Full text of DOI:10.1007/s004170100317

Graefe’s Arch Clin Exp Ophthalmol
(2001) 239:562–566
C L I N I C A L I N V E S T I G AT I O N
DOI 10.1007/s004170100317
Michael Bach
The Freiburg Stereoacuity Test:
automatic measurement of stereo threshold
Christina Schmitt
Miriam Kromeier
Guntram Kommerell
Abstract Background: There is a
need for a stereotest with the follow- to about 1/10. In addition, the two
cies that reduced their visual acuity
Received: 8 February 2001
Revised: 17 May 2001
Accepted: 17 May 2001
Published online: 24 July 2001
© Springer-Verlag 2001
ing properties: (1) Natural viewing
conditions, i.e. stimulus contours
visible for each eye alone, but no or
hardly any cue for monocular detec-
observers with insight into the test
design and two strabismic patients
performed the best PEST procedure
with one eye only. Results: With
tion, and (2) suitability for threshold constant stimuli both observers
determination over a wide range of achieved a stereoacuity of 2.6 arcsec
disparities. To comply with these re- and 3.1 arcsec, respectively, taking a
quirements, we developed the Frei- hit rate of 75% as the threshold. The
burg Stereoacuity Test. Method: The best PEST revealed a stereoacuity of
stimulus configuration is shown on a 2.5 arcsec and 3.0 arcsec, respective-
visual display unit (VDU) using
phase-difference haploscopy with
ly. The scatter transparencies raised
the threshold to 261 and 257, respec-
ferromagnetic liquid crystal shutters. tively. With one eye only, the two
The stereo target consists of a verti-
cal bar that can be presented “in
front of” or “behind” a frame. The
sizes of the bar and the frame are
kept constant relative to the stereo
disparity. Anti-aliasing allows for
observers with insight into the test
design exploited the subtle position
cue and reached a coarse pseudoste-
reopsis. The two strabismic patients
did not utilise the position cue.
Conclusion: The Freiburg Stereoacu-
disparities finer than the pixel raster. ity Test allows determination of ste-
To mask monocular cues the bar is
displaced randomly to the right or
left. The stereo threshold was deter-
mined in two observers with normal
eyes, using first the method of con-
stant stimuli and then the best PEST. Stereoacuity Test is available at
Both procedures were repeated with http://www.ukl.uni-frei-
observers wearing scatter transparen- burg.de/aug/bach/fst/.
reoacuity over a wide range of dis-
parities (1–1000 arcsec). Although
the stimuli can be seen with each eye
alone, monocular depth cues are
sufficiently masked. The Freiburg
M. Bach ( ) · C. Schmitt · M. Kromeier
✉
G. Kommerell
Abteilung Neuroophthalmologie und
Schielbehandlung,
Universitäts-Augenklinik,
Killianstrasse 5, 79106 Freiburg, Germany
e-mail: bach@uni-freiburg.de
Fax: +49-761-2704053
see a cluster of disparate random dots, not standing out
as single elements, but marking a plane as being in front
Introduction
Stereopsis is the ability to detect, on the basis of binocu- of or behind a reference plane. Random-dot tests have
lar disparity, whether a single small feature, e.g., a line, the advantage that they avoid monocular cues, but they
is in front of or behind other features. Julesz [9] called differ considerably from natural viewing conditions.
this ability “local” stereopsis and distinguished it from Moreover, in random-dot tests an ambiguity can occur as
“global” stereopsis that manifests itself in the ability to to which elements belong to each other on the retinae, so

563
that relatively large disparities are required to provide
depth sensation. To avoid these drawbacks we endeav-
oured to develop a test that allows measurement of “lo-
cal” stereoacuity over a wide range of disparities. Mon-
ocular cues had to be obscured.
Conventional methods for measuring stereoacuity suf-
fer from one or more of the following shortcomings: The
number of presentations at each disparity step is small,
the threshold is not systematically estimated, and mon-
ocular cues are present so that stereoacuity cannot be
distinguished from position hyperacuity. Using computer
graphics, we avoided these shortcomings and developed
the “Freiburg Stereoacuity Test”. Our test resembles the
classical three-rod test, but bears three advantages: (1)
statistical evaluation is performed online; (2) monocular
position information is masked by random lateral offset
of the stereo target [17]; (3) the distance between the ste-
reo target and the reference contours is adjusted to mim-
ic typical situations in real life: The reference contours
are close to the stereo target when a high stereoacuity is
required, e.g., when threading a needle, and the reference
contours are distant when gross stereopsis is required,
e.g., when pouring a cup of tea. Increasing the distance
between the stereo target and the reference contours for
large disparities carries the additional advantage that any
overlap of the stereo target with the reference contours is
avoided.
