A new digitised screen test for strabismus measurement

Abstract

Purpose

Our study presents a digitised tangent screen test for ocular motility analysis according to the Harms and Hess tests (measurement of the squint angle in all fields of vision). This test uses an image beamer to display the tangent screen, a position sensor to measure the patient’s head orientation, and a distance sensor to measure the fixation distance. Digital measurement of head orientation allows for a test procedure that eliminates the conventional requirement for a light pointer in the patient’s hand. Thus, the digital screen test is presented, and the uncertainty of the measurement system is evaluated.

Methods

A mathematical relationship was given between the measured squint angles, as well as the angle of diagnostic gaze direction, and the influence quantities on their measurement uncertainty. The individual uncertainties resulted from deviations in the measured values by the position and distance sensors, the calibration of the projection image of the beamer in length units, and the finite image resolution of the beamer. The individual standard uncertainties of the influence quantities were determined. The combined standard measurement uncertainties of the squint and gaze direction angles were given based on the model equation of the error propagation law at the tangent table according to Harms at a test distance of 2.5 m. The patient’s uncertainty contribution to the mobility analysis was not considered.

Results

The combined standard uncertainty of the measurement system (coverage factor k = 2 for 95% confidence level) for the squint angle is ≤ 0.43° for the angle of diagnostic gaze direction ≤ 3.13° at the test distance of 2.5 m. The individual standard uncertainties of the influence quantities on the angles are (k = 1): 1.55°/1.01° (horizontal/vertical angle of the position sensor), 0.19° (distance sensor), 0.06° (calibration of the projection image of the beamer), and 0.02° (image resolution of the beamer). The maximum valid test distance of the digital screen test is 3.8 m.

Conclusion

The digital screen test is compact and can be used at different locations. Compared to the traditional test, the time required for examination via the digitised test is less; additionally, its documentation is simplified. The measurement uncertainty of the diagnostic gaze direction angle is dominated by the sensor drift of the position sensor in the horizontal direction (yaw angle) and is due to the sensor technology. However, this drift error does not affect the squint angle measurement result nor its measurement uncertainty because the measurement principle used here is based on the congruence between the position cross and the fixation object and the confusion principle and compensates for the drift error. The measurement uncertainties of the determined measurement system are the lower limits of the uncertainties in the clinical use of the digital screen test if there are no effects due to significant patient deviations.

Introduction

Binocular vision requires that when fixating on an object, the image appears in both eyes in physiologically identical retinal points in order to create a fusion into a common image [1]. Only an intact fusion allows a spatial visual impression (stereopsis). If the direction of a visual axis deviates from the normal direction during fixation of the object due to misalignment of an eye, stereopsis is disturbed because congruent image superimposition is no longer possible. This misalignment is called strabismus. There are many causes. It may be due to a functional disorder of the eye muscles, refractive errors of the eyes or injuries [2].

Strabismus can be diagnosed with different methods. Simple and orienting examination methods are, for example, the corneal reflection test (Hirschberg test) or the cover test and others [3], [4]. The cover test with prisms allows simple quantitative measurement of the squint angle. With a tangent scale, according to Maddox (Maddox cross) [4], horizontal and vertical squint angles can be measured in the primary direction of gaze. However, a detailed overview of the disturbance of the eye position (and the eye muscles involved) requires the measurement of the misalignment of the eyes when fixating on an object in different directions of gaze [5]. Only in this way can the extent of muscle paralysis (paresis) be determined by measuring unequal squint angles in different directions of gaze (incomitance). Harms [5] described this motility analysis on a tangent screen (Harms tangent screen test): a standardised examination board on the wall with a defined angle grid and a fixation light in its centre, corresponding to the head position of the patient in a given test distance (usually 2.5 m). During the examination, the orientation of both eyes, the orientation of the head and the subjective visual impression of the patient are determined simultaneously in usually nine directions of gaze. This requires further aids, such as a forehead projector on the patient’s head to project a position cross on the board to control the head position, a light pointer (e.g. laser pointer) in the patient’s hand to mark his visual direction and a red filter glass in front of one eye for squint angle measurement according to the visual confusion principle [4], [6]. A light band in the centre of the board, which can be rotated via a servo motor, also allows the measurement of torsional strabismus (cyclotropia). Inaccurate alignment of the forehead projector in the primary position, the unsteady posture of the patient’s head or hand and inaccurate reading of the orientation by the examiner can make the determination of the squint angle up to 5 degrees uncertain [4]. Motility analysis is also possible with the Hess screen test [3], [4], [7]. In contrast to the tangent screen, according to Harms, this is usually performed at close range (usually 0.5 m) and uses glasses with changeable red and green filters and two light pointers. The examiner projects a fixation point on the board with a red-light pointer, and the patient must follow it with a green light pointer. In contrast to the method according to Harms, here, the direction of gaze is not defined by the orientation of the head (forehead projector) but is examined with the head held straight forward.

