A theoretical analysis of three-dimensional eye position measurement using polar cross-correlation

A Robust Method of Eye Torsion Measurement for Medical Applications

The detection of eye torsion is an important element for diagnosis of balance disorders, although it is rarely available in existing eye tracking systems. A novel method is proposed in this paper to provide robust measurement of torsional eye movements. A numerical approach is presented to estimate the iris boundary only according to the gaze direction, so the segmentation of the iris is more robust against occlusions and ambiguities. The perspective distortion of the iris pattern at eccentric eye positions is also corrected, benefiting from the transformation relation that is established for the iris estimation. The angle of the eye torsion is next measured on the unrolled iris patterns via a TM (Template Matching) technique. The principle of the proposed method is validated and its robustness in practice is assessed. A very low mean FPR (False Positive Rate) is reported (i.e., 3.3%) in a gaze test when testing on five participants with very different eye morphologies. The present method always gave correct measurement on the iris patterns with simulated eye torsions and rarely provided mistaken detections in the absence of eye torsion in practical conditions. Therefore, it shows a good potential to be further applied in medical applications.

Knowing what the brain is seeing in three dimensions: A novel, noninvasive, sensitive, accurate, and low-noise technique for measuring ocular torsion

Torsional eye movements are rotations of the eye around the line of sight. Measuring torsion is essential to understanding how the brain controls eye position and how it creates a veridical perception of object orientation in three dimensions. Torsion is also important for diagnosis of many vestibular, neurological, and ophthalmological disorders. Currently, there are multiple devices and methods that produce reliable measurements of horizontal and vertical eye movements. Measuring torsion, however, noninvasively and reliably has been a longstanding challenge, with previous methods lacking real-time capabilities or suffering from intrusive artifacts. We propose a novel method for measuring eye movements in three dimensions using modern computer vision software (OpenCV) and concepts of iris recognition. To measure torsion, we use template matching of the entire iris and automatically account for occlusion of the iris and pupil by the eyelids. The current setup operates binocularly at 100 Hz with noise <0.1° and is accurate within 20° of gaze to the left, to the right, and up and 10° of gaze down. This new method can be widely applicable and fill a gap in many scientific and clinical disciplines.

Central adaptation to repeated galvanic vestibular stimulation: implications for pre-flight astronaut training

Healthy subjects (N = 10) were exposed to 10-min cumulative pseudorandom bilateral bipolar Galvanic vestibular stimulation (GVS) on a weekly basis for 12 weeks (120 min total exposure). During each trial subjects performed computerized dynamic posturography and eye movements were measured using digital video-oculography. Follow up tests were conducted 6 weeks and 6 months after the 12-week adaptation period. Postural performance was significantly impaired during GVS at first exposure, but recovered to baseline over a period of 7–8 weeks (70–80 min GVS exposure). This postural recovery was maintained 6 months after adaptation. In contrast, the roll vestibulo-ocular reflex response to GVS was not attenuated by repeated exposure. This suggests that GVS adaptation did not occur at the vestibular end-organs or involve changes in low-level (brainstem-mediated) vestibulo-ocular or vestibulo-spinal reflexes. Faced with unreliable vestibular input, the cerebellum reweighted sensory input to emphasize veridical extra-vestibular information, such as somatosensation, vision and visceral stretch receptors, to regain postural function. After a period of recovery subjects exhibited dual adaption and the ability to rapidly switch between the perturbed (GVS) and natural vestibular state for up to 6 months.

Three dimensional vestibular ocular reflex testing using a six degrees of freedom motion platform