Fig. 1 Display for the right eye in real size (to be viewed from
4.5 m) when the stereo disparity between the vertical bar and the
frame is 100 arcsec. The pattern with random black-and-white
squares that extends up to the borders of the VDU is trimmed in
the figure. Due to anti-aliasing the edge of the bar is slightly
fuzzy, allowing for disparities finer than the pixel raster (Fig. 2).
To mask monocular cues the bar is shifted randomly to the right or
left; depicted is a displacement to the left. For larger and smaller
stereo disparities, the sizes are scaled proportionally, with the con-
straint that the distance between the bar and the frame is kept at a
minimum of 300 arcsec
(20 arcsec at a distance of 4.5 m). To overcome this limitation,
“anti-aliasing” [2, 16] is applied: The margins of the vertical bar
are smoothed with a gradual transition of the luminance following
a gaussian profile (Fig. 2). The gaussian profile has a standard de-
viation of 2 pixels. The profile can be shifted to the right or left in
steps of less than 1 arcsec at a distance of 4.5 m. The gradual tran-
sition from black to white is acceptable since it merges with the
blur of the retinal image brought about by the optics of the eye.
The optical point-spread function of the eye resembles a gaussian
profile of at least 120 arcsec width at half magnitude [6]. Thus,
any shift of a point stimulus changes the relative illumination of a
Methods
The Freiburg Stereoacuity Test
Technical design
The stimulus is presented at a distance of 4.5 m on a visual display group of neighbouring photoreceptors. The visual system evalu-
unit (VDU), 36 cm wide and 27 cm high, with a resolution of ates these changes, making possible a spatial resolution far beyond
800×600 pixels and a frame rate of 120 Hz. A high luminance the grain of the photoreceptors, as exemplified in the various
(390 cd/m2) is achieved by a special black-and-white VDU forms of hyperacuity, including stereoacuity [6].
(GD403, Richardson Electronics). The screen is driven from the
The anti-aliasing technique requires accurate control of the lu-
mainboard graphics card of a standard computer (Macintosh G4). minance, taking into account the inherent non-linearity of cathode
The software for the generation of the stimulus and the interactive ray tubes. Linearisation of luminance requires a correction with an
determination of the threshold is written in C++.
inverse gamma value [3], which is determined by a simple psy-
Nearly complete separation for the right and left eye is chophysical adjustment task: Two fields are presented adjacent to
achieved by a pair of ferroelectric liquid crystal shutter goggles each other. One is the reference field, consisting of a grid with
(FE1, Cambridge Research Systems). The voltage applied to the black and white stripes. Viewed out of focus it appears as a homo-
liquid crystals controls their transparency. The shutter goggles are geneous grey field with a luminance of 50%. The other field con-
synchronised to the monitor frequency so that images are present- sists of a homogeneous grey. Its luminance has to be equalled by
ed alternately to the right and the left eye. Each eye receives its the operator to that of the reference field. Any residual non-linear-
image at a frequency of 60 Hz, which is just above flicker fusion ity at both ends of the luminance scale is avoided by using only
frequency. Compared to standard liquid crystal shutters [12], the the 5%–95% range. This can easily be tolerated since a reduction
ferroelectric LCDs switch more rapidly (≈50 µs) and have a higher of stereoacuity would be expected only below 50% contrast [10].
on:off ratio (1:500). In a previous version of the Freiburg Stereo-
acuity Test [4, 5] the stimuli for the right and left eyes were sepa- the disparity of the bar. The inner frame width is 8 times the dis-
rated by mirrors instead of shutters.
parity and the inner frame height 10 times the disparity. The
The size of the bar and the frame are kept constant relative to
The stereo target (Fig. 1) consists of a vertical bar that is sur- length of the vertical bar is 70% of the inner frame height so that a
rounded by a frame. Outside of the frame a pattern with random gap remains between the top and the bottom of the bar and the in-
black and white squares (edge length 180 arcsec) is displayed. The ner frame edge. To obtain a sufficient stimulus width and length at
frame and the squares serve as the reference plane. The vertical small disparities, the frame height is kept at least at 3600 arcsec
bar is presented with a stereo disparity up to 1000 arcsec.
and the frame width at 800 arcsec. To mask monocular cues the
In a display with sharp black-and-white edges the stimulus po- bar is not centred with respect to the frame but placed randomly,
sition would be limited to steps according to the size of the pixels trial by trial, to the right or left of the centre by the amount of the

564
the test during its development in many preliminary sessions and
knew all technical details. Observer 1 was 32 years of age. Her re-
fraction was –0.5 cyl 170° in the right and –0.25 sph in the left
eye, but she never wore her correction. Her uncorrected visual
acuity was 1.1 in the right eye and 1.4 in the left eye. Observer 2
was 65 years of age. His refraction was +1.25 cyl 90° in both eyes,
and he always wore his correction. His corrected visual acuity was
2.0 in the right and 1.9 in the left eye.
