Observers frequently underestimate the in-depth slant of rectangles under reduction conditions. This also occurs for slanted rectangles depicted on a flat display medium. Perrone (1982) provides a model for judged slant based upon properties of the 2-D trapezoidal projection of the rectangle. Two important parameters of this model are the angle of convergence of the sides of the trapezoid and the projected length of the trapezoid. This model was tested using a range of stimulus rectangles and found that the model failed to predict some of the major trends in the data. However, when the projected width of the base of the trapezoid projection was used in the model, instead of the projected length, excellent agreement between the theoretical and obtained slant judgements resulted. The good fit between the experimental data and the new model predictions indicates that perceived slant estimates are highly correlated with specifiable features in the stimulus display

Perrone, John A.

Wenderoth, Peter

English

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/ 90-22926
VISUAL SLANT UNDERESTIMATION
John A. Perrone
NASA Ames Research Center
Moffett Field, California
Peter Wenderoth
University of Sydney
Sydney, Australia
SUMMARY
Observers frequently underestimate the in-depth slant of rectangles under reduction conditions.
This also occurs for slanted rectangles depicted on a flat display medium. Perrone (1982) provides
a model for judged slant based upon properties of the two-dimensional trapezoidal projection of the
rectangle. Two important parameters of this model are the angle of convergence of the sides of the
trapezoid and the projected length of the trapezoid. We tested this model using a range of stimulus
rectangles and found that the model failed to predict some of the major trends in the data. How-
ever, when the projected width of the base of the trapezoidal projection was used in the model,
instead of the projected length, excellent agreement between the theoretical and obtained slant
judgments resulted. The good fit between the experimental data and the new model predictions
indicates that perceived slant estimates are highly correlated with specifiable features in the stimulus
display.
INTRODUCTION
Attempts at depicting surfaces slanted in depth on a flat display medium are often hampered by
a common perceptual illusion which results in underestimation of the true depth. Surfaces appear
to lie closer to the fronto-parallel plane than the perspective projection dictates. This has been a
common finding in a wide range of experiments involving slant perception, starting with Gibson's
study (1950) on texture gradients (e.g., Clark, Smith and Rabe, 1955; Gruber and Clark, 1956;
Smith, 1956; Flock, 1965; Freeman, 1965; Braunstein, 1968; Wenderoth, 1970).
The mode of viewing slanted surfaces under the conditions used in slant perception experi-
ments differs from the way we normally encounter visual slant in our environment (Perrone,
1980). Cutting and Millard (1984) has also questioned the use of slant as a variable in the under-
standing of surface perception. However, slant underestimation remains an interesting phe-
nomenon because the information is present in the stimulus display for the veridical perception of
slant (Perrone, 1982), yet apparently the human visual system does not use that information
correctly.
Theories attempting to explain the underestimation are rare. Gogel (1965) applied his
"equidistance tendency" theory to slant underestimation effects and Lumsden (1980) speculated
that truncation of the visual field by the use of an aperture may be a factor causing underestimation.
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https://ntrs.nasa.gov/search.jsp?R=19900013610 2020-03-19T22:08:14+00:00Z
Perrone(1980,1982)hasproposedseveralmodelsof slantperceptionwhichattemptto
accountfor theslantunderestimation.Thispapertestsandmodifiesoneof thesemodels.Our aim
is topinpoint thestimulusfeaturesusedbyobserverswhenmakingvisualslantestimates.This
wouldprovideusefulinsightsinto areassuchasspatialorientation,pictureperception,andpilot
night-landingerrors(Perrone,1984).
MODEL OF SLANT UNDERESTIMATION
The slant angle a), is obtainable from the two-dimensional projection of the surface onto the
retina. (For a technique using perspective lines, see Freeman, 1966; Perrone, 1982.)
The slant angle is found from the two-dimensional variables given in figure 1 using:
0 = tan- 1(tan rc/X)f
This equation states that the slant angle, O, can be derived from the angle of convergence (_) of the
perspective line in the projection, and the distance, X, from the center of the projection out to the
perspective line. In equation 1, f is a known constant and it is the arbitrary distance from the eye to
the theoretical projection plane used to analyze the array of light reaching the eye.
The convergence angle of perspective lines, re, can give the slant angle 0 as long as the correct
distance X is used. Using a value of X greater than the true value will result in a calculated slant
angle less than the actual slant angle, i.e., slant underestimation. Perrone (1980, 1982) proposed a
model which suggested that deviation of the perceived straight-ahead direction results in a judg-
ment of slant based on an incorrect value of X.
Two versions of the model have been proposed:
Model A. Perrone (1982) suggested that because of the reduced viewing conditions and
because of the unusual form of the presenting slant, the observer's perceived straight-ahead direc-
tion deviates from the true straight-ahead (fig. 2) and that the visual system uses the length X'