In clinical practice, these traditional examination methods often require time and longstanding practical experience on the part of the examiner. Automated auxiliary systems in the measurement and diagnosis of strabismus can, therefore, contribute to improvement [8], [9], [10], [11], [12], [13], [14]. Leitritz et al. [15] presented a digital variant of the Harms tangent screen test with a projection to the wall by a beamer, a pointer for the patient (computer mouse) and two-colour glasses with an integrated forehead projector as a fixation aid for head orientation.

In this paper, a further developed digitised tangent screen test (hereinafter referred to as the digital screen test) according to Leitritz et al. [15] is presented, and the measurement uncertainty of the instrument is validated (exclusive of the uncertainty through the patient). In contrast to the digital variant reported by Leitritz et al. [15], here, the head orientation is measured by an absolute position sensor, which allows a test procedure without a light pointer in the patient’s hand. A distance sensor allows an individual scaling of the tangent screen and thus the examination of strabismus for different test distances.

Material and methods

Digital equipment

For the digital screen test, an ultra-short throw beamer (Casio XJ-UT351WN, Casio Computer Co. LTD, Japan, resolution 1280 × 800, brightness 3500 lm) and a mini-computer (Intel® NUC-Kit NUC7i5BNH), which is connected via Bluetooth to a microcontroller (Adafruit Feather M0 Bluefruit LE) on the glasses frame, are used (Fig. 1). Integrated on the controller board are a position sensor (absolute orientation sensor BNO055, Bosch Sensortec GmbH, Germany) and a distance sensor (time-of-flight sensor VL53L1X, STMicroelectronics N.V., Netherlands). The position sensor measures the roll, pitch and yaw angle in degrees, which are used to determine the orientation of the head in three-dimensional space. A position cross then visualises the head orientation on the projected screen and is updated with 50 fps. In the primary gaze direction, i.e. with the head and body posture straight and viewing straight ahead, the distance sensor defines the test distance and the beamer projects a tangent screen with a black background in the appropriate size according to the equations given in the appendix. The glasses have a blue (right eye) and a red (left eye) filter glass corresponding to the equivalent colour spectrum of the beamer and have a maximum transmission at wavelengths of 480 nm (blue) and 700 nm (red) [15]. A digital fixation object in the centre of the screen is displayed in blue or red, depending on the choice for the fixating eye; the eye to be examined sees the position cross in the contrary colour. Before starting a motility analysis with the digital screen test, a calibration of the projected image with pixels in horizontal and vertical length units is required for a selected (or changed) location of the beamer. The adjustments to the beamer, such as contrast, sharpness, colour, brightness and image alignment, are to be made according to the instructions in the manufacturer’s manual.

Figure 1.

Digital screen test in motility analysis according to Harms (here, using a dummy head and camera tripod with swivel joint for measurement uncertainty analysis). A beamer projects the tangent grid on the wall. The patient fixes with the left eye (red glass) the centre of the chart (red square). The right eye to be examined sees only the position cross (blue cross) with the given gaze direction and head orientation. The small picture shows the glasses with the colour filters and the measuring unit with the controller, position sensor and distance sensor attached to their frame. The diode laser is used to analyse the measurement accuracy of the position sensor and is not part of the digital screen test.