The vestibular organ is a sensor that measures angular and linear accelerations with six degrees of freedom (6DF). Complete or partial defects in the vestibular organ results in mild to severe equilibrium problems, such as vertigo, dizziness, oscillopsia, gait unsteadiness nausea and/or vomiting. A good and frequently used measure to quantify gaze stabilization is the gain, which is defined as the magnitude of compensatory eye movements with respect to imposed head movements. To test vestibular function more fully one has to realize that 3D VOR ideally generates compensatory ocular rotations not only with a magnitude (gain) equal and opposite to the head rotation but also about an axis that is co-linear with the head rotation axis (alignment). Abnormal vestibular function thus results in changes in gain and changes in alignment of the 3D VOR response. Here we describe a method to measure 3D VOR using whole body rotation on a 6DF motion platform. Although the method also allows testing translation VOR responses (1), we limit ourselves to a discussion of the method to measure 3D angular VOR. In addition, we restrict ourselves here to description of data collected in healthy subjects in response to angular sinusoidal and impulse stimulation. Subjects are sitting upright and receive whole-body small amplitude sinusoidal and constant acceleration impulses. Sinusoidal stimuli (f = 1 Hz, A = 4°) were delivered about the vertical axis and about axes in the horizontal plane varying between roll and pitch at increments of 22.5° in azimuth. Impulses were delivered in yaw, roll and pitch and in the vertical canal planes. Eye movements were measured using the scleral search coil technique (2). Search coil signals were sampled at a frequency of 1 kHz. The input-output ratio (gain) and misalignment (co-linearity) of the 3D VOR were calculated from the eye coil signals (3). Gain and co-linearity of 3D VOR depended on the orientation of the stimulus axis. Systematic deviations were found in particular during horizontal axis stimulation. In the light the eye rotation axis was properly aligned with the stimulus axis at orientations 0° and 90° azimuth, but gradually deviated more and more towards 45° azimuth. The systematic deviations in misalignment for intermediate axes can be explained by a low gain for torsion (X-axis or roll-axis rotation) and a high gain for vertical eye movements (Y-axis or pitch-axis rotation (see Figure 2). Because intermediate axis stimulation leads a compensatory response based on vector summation of the individual eye rotation components, the net response axis will deviate because the gain for X- and Y-axis are different. In darkness the gain of all eye rotation components had lower values. The result was that the misalignment in darkness and for impulses had different peaks and troughs than in the light: its minimum value was reached for pitch axis stimulation and its maximum for roll axis stimulation.

CASE PRESENTATION

Nine subjects participated in the experiment. All subjects gave their informed consent. The experimental procedure was approved by the Medical Ethics Committee of Erasmus University Medical Center and adhered to the Declaration of Helsinki for research involving human subjects. Six subjects served as controls. Three subjects had a unilateral vestibular impairment due to a vestibular schwannoma. The age of control subjects (six males and three females) ranged from 22 to 55 years. None of the controls had visual or vestibular complaints due to neurological, cardio vascular and ophthalmic disorders. The age of the patients with schwannoma varied between 44 and 64 years (two males and one female). All schwannoma subjects were under medical surveillance and/or had received treatment by a multidisciplinary team consisting of an othorhinolaryngologist and a neurosurgeon of the Erasmus University Medical Center. Tested patients all had a right side vestibular schwannoma and underwent a wait and watch policy (Table 1; subjects N1-N3) after being diagnosed with vestibular schwannoma. Their tumors had been stabile for over 8-10 years on magnetic resonance imaging.

Measuring torsional eye movements by tracking stable iris features

We propose a new method to measure torsional eye movements from videos taken of the eye. In this method, we track iris features that have been identified as Maximally Stable Volumes. These features, which are stable over time, are dark regions with bright borders that are steep in intensity. The advantage of Maximally Stable Volumes is that they are robust to nonuniform illumination and to large changes in eye and camera position. The method performs well even when the iris is partially occluded by reflections or eyelids, and is faster than cross-correlation. In addition, it is possible to use the method on videos of macaque eyes taken in the infrared, where the iris appears almost featureless.

Peaks and troughs of three-dimensional vestibulo-ocular reflex in humans

The three-dimensional vestibulo-ocular reflex (3D VOR) ideally generates compensatory ocular rotations not only with a magnitude equal and opposite to the head rotation but also about an axis that is collinear with the head rotation axis. Vestibulo-ocular responses only partially fulfill this ideal behavior. Because animal studies have shown that vestibular stimulation about particular axes may lead to suboptimal compensatory responses, we investigated in healthy subjects the peaks and troughs in 3D VOR stabilization in terms of gain and alignment of the 3D vestibulo-ocular response. Six healthy upright sitting subjects underwent whole body small amplitude sinusoidal and constant acceleration transients delivered by a six-degree-of-freedom motion platform. Subjects were oscillated about the vertical axis and about axes in the horizontal plane varying between roll and pitch at increments of 22.5° in azimuth. Transients were delivered in yaw, roll, and pitch and in the vertical canal planes. Eye movements were recorded in with 3D search coils. Eye coil signals were converted to rotation vectors, from which we calculated gain and misalignment. During horizontal axis stimulation, systematic deviations were found. In the light, misalignment of the 3D VOR had a maximum misalignment at about 45°. These deviations in misalignment can be explained by vector summation of the eye rotation components with a low gain for torsion and high gain for vertical. In the dark and in response to transients, gain of all components had lower values. Misalignment in darkness and for transients had different peaks and troughs than in the light: its minimum was during pitch axis stimulation and its maximum during roll axis stimulation. We show that the relatively large misalignment for roll in darkness is due to a horizontal eye movement component that is only present in darkness. In combination with the relatively low torsion gain, this horizontal component has a relative large effect on the alignment of the eye rotation axis with respect to the head rotation axis.