Strabismic patient 1 was 48 years old. He had been strabismic
since early childhood and had esotropia of about 5°. The leading
left eye was emmetropic with a visual acuity of 1.6. The squinting
right eye was slightly hyperopic and had a visual acuity of 0.1.
Strabismic patient 2 was 16 years old. At the age of 6 months his
parents noted a left esotropia, which was treated with spectacles
and occlusion. Beginning at the age of 14 years the esotropia grad-
ually turned to an exotropia of 16°. The leading right eye was em-
metropic with a visual acuity of 1.5. The visual acuity of the
squinting eye was 0.8 with a correcting lens of –0.5 D.
All visual acuities were measured with the automatic Freiburg
Acuity Test [1].
Stereo threshold with constant stimuli
We used a two-alternative forced-choice paradigm without instan-
taneous feedback about the correctness of the responses. In a first
run, we varied the disparity between 1 and 1000 arcsec on a scale
with steps of 0.3 log units (factor 2.0). In five further runs we var-
ied the disparity in steps of 0.1 log units (factor 1.26) between
1 arcsec and a level at which 100% correct answers had been
reached in the first run. In each run, each disparity was presented
10 times in random order. The runs were separated by at least
10 min of rest. Analysis was based on a maximum likelihood fit to
a psychometric function (Fig. 4), as described by Meigen et al.
[14].
Fig. 2 Luminance across the vertical bar, following a gaussian
profile. In the example depicted here the horizontal position of the
profiles for the two eyes differs by one fourth of a pixel, corre-
sponding to 5 arcsec. For disparities above 125 arcsec a plateau is
interspersed between the two slopes of the gaussian profile. The
luminances indicated in the figure refer to our experimental set-
up. The VDU was surrounded by cardboard of about 0.5 m width
with a luminance of 11 cd/m2. All values were measured through
the shutter goggle
Stereo threshold with an adaptive staircase procedure
actual disparity. The distance between the bar and the left or right
inner frame edge is 3 times the disparity, but is clamped to
300 arcsec for disparities below 100 arcsec to comply with the
spatial requirements for fine stereoacuity [13].
Since the determination of a threshold with constant stimuli is
very time consuming, we examined whether a more rapid proce-
dure could be applied. For this purpose we implemented the best
PEST [8, 11]). The best PEST assumes that the psychometric
function has a sigmoid form and takes the point where the slope is
steepest as the threshold. Following Liebermann and Pentland [11]
we chose a logistic function. It describes the hit rate P depending
on the stereo disparity as follows:
Procedure
The stereo disparities are chosen on a logarithmic scale, as used
for instance by Norcia et al. [15]. The stimulus duration ends
when the observer makes his or her choice by pressing either the
“in front” or the “behind” button on a response box, a standard nu-
merical USB keypad, connected to the computer by an extension
cord. (It turned out that observers responded above threshold after
about 1 s, below threshold after about 1–5 s). The next stimulus is
presented after an interval of 0.5 s in which the random-dot pat-
tern covers the whole field.
Various strategies for threshold determination can be imple-
mented in the Freiburg Stereoacuity Test, for instance the method
of constant stimuli and adaptive staircase procedures such as the
best PEST (parameter estimation by sequential testing [11]).
where pguess is 1/2, d0 is stereo threshold, s is slope, and d is dis-
parity.
We set the slope of the psychometric function to 3 and as-
sumed it to be constant across stereo acuities on a logarithmic
scale. After each response the best PEST calculates the most likely
threshold on the basis of all previous responses and sets the next
stimulus accordingly (Fig. 3). The best PEST provides a feedback
in that the stimulus decreases when the foregoing answer was cor-
rect. This feedback is so subtle, however, that it can be utilised on-
ly by sophisticated observers.
We limited the best PEST to 100 stimulus exposures. After the
initial 12 and then after every fifth real stimulus we presented a
“bonus” with a disparity five times the current estimation of the
threshold to maintain the observer's motivation.
