30, 97-110.