Cyclorotation models for eyes and cameras

The human visual system obeys Listing's law, which means that the cyclorotation 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 conveniently be formulated in terms of rotation matrices. The function that defines the observed cyclorotation 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 simplest 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 condition is important for robotic models of human vision, which typically do not reproduce the mechanics of the oculomotor system.

Evaluation tests for eye tracking systems

In this paper, we describe the tests developed by our research team to evaluate the performance of eye tracking algorithms to quantify vertical and horizontal eye movements. To perform the tests, we created a special library to generate synthetic eye images, where we can control most characteristics of the eyes. It is possible to simulate the different situations observed during the acquisition of eye images of a real subject.

Robust measurement of ocular torsion using iterative Lucas-Kanade

We present a new method to measure ocular torsion using Lucas-Kanade method. After pixels of iris annulus around a pupil have been converted into Cartesian coordinates, 30 features on the iris was selected then the features were tracked using the iterative Lucas-Kanade algorithm to calculate torsional shift. The results show that a precision of the method is higher than those measured by a conventional cross-correlation and by a template matching method. The suggested method showed 0.03 degrees mean error with 0.15 degrees maximum error. Particularly, the method was robust to change of pupil size and misalignment of pupil location. Processing time was also fast enough to be implemented in a real-time system.

Measurement of ocular torsion using iterative Lucas-Kanade optical flow method

This paper presents a new method for measuring ocular torsion using the optical flow. Iris image is cropped and transformed into rectangular image to make a orientation invariant image. Feature points were selected at iris region from a reference and a target image, and then shift of each feature was calculated using iterative Lucas-Kanade method. The feature points were selected according to the strength of corner on the iris image. The accuracy of the algorithm was tested using printed eye image and compared with traditional cross-correlation method. Measurement error was less than 0.15 degree.

Measurement of ocular torsion using iterative optical flow

This work represents a new method for measuring ocular torsion using optical flow. Feature points are obtained from reference and current image, and the relative optical flows of each point are calculated. The feature points are selected according to the strength of corner on the signature of iris. This method is robust and effective in calculation.

A method for size estimation of amorphous pupil in 3-dimensional geometry

Measuring pupil size is a noninvasive method for evaluation of autonomic nervous system (ANS) activity because the pupil is exclusively controlled by the autonomic nervous system. To evaluate the activity of ANS, an accurate pupil size calculation method different from those of pupillometer and eye tracker should be suggested. This paper presents a three-dimensional model of eye and pupil and also suggests an accurate estimation method of pupil size from images projected on two-dimensional image plane with distortions.

Vestibular function testing

PURPOSE OF REVIEW

This review provides an overview of vestibular function testing and highlights the new techniques that have emerged during the past 5 years.

RECENT FINDINGS

Since the introduction of video-oculography as an alternative to electro-oculography for the assessment of vestibular-induced eye movements, the investigation of the utricle has become a part of vestibular function testing, using unilateral centrifugation. Vestibular evoked myogenic potentials have become an important test for assessing saccular function, although further standardization and methodological issues remain to be clarified. Galvanic stimulation of the labyrinth also is an evolving test that may become useful diagnostically.

SUMMARY

A basic vestibular function testing battery that includes ocular motor tests, caloric testing, positional testing, and earth-vertical axis rotational testing focuses on the horizontal semicircular canal. Newer methods to investigate the otolith organs are being developed. These new tests, when combined with standard testing, will provide a more comprehensive assessment of the complex vestibular organ.