Digital screen test

The patient is placed at a given distance in front of the projected screen and looks with the leading eye (hereafter, for example, the right eye) through the blue filter of the glasses at the blue-coloured fixation object, thus defining the primary direction of gaze. The examiner confirms the head orientation and the test distance in the software. The displayed red position cross, which in the example is only seen with the left eye through the red filter, now defines relative to this direction any vertical and horizontal deviation of the head orientation by the measured pitch and yaw angle. If the two visual axes do not intersect at the location of the blue fixation object, i.e. there is a misalignment of the left eye, the patient sees the red cross on the screen at a different position (confusion principle). In the conventional test, according to Harms or Hess, the patient must mark this position (light pointer in hand), and the examiner notes the horizontal and vertical angle deviation relative to the central fixation object as the squint angle. In the digital screen test, on the other hand, the patient is invited to bring the red position cross into alignment with the blue fixation object by moving his head. In doing so, the patient, while continuing to fixate the central object, will have to adopt a head orientation that is opposite to the squint angle (e.g. a lateral head rotation of 5° to the right corresponds to a squint angle of 5° temporally on the left eye). If the objects are congruent, the patient can inform the examiner, and the examiner confirms the squint angles in the software. This test can be performed for any diagnostic direction of gaze. For example, if the strabismus is to be examined in tertiary gaze direction up/right with 15°, the coordinates of the position cross are set by these angles on the screen by the software. The direction of gaze is then established by the patient’s head movement in the opposite direction when congruence is requested. A deviation of 15° downwards/leftwards then corresponds to the vertical/horizontal squint angle for this direction of gaze. If the misalignment of the right eye is to be examined, the colours of the projection image are to be alternated by the software. In order to avoid a lateral head tilt of the patient during the motility analysis, the measured roll angle is given as a control. The cyclotropy can be examined in the same way. A digital bar defined in the direction of gaze (instead of a position cross), which assumes the angle of the lateral head tilt (roll angle) and has the colour of the filter glass on the eye to be examined, is to be guided by the patient to the centre of the projection screen and must then be placed parallel to a horizontal bar with a contrary colour. The roll angle then equals the torsional squint angle of the eye.

Measurement uncertainty

The measured angle α in the motility analysis depends not only on the patient’s squint angle but also on other external quantities that influence the measurement result with measurement uncertainty. The influence quantities given in the following are mathematically derived from the measurement equation (A.1) of the tangent screen according to Harms in the appendix. The uncertainty evaluation follows the rules of the Guide to the Expression of Uncertainty in Measurement (GUM) [16].

The causes of the uncertainty can be formulated for the digital screen test by an additive model:

$$\alpha ={\alpha }_{\mathit{ind}}+\delta {\alpha }_{\mathit{ind}}+\delta {\alpha }_d+\delta {\alpha }_{\mathit{cal}}+\delta {\alpha }_{\mathit{grid}}+\delta {\alpha }_{\mathit{pat}}$$ (1)

Here α_ind is the corrected display value (see Appendix) and δα_ind is the deviation of the sensor value pitch or yaw angle, δα_d is the deviation due to the sensor value distance, δα_cal is the deviation due to the finite accuracy of the calibration of the projection screen, δα_grid is the deviation due to the finite image resolution of the projector and δα_pat is the deviation due to the unsteady head and body posture of the patient. With the angle of the gaze direction as the display value α_ind, equation (1) also gives the measurement uncertainty of the diagnostic gaze direction. All mentioned influence quantities δα describe fluctuating deviations (mean value of δα = 0) and are given by the standard uncertainties u(δα). The influence quantities δα have an additive effect on the measurement result according to the model equation (1). The uncertainty of the measured squint angle (combined standard uncertainty u_c) is, therefore, the positive sum of the standard variances u²(δα) according to the rules of Gaussian of error propagation law [16]. The uncertainty u_c is given here with the coverage factor k = 2 for the 95% confidence interval.

The model equation (1) describes the measurement uncertainty due to the measurement system (δα_ind, δα_d, δα_cal, δα_grid) and the measurement process with the patient (δα_pat). Only the influence quantities of the measuring system are investigated here.