Validation
Subjects
We examined the suitability of the Freiburg Stereoacuity Test in
two observers with normal eyes (authors CS and GK) and in two
cooperative patients who had been strabismic from birth and had
never experienced stereopsis. The two observers had performed
Observers 1 and 2 underwent the best PEST procedure 6 times
after having passed the tests with constant stimuli.

565
Fig. 3 Best PEST: sequence of 100 stereo disparities presented to Fig. 4 Psychometric functions obtained with constant stimuli (ob-
observer 1 and strabismic patient 1. If a response is correct, the server 1). The solid line indicates the 75% hit rate reached at
next stimulus is harder to detect, and vice versa. Thus, the thresh- 2.6 arcsec without scatter transparencies and at 261 arcsec with
old is gradually approached. In the depicted example, observer 1 scatter transparencies. The dashed lines represent the 95% confi-
reached a stereoacuity of 2.3 arcsec. Strabismic patient 1 respond- dence intervals. The best PEST results are depicted as crosses (av-
ed by mere chance so that the best PEST reacted with the largest erage of the 6 runs) and horizontal bars (95% confidence inter-
possible disparity
vals)
Stereoacuity with scatter transparencies
Both the constant stimuli procedure and the best PEST were re-
peated with scatter transparencies in front of the eyes which re-
duced the visual acuity in observer 1 from 1.1 to 0.16 in the right
eye and from 1.4 to 0.11 in the left eye, and in observer 2 from
2.0 to 0.18 in the right eye and from 1.9 to 0.14 in the left eye.
Monocular controls
Observers 1 and 2, who had performed the whole sequence of ex-
periments described above and knew all the details of the test de-
sign, performed the best PEST with their left eye occluded. The
two strabismic patients performed the best PEST 4 times with
their squinting eye occluded.
Fig. 5 Psychometric functions obtained with constant stimuli and
the results of the best PEST (observer 2). The 75% hit rate was
reached at 3.1 arcsec without scatter transparencies and at 257 arc-
sec with scatter transparencies
Results
The procedure with constant stimuli yielded sigmoid
psychometric functions between hit rates of 50%, the
chance level in the two-alternative test, and 100%
With one eye only, the two observers, utilising the po-
(Figs. 4, 5). Applying the conventional 75% hit rate for sition cue, achieved a coarse pseudostereopsis: observ-
the threshold, observer 1 reached a stereoacuity of er 1 at 56 arcsec, observer 2 at 26 arcsec. The two stra-
2.6 arcsec without and 261 arcsec with the scatter trans- bismic patients responded by chance.
parencies. The values for observer 2 were 3.1 arcsec and
257 arcsec, respectively. With best PEST the two observ-
ers reached very similar values (Figs. 4, 5). Observer 1 Discussion
reached an average of 2.5 arcsec (95% confidence inter-
val 2.3–2.7 arcsec) without scatter transparencies and an The Freiburg Stereoacuity Test allows presentation of
average of 254 arcsec (95% confidence interval stimuli over a wide range of disparities. Using the anti-
225–286 arcsec) with scatter transparencies. Observer 2 aliasing technique, a disparity as small as 1 arcsec can be
reached an average of 3.0 arcsec (95% confidence inter- realised on a standard monitor with resolution of
val 2.4–3.7 arcsec) without scatter transparencies and an 800×600 pixels at a distance of 4.5 m. By adjusting the
average of 327 arcsec (95% confidence interval distance between the stereo target and the reference
261–411 arcsec) with scatter transparencies.
frame to values as small as 300 arcsec very high stereo-

566
acuity can be reached, as exemplified by observer 1,
As opposed to random-dot stereograms, the Freiburg
whose stereoacuity of 2.6 arcsec is in the region typical- Stereoacuity Test uses a stimulus that can also be seen
ly reached by trained observers in differentiation tasks monocularly. This bears the advantage that the viewing
between “in front” and “behind” [7]. On the other hand, condition is natural and that very small stereodisparities
gross stereoacuities of around 1000 arcsec can also be can be detected, but also entails the disadvantage that the
measured with the Freiburg Stereoacuity Test, allowing, lateral position of the stereo target provides a monocular
for instance, quantification of the detrimental effect of cue for depth. In the Freiburg Stereoacuity Test this
optical blur. This was demonstrated in observers 1 and 2, monocular cue is masked by randomly placing the target
whose stereoacuity was reduced by scatter transparencies to the right or left. The two observers who knew all the
to about 250 arcsec.