Computation of gaze orientation under unrestrained head movements

Given the high relevance of visual input to human behavior, it is often important to precisely monitor the spatial orientation of the visual axis. One popular and accurate technique for measuring gaze orientation is based on the dual search coil. This technique does not allow for very large displacements of the subject, however, and is not robust with respect to translations of the head. More recently, less invasive procedures have been developed that record eye movements with camera-based systems attached to a helmet worn by the subject. Computational algorithms have also been developed that can calibrate eye orientation when the head's position is fixed. Given that camera-based systems measure the eye's position in its orbit, however, the reconstruction of gaze orientation is not as straightforward when the head is allowed to move. In this paper, we propose a new algorithm and calibration method to compute gaze orientation under unrestrained head conditions. Our method requires only the accurate measurement of orbital eye position (for instance, with a camera-based system), and the position of three points on the head. The calculations are expressed in terms of linear algebra, so can easily be interpreted and related to the geometry of the human body. Our calibration method has been tested experimentally and validated against independent data, proving that is it robust even under large translations, rotations, and torsions of the head.

A new method for accurate and fast measurement of 3D eye movements

Videooculography (VOG) is an eye movement measurement method used in the objective evaluation of vestibulo-ocular reflexes (VOR). An important requirement of VOG is to accurately estimate pupil center and ocular torsion, irrespective of drooping eyelids, eyelashes, corneal reflection, and blinking. Finding the accurate center of the pupil is particularly important in three-dimensional VOG, since otherwise, significant errors can occur in measuring torsional eye movement. A fast algorithm was proposed to accurately ascertain the pupil center, in spite of the complicating factors mentioned above. In this study, real-time three-dimensional VOG, which can measure horizontal, vertical, and torsional eye movements and calculate the pupil radius, was implemented using the proposed method. When the pupil radius was determined, the vertical position was measured within an error margin of less than 3%, even though only 10% of the pupil was visible. The time required to measure both three-dimensional eye movements and the pupil radius was less than 16 ms. Thus, eye movements can be measured in real-time. The resolutions of horizontal, vertical, and torsional eye movement were 0.2 degrees, 0.2 degrees, and 0.1 degrees, respectively, with maximum ranges of +/- 35 degrees, +/- 25 degrees, and +/- 18 degrees.

Functional Assessment of Head???Eye Coordination During Vehicle Operation

PURPOSE

Visual impairment, resulting from ocular abnormalities or brain lesions, can significantly affect driving performance. The impact of vestibulopathy on head-eye coordination is also a concern in vehicle operation safety, yet to date there has been little functional research in this area. An understanding of decrements in driving ability resulting from visual and vestibular pathology, plus the differences in visual strategies used by novice and experienced drivers, would benefit from an objective analysis of head-eye coordination during vehicle operation.

METHODS

We have developed a laptop-based system for measuring eye, head, and vehicle movement in real time. Digital video cameras mounted on lightweight swimming goggles are used to provide images of the eye and scene, allowing assessment of gaze. In addition, the use of inertial measurement units to simultaneously transduce head and vehicle movement allows us to evaluate the vestibular contribution to stable vision.

RESULTS

Data was obtained from a flight simulator and while driving a car. During banking turns in the flight simulator, there was a sustained roll tilt of the head and eyes toward the scene-derived visual vertical with a combined gain of approximately 25%. One of the most complex visual tasks when driving was exiting a multistory car park, which involved the scanning of hundreds of parked vehicles with an average fixation time of approximately 100 ms. The vertical vestibulo-ocular reflex was also found to make a significant contribution to the maintenance of dynamic visual acuity even while driving on paved surfaces.

CONCLUSION

These results demonstrate the viability of functional assessment of head-eye coordination during vehicle operation, and potential applications of this technology to driver assessment are discussed. Analysis of both active and reflex contributions to gaze may provide a clearer understanding of the impact of visual and vestibular impairment on driving ability.