Influence quantity δα_ind: The repeatability of the pitch and yaw angle of the BNO055 sensor is examined during the motility analysis in the direction of gaze at 15° according to the test procedure described. For this purpose, a laser is fixed on the rotation point of the glasses (see Fig. 1), and its beam is positioned in the centre of the digital screen instead of the position cross, under the condition that the roll angle is zero degrees. The displayed angular value of the cross, when the laser spot is congruent with the fixation object, is then the angular deviation δα_sens of the sensor. The measurement uncertainty of the angle comparison depends only on the accuracy of the position of the laser spot in the centre of the projection screen. According to equation (A.5), it is approximately δα_spot = ±0.14° in the direction of gaze at 0° and distance at 2.5 m with an estimated reading accuracy of ±2 pixels. The measurement uncertainty of the angles is given as the combined standard uncertainty of the pitch and yaw angle measurement and the deviation of the angle comparison (error limit δα_spot = ±0.14°, equal distribution assumed [16]).

Influence quantity δα_d: The measurement uncertainty of the squint angle as a function of the distance measurement with the sensor VL53L1X is given by equation (A.3) and has a maximum in the secondary gaze direction. The repeatability of the sensor value d_ind is investigated in orthogonal distances in steps of 0.2 m from 0.4 m to 4.4 m to the projection screen. The standard of comparison is a folding ruler with accuracy class III defined in Measuring Instruments Directive 2004/22/EG [17].

Influence quantity δα_cal: The measurement uncertainty of the squint angle as a function of the finite accuracy of the calibration of the beamer (scaling factors c_x and c_y, see Appendix) is given by equation (A.4) and has a maximum in the secondary gaze direction. The factors are determined by applying and measuring with a folding ruler (length 2 m, accuracy class III) at two digital marks specified in pixels at the projected screen. The measurement uncertainty of the factors is given by the folding ruler and the reading accuracy. The reading accuracy will be estimated.

Influence quantity δα_grid: The measurement uncertainty of the squint angle as a function of the finite image resolution of the beamer is given by equation (A.5) and has a maximum in the primary gaze direction. The horizontal and vertical coordinates of the tangent table are rounded to integer values. The maximum rounding error is δp = ±0.5 pixels.

The experimental investigations are realised with darkened ambient lighting and on a white projection wall. The height of the projection area is 2.35 m (width 2.35 mm × 1280/800 = 3.76 mm), according to the commercially available Harms tangent screen (size approx. 2.35 m × 2.35 m, test distance 2.50 m and max. scaling 25° × 25°). In the uncertainty evaluation, the scaling factors are defined as c_x/y = 2350 mm/800 pixels = 2.938 mm/pixel. To determine the standard uncertainty of the squint angles (and the direction of gaze), the influence quantities with maximum deviation (horizontal or vertical angle, viewing direction α_gaze = 0° or 25°) at a test distance d = 2.5 m are specified.

Results

Influence quantity position:Fig. 2 shows the result of the yaw and pitch angle measurement of the BNO055 sensor in a motility analysis in 9 gaze directions at 15° and their repeatability (N = 21). The measurement starts with the central gaze direction at 0°/0° and is continued counterclockwise starting with the evaluation at 15°/0°. The measurement deviations are shown in the given directions of gaze, but they are determined from the congruence between the fixation and position object (here, laser beam at the centre 0°/0°). The standard uncertainty u_sens is, therefore, to be determined from the sample variance with a mean value of zero. The maximum standard uncertainties of the angles from the nine gaze directions are given in Table 1.

Figure 2.

Horizontal (yaw) and vertical (pitch) angular deviations of the position sensor BNO055 in 9 directions of gaze (N = 21). The measurement starts with the central (0°/0°), then with the primary (15°/0°) direction of gaze and continues counterclockwise. The values in brackets indicate the horizontal/vertical standard uncertainty.

Table 1.

Maximum standard uncertainty u_sens of the pitch and roll angle (N = 21), median values as well as minimum and maximum deviations of the position sensor BNO055 from the 9 gaze directions of the motility analysis at 15°.