details of the test were able to utilise the position cue,
The logarithmic scale used in our design has proved achieving a coarse pseudostereoacuity of about 40 arc-
to be adequate since the shape of the psychometric func- sec, about 10 times worse than their true stereoacuity.
tion was quite similar when the stereoacuity was deter- Nonetheless, the masking can be considered sufficient
mined without scatter transparencies at about 2.5 arcsec since two cooperative strabismic patients who had never
and with scatter transparencies at about 250 arcsec. The experienced stereopsis and thus were used to relying
alternative, a linear scale, would have resulted in very solely on monocular cues for depth distinction were un-
different shapes of the psychometric function at high and able to solve the test.
low stereoacuities; the curve would have been quite flat
at 250 arcsec.
In conclusion, the Freiburg Stereoacuity Test fulfils
requirements that reach far beyond those conventionally
Implementing the best PEST demonstrated that the requested for clinical stereo tests. Hence, the Freiburg
Freiburg Stereoacuity Test allows measurement of ste- Stereoacuity Test appears suitable for answering
reoacuity with a time-saving adaptive staircase proce- questions such as whether or not a given therapeutic
dure. The average of the best PEST results was in good intervention improves stereoacuity. The Freiburg Ste-
agreement with the 75% correct answers used as the reoacuity Test is available at http://www.ukl.uni-frei-
threshold in the two-alternative forced choice paradigm burg.de/aug/bach/fst/.
with constant stimuli. Moreover, the reproducibility of
the best PEST, apparent in a small 95% confidence inter-
val, is very good in trained observers.
Acknowledgement Supported by the Deutsche Forschungsge-
meinschaft (KO 761/1-1).
References
1. Bach M (1996) The Freiburg Visual
Acuity Test – automatic measurement
of visual acuity. Optom Vision Sci
73:49–53 http://www.ukl.uni-frei-
burg.de/aug/bach/fst/
2. Bach M (1997) Anti-aliasing and
dithering in the 'Freiburg Visual Acuity
Test'. Spatial Vision 11:85–89
3. Bach M, Meigen T, Strasburger H
(1997) Raster-scan cathode ray tubes
for vision research – limits of resolu-
tion in space, time and intensity, and
some solutions. Spatial Vision
10:403–414
6. Fahle M, Troscianko T (1991) Limiting 13. McKee SP (1983) The spatial require-
factors of texture and stereoscopic
depth perception in humans. In:
Kulikowski J, Walsh V, Murray IJ
(eds): Vision and visual dysfunction,
vol 5. Macmillan, London, pp 120–132
7. Fahle M, Henke-Fahle S, Harris J
(1994) Definition of thresholds for ste-
reoscopic depth. Br J Ophthalmol
78:572–576
8. Harvey LO Jr (1986) Efficient estima-
tion of sensory thresholds. Behav Res
Methods Instrum Computers
18:623–632
9. Julesz B (1971) Foundations of cyclo-
ments for fine stereoacuity. Vision Res
23:191–198
14. Meigen T, Lagrèze W, Bach M (1994)
Asymmetries in preattentive line detec-
tion. Vision Res 34:3103–3109
15. Norcia AM, Sutter EE, Tyler CW
(1985) Electrophysiological evidence
for the existence of coarse and fine dis-
parity mechanisms in human. Vision
Res 25:1603–1611
16. Rohaly AM, Wilson HR (1999) The ef-
fects of contrast on perceived depth
and depth discrimination. Vision Res
39:9–18
4. Bömer T, Gentner C, Jedynak A,
Kommerell G (1993) Der Freiburger
Stereotest zur Beurteilung des Stereo-
sehens bei Führerscheinbewerbern.
Klin Monatsbl Augenheilkd
pean perception. University of Chicago 17. Westheimer G, McKee SP (1980)
Press, Chicago
Stereogram design for testing local ste-
reopsis. Invest Ophthalmol Vis Sci
19:802–809
10. Legge GE, Gu YC (1989) Stereopsis
and contrast. Vision Res 29:989–1004
11. Lieberman HR, Pentland AP (1982)
Microcomputer-based estimation of
psychophysical thresholds: the Best
PEST. Behav Res Methods Instrum
14:21–25
202:511–519
5. Bömer T, Dölp R, Kommerell G (1995)
The psychometric function for detec-
tion of stereo disparity in normal sub-
jects. Ophthalmologe 92:120–124
12. Lindblom B, Frisén L (1988) Measur-
ing stereo acuity with liquid crystal
shutters and computer graphics. Neuro-
ophthalmology 8:283–287