Vestibular Perception and Action Employ Qualitatively Different Mechanisms. I. Frequency Response of VOR and Perceptual Responses DuringTranslationandTilt

To investigate the neural mechanisms that humans use to process the ambiguous force measured by the otolith organs, we measured vestibuloocular reflexes (VORs) and perceptions of tilt and translation. One primary goal was to determine if the same, or different, mechanisms contribute to vestibular perception and action. We used motion paradigms that provided identical sinusoidal inter-aural otolith cues across a broad frequency range. We accomplished this by sinusoidally tilting (20°, 0.005–0.7 Hz) subjects in roll about an earth-horizontal, head-centered, rotation axis (“ Tilt”) or sinusoidally accelerating (3.3 m/s2, 0.005–0.7 Hz) subjects along their inter-aural axis (“ Translation”). While identical inter-aural otolith cues were provided by these motion paradigms, the canal cues were substantially different because roll rotations were present during Tilt but not during Translation. We found that perception was dependent on canal cues because the reported perceptions of both roll tilt and inter-aural translation were substantially different during Translation and Tilt. These findings match internal model predictions that rotational cues from the canals influence the neural processing of otolith cues. We also found horizontal translational VORs at frequencies >0.2 Hz during both Translation and Tilt. These responses were dependent on otolith cues and match simple filtering predictions that translational VORs include contributions via simple high-pass filtering of otolith cues. More generally, these findings demonstrate that internal models govern human vestibular “perception” across a broad range of frequencies and that simple high-pass filters contribute to human horizontal translational VORs (“action”) at frequencies above ∼0.2 Hz.

Convergence reduces ocular counterroll (OCR) during static roll-tilt

When humans are roll-tilted around the naso-occipital axis, both eyes roll or tort in the opposite direction to roll-tilt, a phenomenon known as ocular counterroll (OCR). While the magnitude of OCR is primarily determined by vestibular, somatosensory, and proprioceptive input, direction of gaze also plays a major role. The aim of this study was to measure the interaction between some of these factors in the control of OCR. Videooculography was used to measure 3D eye position during maintained whole body (en bloc) static roll-tilt in darkness, while subjects fixated first on a distant (at 130 cm) and then a near (at 30 cm) head-fixed target aligned with the subject's midline. We found that while converging on the near target, human subjects displayed a significant reduction in OCR for both directions of roll-tilt--i.e. the interaction between OCR and vergence was not simple addition or subtraction of torsion induced by vergence with torsion induced by roll-tilt. To remove the possibility that the OCR reduction may be associated with the changed horizontal position of the eye in the orbit during symmetric convergence, we ran an experiment using asymmetric convergence in which the distant and near targets were aligned directly in front of one eye. We found the magnitude of OCR in this asymmetric convergence case was also reduced for near viewing by about the same amount as in the symmetric vergence condition, confirming that the convergence command rather than horizontal position of the eye underlies the OCR reduction, since there was no horizontal movement of the aligned eye in the orbit between fixation on the distant and near targets. Increasing vergence from 130 to 30 cm reduced OCR gain by around 35% on average. That reduction was equal in both eyes and occurred in both the symmetric and asymmetric convergence conditions. These results demonstrate the important role vergence plays in determining ocular counterroll during roll-tilt and may support the contention that vergence acts to reduce the conflict facing a stereopsis-generating mechanism.

Robust and real-time torsional eye position calculation using a template-matching technique

Computation of eye rotation about the line of sight (torsion) using image processing techniques has traditionally used cross-correlation of iral signatures sampled from circular arcs centered on the pupil. We have developed a new algorithm that utilizes a template-matching technique to calculate torsional eye position. Iral signatures are obtained from two annuli centered on the pupil center. By assuming that torsional rotation of the eye is constrained between successive video frames (<2 degrees), only a small window of the previous reference signature is necessary to determine relative torsional eye displacement. This dramatically reduces the number of pixels needed for computing torsion. This algorithm is considerably faster, attains a higher accuracy, and exhibits considerably less noise than the cross-correlation technique. Running on a 800 MHz Intel-based Dual Processor Pentium III, with a Matrox frame grabber, the system is capable of processing three-dimensional eye position at a rate of 120 frames/s.

Improving Calibration of 3-D Video Oculography Systems

Eye movement recordings with video-based techniques have become very popular, as long as they are restricted to the horizontal and vertical movements of the eye. Reliable measurement of the torsional component of eye movements, which is especially important in the diagnosis and investigation of pathologies, has remained a coveted goal. One of the main reasons is unresolved technical difficulties in the analysis of video-based images of the eye. Based on simulations, we present solutions to two of the primary problems: a robust and reliable calibration of horizontal and vertical eye movement recordings, and the extraction of suitable iris patterns for the determination of the torsional eye position component.