Direction	usens in °	min. δαsens in °	max. δαsens in °	Median in °
horizontal (yaw)	±1.55	−3.51	+1.19	−0.64
vertical (pitch)	±1.01	−1.30	−0.04	−1.09

Influence quantity distance:Fig. 3 shows the result of the distance measurement of the sensor VL53L1X and their repeatability (N = 18) in the range of 0.4 m to 4.4 m. There is a systematic measurement deviation δd < 0 mm (mean value not equal to zero), which requires a correction $\delta \overline{d}$ of the display value with $d=d_{\mathit{ind}}-\delta \overline{d}$ in the screen test, however, the sensor is used without correction. The measurement deviation is then evaluated with an error limit in the intended distance range, assuming a uniform distribution [16]. In the range from 0.4 m to 3.8 m, the maximum uncertainty δd_limit = 37 mm (as per manufacturer: 20 mm [18]). The standard uncertainty u_d is according to equation (A.3):

$$u_d=\frac{180°}{\pi }\sin \left(2{\alpha }_{\mathit{gaze}}\right)·\frac{\delta d_{\lim it}}{2\sqrt{3}·d}⩽\frac{0.469°·m}{d}=\pm 0.19°({\alpha }_{\mathit{gaze}}=25°,d=2.5m)$$ (2)

Figure 3.

Measured distances and their repeatability (N = 18) of the VL53L1X sensor. With the error limit of 37 mm, the valid distance range is 0.4 m–3.8 m.

Influence quantity calibration: The error limit of the folding ruler is δl = 0.6 mm + 0.4·L [17]. L is the value of the length l to be measured rounded up to the nearest full meter. Measuring digital marks with l > 2 m on the projection screen is not practicable. It is valid L ≤ 2 m and δl ≤ 1.4 mm. The reading accuracy at the two marks is estimated with an error limit of 4 mm. Here it is considered that the projection of a digital marker cannot be localised to within one pixel due to the imaging quality of the beam. For the error limit, uniform distribution is assumed. The standard uncertainty of the length l is $u_l⩽\left(1.4mm+2·4mm\right)/\sqrt{3}=\pm 5.427mm$. The standard uncertainty of the scaling factors c_x/y is maximal with u_c = u_l/800 pixels = ±0.007 mm/pixel in the vertical direction. The standard uncertainty u_cal of the squint angle is according to equation (A.3):

$$u_{\mathit{cal}}=\frac{180°}{\pi }\sin \left(2{\alpha }_{\mathit{gaze}}\right)·\frac{u_c}{2c_{x/y}}⩽\pm 0.06°({\alpha }_{\mathit{gaze}}=25°)$$ (3)

Influence quantity resolution: With the error limit δp = ±0.5 pixel (uniform distribution) and in primary gaze direction (α_gaze = 0°), the standard uncertainty u_grid of the squint angle is according to equation (A.5):

$$u_{\mathit{grid}}=\frac{180°}{\pi }{\cos }^2\left({\alpha }_{\mathit{gaze}}\right)·\frac{c_{x/y}}{d}·\frac{\delta p}{\sqrt{3}}⩽\frac{0.049°·m}{d}=\pm 0.02°({\alpha }_{\mathit{gaze}}=0°,d=2.5m)$$ (4)

Uncertainty of the measurement system:Table 2 shows the summary of the uncertainty contributions of the influence quantities to the standard uncertainty of the measurement system in the motility analysis. The combined standard uncertainty u_c according to equation (5) applies to the squint angle measurement as well as to the diagnostic direction of gaze and is given for a confidence interval of 95% (coverage factor k = 2).

$$u_c=k·\sqrt{u_{\mathit{ind}}^2+u_d^2+u_{\mathit{cal}}^2+u_{\mathit{grid}}^2}$$ (5)

Table 2.

Measurement uncertainty budget of the digital screen test measurement system. The combined standard uncertainty u_c of the squint angles and the diagnostic direction of gaze are calculated from the estimated standard uncertainties u(δα) of the influence quantities according to equation (5).