(equal to the projected length Y) instead of X.
It is proposed that the visual system is attempting to measure the change in width over a square
area of the projection plane, determined by Y, but because there are no perspective lines a distance
X' out from c', the outside edge of the rectangle is used instead. When X' is substituted into
equation (1) instead of X, the equation for perceived slant becomes 13= tan-l(tan_/X')f. How-
ever, in order to use this equation for predicting perceived slant, we need to replace the two-
dimensional variables (_ and X') with the three-dimensional parameters of the stimulus situation.
This gives the following equation for perceived slant:
13=tan_l [Wsin0(D2-L2sin20)]4 L D2cos20
(1)
(2)
0 = actual slant
W = actual width of rectangle
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L = half thetotallengthof rectangle
D = distancefrom eyeto centerof rotation
To date,Perrone(1982)hasshownhowthis sortof analysisprovidesacceptablefits to data
collectedbyothers(e.g.,Clark,Smith,andRabe,1955;Smith,1956),but thesestudieswere
designedto investigateotheraspectsof slantperceptionandsodid not involvedirectmanipulation
of thevariablesintegralto themodel.
Oneproblemwith thisversionof themodelis thatit predictsthatslantoverestimation will
occur when the projected height of the test rectangle (Y) becomes less than the projected half-width
at the axis of rotation (X). However, there have been no published accounts of slant overestima-
tion occurring, but this may simply be because nobody has used test rectangles with the appro-
priate length-to-width ratio.
Model B. (Modified version of Model A). This version proposes that the total base width of
the rectangle (Xb) is used in the evaluation of the slant angle instead of X. This new form of the
model can be interpreted as saying that the observers are basing their slant estimates on the con-
vergence angle, re, of perspective lines which they believe to be twice the true distance out from the
center. It may be that it is a difficult and unnatural task for the observer to judge the slant of a
surface which is centered on the median plane of the eye. It is easier if we have a side view or at
least a more oblique view of the slanted surface. The observers may resort to making their judg-
ments on the basis that they have a more extreme or displaced viewpoint than is in fact the case.
Their interpretation of the slant of the rectangle may be based on an assumed view of the rectangle
which is displaced or rotated relative to its true position.
When this error is combined with the proposed deviation of the perceived straight-ahead
(Perrone 1982), the result may be the erroneous use of the total base width of the projected trape-
zoid rather than the correct half-width at the axis of rotation. When the total projected base width
of a slanted rectangle is used to estimate theta from equation 1, the predicted perceived slant angle
is found using
13= tan -1 [tan0 (D;L sin 0).]2 (3)
0 = actual slant
L = half the total length of rectangle
D = distance from eye to center of rotation
TESTING THE MODEL
An experiment was designed to verify which of the two cases (equation 2 or equation 3) best
models the data from human observers in the slant perception task. If it can be established that
specific features of the stimulus display are being used in the slant estimation process, then the
more difficult task of discovering why these particular variables are being used can be attempted.
The model provides a means of narrowing down the choice of possible variables and the combi-
nation in which they are used.
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Experiment
The stimuli were computer-generated two-dimensional perspective representations of rectan-
gular outline figures, presented on a CRT and viewed monocularly through an aperture. These
figures represented rectangles measuring 25 cm wide with the following lengths: 50 cm (condition
1), 25 cm (condition 2), and 15 cm (condition 3). These were depicted to be at a distance of 57 cm
from the subject's eye and slanted backwards away from the observer by varying angles of slant.
The actual slant angles used were 20 ° 40 °, 60 ° and 80 ° measured from the vertical.
The subject reproduced the judged slant of the rectangle on a response device which was
located 90 ° to the right and positioned at eye level. The response device consisted of a thin black
line inscribed on a clear plexiglass strip which was mounted on a circular white metal disk 23 cm in
diameter. Vertical and horizontal black lines were drawn on the disk to provide anchor points
(Wenderoth, 1970). Subjects were 10 paid volunteers, naive as to the aims of the experiment.
Predictions
If Model A is correct, then the slant estimates for the three different conditions should lie along
three distinct curves given in figure 3a. For some of the stimulus conditions, the subjects should
judge the rectangle to be slanted farther back from the fronto-parallel plane than the true position
(slant overestimation). This corresponds to any region of the curves which lies above the dotted
line in figure 3a. If a Model B is correct, the slant estimates for all three conditions should all lie
on approximately the same curve of the shape shown in figure 3b. No slant overestimation should
occur.
Results
The data from the 10 subjects have been plotted in figure 4 along with the predictions from
Model B. For the case in which a tall narrow rectangle was used (Condition 1), the results are
similar to those obtained in past slant perception experiments which used rectangles with a length-