Perception of tilt and ocular torsion of normal human subjects during eccentric rotation

HYPOTHESIS

The authors hypothesized that a nonvisual measure of the subjective horizontal during eccentric rotation is a reliable method for evaluating the spatial orientation of normal subjects during such rotation.

BACKGROUND

Eccentric rotation is a promising tool for the evaluation of otolithic function in healthy subjects and in patients. Although eye torsion is an objective measurement, it is also affected by angular acceleration/deceleration. Subjective horizontal is more accurately related to the changes in linear acceleration, but visual judgments of orientation are confounded by ocular torsion.

METHODS

20 subjects were tested during eccentric yaw rotation in both directions, generating centripetal acceleration directed along the interaural axis ranging from 0.38 g to 1 g. Perception of body tilt in roll and in pitch was recorded in darkness using verbal reports and a somatosensory plate that the subjects maintained parallel to the perceived horizon. Torsion of the eyes was recorded by a video camera.

RESULTS

Perceived roll tilt was close to the tilt of the gravito-inertial acceleration vector. However, there were differences in perceived roll and pitch tilt between facing-motion and back-to-motion headings, presumably related to the direction of the tangential acceleration. Ocular torsion was dependent on both angular and centripetal accelerations.

CONCLUSIONS

Measurement of perceived roll and pitch tilt using a somatosensory plate is a reproducible method for quantifying the effects of linear accelerations during eccentric rotation. This method may prove useful for the diagnosis of otolithic dysfunction in dizziness and in patients with vestibular disorders.

Robust pupil center detection using a curvature algorithm

Determining the pupil center is fundamental for calculating eye orientation in video-based systems. Existing techniques are error prone and not robust because eyelids, eyelashes, corneal reflections or shadows in many instances occlude the pupil. We have developed a new algorithm which utilizes curvature characteristics of the pupil boundary to eliminate these artifacts. Pupil center is computed based solely on points related to the pupil boundary. For each boundary point, a curvature value is computed. Occlusion of the boundary induces characteristic peaks in the curvature function. Curvature values for normal pupil sizes were determined and a threshold was found which together with heuristics discriminated normal from abnormal curvature. Remaining boundary points were fit with an ellipse using a least squares error criterion. The center of the ellipse is an estimate of the pupil center. This technique is robust and accurately estimates pupil center with less than 40% of the pupil boundary points visible.

Vestibulo-oculomotor research and measurement technology for the space station era

Recent improvements in measurement techniques and mathematical representations for eye, head and body movement have enhanced our appreciation of the complexity of spatial orientation and locomotion in three-dimensional space. The shortcomings of present measurement techniques, and their solution with emerging technologies are described. The prolonged microgravity conditions on the space station provide a unique opportunity to investigate these three-dimensional aspects of the vestibular and oculomotor systems, and in particular, the role of the otolith afferences. While the canal-ocular responses and their central pathways are reasonably well understood, the community has only recently become aware of the variety of functions fulfilled by otolith-mediated information, i.e., translational otolith-ocular reflex, inertial processing, gravitational reference, vergence control. Recent results, largely from experiments performed on the Mir Station, where the emphasis was on the otolith contribution to the vestibulo-ocular response mechanisms, are reviewed.

Clinical use of Three-Dimensional Video Measurements of Eye Movements

Noninvasive measurements of three-dimensional eye position can be accurately achieved with video methods. A case study showing the potential clinical benefit of these enhanced measurements is presented along with some thoughts about technological advances, essential for clinical application, that are likely to occur in the next several years. (Otolaryngol Head Neck Surg 1998;118:S35-S38)

A geometric basis for measurement of three-dimensional eye position using image processing

Polar cross correlation is commonly used for determination of ocular torsion from video images, but breaks down at eccentric positions if the spherical geometry of the eye is not considered. We have extended this method to allow three-dimensional eye position measurement over a range of +/- 20 deg by determining the correct projection of the eye onto the image plane of the camera. We also determine the orientation of the camera with respect to the eye, allowing eye position to be represented in appropriate head-fixed coordinates. These algorithms have been validated using both in vitro and in vivo measures of eye position.