Information about influence quantities					Standard uncertainties of the influence quantities
Description	Quantity	Unit	Value or Deviation	Comment	Distribution of deviation	Factor c	Equation u(δα)	Standard uncertainty u(δα)	Variance u2 (Contribution to squint angle)	Variance u2 (Contribution to gaze angle)
Squint angle	αind	°	= α sens	Value of pitch or yaw angle of the position sensor with congruence condition
Deviation in the angle validation method	δαspot	°	0.14	Reading accuracy of the laser spot in the angular validation of the position sensor (see text)	equal	$1/\sqrt{3}$	$c·\delta {\alpha }_{\mathit{spot}}$	0.08°	0.0064	0.0064
Gaze angle	αind	°	15	Given gaze direction in eq. (1) (see text)
Deviation of the position sensor	δαind	°	1.55	Experimental result δαsens from the motility analyse according to Harms	normal	$1$	$c·\delta {\alpha }_{\mathit{ind}}$	1.55°	0	2.4025
Distance	d	m	2.5	Test distance according to Harms
Deviation of distance	δd	mm	37	Maximum systematic deviation. Set as error limit (see text)	equal	$1/\sqrt{3}$	Eq. (2)	0.19°	0.0361	0.0361
Calibration of projection	cx, cy	mm/pixel	2.938	Image size (W × H): 3.76 m × 2.35 m
Accuracy of calibration	δcx/y	mm/pixel	0.007	estimated (see text)	equal	$1/\sqrt{3}$	Eq. (3)	0.06°	0.0036	0.0036
Resolution of beamer	px , py	pixel	1280, 800	Manufacturer's information
Deviation of grid	δp	pixel	0.5	Rounded to integer value	equal	$1/\sqrt{3}$	Eq. (4)	0.02°	0.0004	0.0004
							Total result:	uc2 =	0.0465	2.4490
								uc (k=1) =	0.214°	1.565°
								uc (k=2) =	0.428°	3.130°

For the squint angle measurement, if the fixation objects are congruent, the measurement uncertainty u_ind = 0 (no influence by deviation δα_sens of the position sensor, see equation (A.6)). However, the measurement deviation δα_spot in the alignment of the laser spot in the angular validation of the position sensor is considered. For the combined standard uncertainty of the diagnostic direction of gaze, consider the deviation δα_sens of the position sensor: u_ind = u_sens. In the uncertainty budget, the maximum standard uncertainty of the horizontal angle was chosen for u_sens (Table 1). The complete and rounded measurement results (horizontal and vertical) of the measurement system are for the ocular motility analysis according to Harms at a distance of 2.5 m:

$$\begin{array}{ccc}\mathit{Squ}intangle\hfill & \alpha ={\alpha }_{\mathit{ind}}\pm 0.43°\hfill & (k=2)\hfill \\ \mathit{Direction}ofgaze\hfill & \alpha ={\alpha }_{\mathit{ind}}\pm 3.13°\hfill & (k=2)\hfill \end{array}$$

Discussion

Presented is a digital screen test examining strabismus in all gaze directions on a tangent screen. A short distance beamer is used to digitally display the tangent screen. A distance sensor allows the examination of strabismus for different test distances for the patient to the tangent screen. A position sensor digitally measures the patient’s head orientation, allowing a test procedure without a light pointer in the patient’s hand to indicate the squint angle. The procedure of the digital screen test was previously described, revealing compact, movable equipment. An ocular motility analysis can be performed via the digital screen test on a projected tangent screen according to either Harms [5] or Hess [7]. Compared to the traditional examination, the digital screen test requires less time, and the documentation is simplified by the integration of the patient database into the software [15].

The measurement uncertainty of the digital screen test was investigated in a motility analysis on the Harms tangent screen by the influence quantities position sensor, distance sensor, calibration and finite image resolution of the beamer (without uncertainty due to the patient). The expanded standard uncertainty (k = 2, 95% confidence level) of the measurement system was determined with u_c ≤ 0.43° for the squint angles and u_c ≤ 3.13° for the diagnostic gaze direction at a test distance d = 2.5 m, whereby the largest part of the uncertainty budget has the influence quantity “position sensor” for the gaze direction (see Table 2). In the uncertainty budget of the squint angle measurement, the deviation of the influence quantity “position sensor” does not have an effect because the measurement principle is based on the congruence between the position cross and fixation object and the confusion principle. This is described mathematically in the last section of the appendix. The determined measurement uncertainty of the squint angle and the angle of diagnostic gaze direction are valid up to a fixation distance of 3.8 m. For a larger distance, the uncertainty of the distance measurement increases disproportionately in the uncertainty budget (see Fig. 3). This corresponds with the maximum possible range of the distance sensor VL53L1X with approx. 4 m for dark ambient lighting [18]. At greater distances, the sensor has a higher repeatability error.