to-width ratio greater than one, (e.g., Smith, 1956). For this condition, both Model A and B give
reasonable predictions for the smaller test angles (see C1 predictions in fig. 3a). However, for the
remaining conditions, the data depart greatly from the Model A predictions and none of the pre-
dicted overestimation of slant occurred.
The mean absolute error between the Model A predictions and the data over the three conditions
was 13.9 °, (sd = 8.1). For Model B, on the other hand, the mean absolute error was only 2.6 °,
(sd = 1.9). The mean absolute errors from Model A are significantly greater than those from
Model B, (t = 4.5, p < 0.05, 22df) and represent a worse fit between the model predictions and
data.
CONCLUSIONS
Slant underestimation Model A (Perrone 1982) incorrectly predicts overestimation to occur for
rectangles which have a projected length less than half of the base width. In fact, the influence of
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theprojectedlengthof therectangleonslantjudgmentsis minimal. However,ModelB provides
anexcellentfit betweentheexperimentaldataandthepredictions.Thesepredictionsarebasedon
measurablefeaturesof theexperimentalconfiguration.Thereareno freeparameters.Model B
statesthatthetotalprojectedbasewidth of therectangleisusedinsteadof half theprojectedwidth
attheaxisof rotation.Twoparametersof thetwo-dimensionalprojectionareimportantin theslant
estimationprocess:(1) theangleof convergenceof perspectivelinesand(2) thedistanceof the
perspectivelinesfrom thecenterof theprojection.Thesuccessof ModelB suggeststhehuman
observersmakeerrorsin slantestimatesbecausetheymisperceivethissecondparameter.
Thequestionremainsasto whyhumanobserversuse"incorrect"featuresof thestimulusin
theirassessmentof theslantangle. It hasbeenshownthatthecorrectslantangleisobtainable
from theappropriateuseof thevariablesgivenin equation1. Thesevariablesareknownto be
presentin thetwo-dimensionalstimulusreachingtheobserver'seye. Theexperimentaldataare
consistentwith theproposalthatthetotalbasewidth of thetrapezoidalprojectionisusedinsteadof
half theprojectedwidth attheaxisof rotation. However,it doesnot shedanylight asto why this
shouldbethecase.
Furtherresearchis requiredbeforewecanconcludetheactualmechanismsusedby thehuman
visualsystemin makingslantestimates.In themeantime,sufficientevidenceexiststo conclude
thatslantjudgmentsby anobserverarehighlycorrelatedwith specificmeasurablefeaturesin the
two-dimensionalarrayof light reachingtheobserver'seye.Theslantestimatesexhibita large
amountof errorandoftengreatlyunderestimatethetrueslantangle. Thispapershowsthatsuch
errorscannotbeattributedto thefact thatinsufficientinformationexistsin thestimulusfor veridical
slantjudgments.Theinformationisavailable,but is incorrectlyused.
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REFERENCES
Braunstein, M. L.: Motion and texture as sources of slant information. J Exper. Psychol., vol. 8,
1968, pp. 584-590.
Clark, W. C.; Smith, A. H.; and Rabe, A.: Retinal gradients of outline as a stimulus for slant.
Canadian J. Psychol., vol. 9, 1955, pp. 247-253.
Cutting, J. C.; and Millard, R. T.: Three gradients and the perception of flat and curved surfaces.
J. Exper. Psychol.: General, vol. 113, 1984, pp. 198-216.
Flock, H. R.: Optical texture and linear perspective as stimuli for slant perception. Psychol. Rev.,
vol. 72, 1965, pp. 505-514.
Freeman, R. B., Jr.: Ecological optics and visual slant. Pyschol. Rev., vol. 72, 1965, pp. 501-
504.
Freeman, R. B., Jr.: Function of cues in the perceptual learning of visual slant. Psychol.
Monogr.: Gen. Appl., vol. 80, 1966, (2) whole no. 610.
Gibson, J. J.: The perception of surfaces. Amer. J. Psycho1., vol. 63, 1950, pp. 367-384.
Gogel, W. C.: Equidistance tendency and its consequences. Psycho1. Bull., vol. 64, 1965,
pp. 153-163.
Gruber, H. E.; and Clark, W. C.: Perception of slanted surfaces. Percept. Motor Skills, vol. 16,
1956, pp. 97-106.
Lumsden, E. A.: Problems of magnification and minification: An explanation of the distortions of
distance, slant, shape and velocity. In M. A. Hagen (Ed.) The Perception of Pictures, vol. 1,
1980, (pp. 91-134). New York: Academic Press.
Perrone, J. A.: Slant underestimation: A model based on the size of the viewing aperture. Per-
ception, vol. 9, 1980, pp. 285-302.
Perrone, J. A.: Visual slant underestimation: A general model. Perception, vol. 11, 1982, pp.
641-654.
Perrone, J. A.: Visual slant misperception and the 'black-hole' landing situation. Av. Space
Environ. Medicine, vol. 55, 1984, pp. 1020-5.
Smith, A. H. Gradients of outline convergence and distortion as stimuli for slant. Canadian J.
Psychol., vol. 10, 1956, pp. 211-218.
Wenderoth, P. M.: A visual spatial after-effect of surface slant. Amer. J. Pyschol., vol. 83,
1970, pp. 576-590.
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Figure 1.- The two-dimensional information reaching the eye is analyzed on a theoretical projec-
tion plane an arbitrary distance f from the eye. All measurements on the projection plane are
made within the plane of the page.
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true straight-ahead
apparent straight-ahead
A v
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Figure 2.- Deviation of the perceived straight-ahead results in the analysis being carried out about
c' instead of c. Model A states that the length X' (equal to Y) is used instead of X. Model B
proposes that X b is used instead of X.
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8-9



Visual slant underestimation

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