The position sensor BNO055 uses three triaxial sensors (accelerometer, gyroscope and geomagnetic sensor), whose measured values are fused in the proprietary algorithm and must be calibrated at the usage site in a time-consuming manner (figure-8 motion etc.) [19]. As such, for a selected usage site, a previously saved valid calibration profile of the sensor is loaded from the host system at the beginning of an investigation to ensure quick deployment of the screen test. However, the measurement uncertainty of the horizontal (yaw) angle increases throughout the examination due to long-term drift (see Fig. 2) and is included here in the measurement uncertainty of the gaze direction angles. This phenomenon and its causes are described in the literature [20], [21], [22]. In addition to the sensor technology, the accuracy is also determined by the fusion and calibration algorithm. Furthermore, a stable magnetic field in the usage site environment is important for maintaining measurement reliability [23], [24]. Electromagnetic interference fields or ferromagnetic substances in the near vicinity of the sensor can lead to larger measurement deviations. This is to be considered for a chosen location of the digital screen test.

The position sensor measures the three rotations (roll, pitch and yaw angles) of the patient’s head (body-frame) as absolute values in a fixed reference system (world-frame). The three angles are thus independent of each other for each head orientation. Therefore, there is no dependence between the measured horizontal and vertical angles and their uncertainties, and the model equation (1) does not contain a correlation term of the measured quantities. Due to the measurement principle of congruence between the fixation and position marks, the squint angles and the gaze direction angles and their measurement accuracy do not at all depend on a laterally tilted projection of the tangent scale of the beamer (the tangent scale does not even have to be displayed in the measurement process). The spatial relationship between the patient and the beamer is only given by the test distance to the image projection and the image size. Further influence quantities on the measurement result by the beamer are not considered in equation (1).

In the digital screen test, the result of the squint angles in the mobility analysis is influenced by the uncertainty δα_pat of the patient (measurement process) as well as the uncertainty of the instrument (measurement system) according to equation (1). This uncertainty is dependent on the patient’s cooperation, unsteady head, and body or forced head posture and can take on different values in the measurement process. Nevertheless, the uncertainty in the measurement process can never be less than the uncertainty of the measurement system. The measurement uncertainties determined here are, therefore, the lower limits of the uncertainties in the clinical use of the digital screen test if no other significant deviations act via the measurement process.

In the performance of the traditional test, a lateral head tilt of the patient leads to systematic deviations in the result of the strabismus analysis since the horizontally and vertically measured angles are defined in the global reference system and do not correspond to the real strabismus angles of the patient in the “rotated” system. Therefore, the examiner must usually correct the patient’s head during the examination, which is also necessary without further actions in the digitised version presented here. Alternatively, in the digital screen test, the roll angle of the position sensor is made available in the measurement process, and it is possible to correct the systematic deviation by a mathematical coordinate transformation between the world-frame and body-frame.

The traditional test, according to Harms, is reported in clinical use with an uncertainty of the measured squint angle of up to 5° [4]. For a comparison of the measurement accuracy to the standard procedure, the digitalised test must be evaluated in use. For this purpose, a clinical study utilising the digital screen test and the Harms tangent screen test on patients with strabismus has already been started.

Funding

This work was funded by the Dr Ernst and Wilma Mueller Foundation. At this point, we would like to thank them for their support.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix.

Measurement equation: The definition of the coordinates of the tangent screen according to Harms and Hess are

$$\begin{array}{ccc}x=d·\tan \left(\alpha \right),\hfill & y=d·\tan \left(\beta \right)\hfill & (Harms)\hfill \\ x=d·\tan \left(\alpha \right)/\cos \left(\beta \right),\hfill & y=d·\tan \left(\beta \right)/\cos \left(\alpha \right)\hfill & (Hess)\hfill \end{array}$$ (A.1)

It is d the test distance of the patient to the screen in primary gaze direction (the origin), x and y the horizontal and vertical distance on the screen at angles α and β relative to the origin. The tangent scales of the screens are usually divided into 5 degrees. In the digital screen test, the coordinates (x, y) are a function of the pixel coordinates (p_x, p_y) of the image beamer: x = c_x·p_x and y = c_y·p_y, with the scaling factors c_x and c_y in the unit m/pixel.

Derivation of the influence quantities: At the tangent screen according to Harms is the distance deviation of the horizontal coordinate are

$$\begin{array}{c}\delta x=\frac{\partial x}{\partial d}·\delta d=\tan \left(\alpha \right)·\delta dand\hfill \\ \delta x=\frac{\partial x}{\partial \alpha }·\delta \alpha =\frac{d}{{\cos }^2\left(\alpha \right)}·\delta \alpha \hfill \end{array}$$ (A.2)

The angular deviation δα can be given as a function of the distance deviation δd:

$$\delta {\alpha }_d=\frac{{\cos }^2\left(\alpha \right)}{d}·\delta x={\cos }^2\left(\alpha \right)·\tan \left(\alpha \right)·\frac{\delta d}{d}=\sin \left(\alpha \right)·\cos \left(\alpha \right)·\frac{\delta d}{d}=\sin \left(2\alpha \right)·\frac{\delta d}{2d}$$ (A.3)

With the scaling factor c_x, the distance x = c_x·p_x and their deviation δx = p_x·δc_x = x/c_x·δc_x. The angular deviation δα can be given as a function of the factor deviation δc_x:

$$\delta {\alpha }_{\mathit{cal}}=\frac{{\cos }^2\left(\alpha \right)}{d}·\frac{x}{c_x}·\delta c_x={\cos }^2\left(\alpha \right)·\tan \left(\alpha \right)·\frac{\delta c_x}{c_x}=\sin \left(2\alpha \right)·\frac{\delta c_x}{2c_x}$$ (A.4)

With δx = c_x·δp_x the angular deviation δα is a function of the image resolution δp_x for given parameters d and c_x:

$$\delta {\alpha }_{\mathit{grid}}={\cos }^2\left(\alpha \right)·\frac{c_x}{d}·\delta p_x$$ (A.5)

Equations (A.3), A.4 and A.5 appropriately apply to the vertical angular deviation δβ. The angle values are to be given here in the unit radian.

Measurement principle: In equation (1), the displayed squint angle α_ind = α_sens + α_gaze is equal to the measured angle α_sens of the position sensor at the congruence of the fixation marks, corrected for the preset gaze direction α_gaze. Without loss of generality, let the patient’s squint angle be zero, his head position in the primary gaze direction (α_sens = 0°) and the gaze direction α_gaze (e.g. 15°) be preset. The sensor has a measurement deviation δα_sens (e.g. 1°), and it is α_sens = δα_sens. The position cross is then visualised according to equation (A.1) with the angle α = δα_sens + α_gaze = 16°. In order to bring this position cross into the primary direction (congruence condition), a head rotation angle -α (in the example to the left with 16°) must now be performed. With congruence, the value of the sensor is then δα_sens + α = δα_sens – (δα_sens + α_gaze) = –α_gaze and independent of the value of the measurement deviation δα_sens. The squint angle is α_ind = 0° with the above precondition. The deviation of the squint angle is not a function of the angular deviation δα_sens of the position sensor if the fixation objects are congruent:

$$\delta {\alpha }_{\mathit{ind}}=\frac{\partial {\alpha }_{\mathit{ind}}}{\partial {\alpha }_{\mathit{sens}}}·\delta {\alpha }_{\mathit{sens}}=0·\delta {\alpha }_{\mathit{sens}}=0$$ (A.6)

The measurement deviation of the position sensor only includes the diagnostic gaze direction!