Neurocomputing 44–46 (2002) 907 – 914

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Why is binocular visual space distorted
compared to physical space ? Taro Maedaa; b; ∗ , Eimei Oyamac , Susumu Tachia
a School of Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku,113-8656 Tokyo, Japanb PREST, JST, Honmachi 4-1-8, Kawaguchi-shi, Saitama 332-0012, Japanc National Institute of Advanced Industrial Science and Technology, Namiki 1-2, Tsukuba-shi,Ibaraki 305–8564, JapanAbstract
We propose the independent scalar learning elements summation (ISLES) model, a neuralnetwork model for the developmental learning of binocular visual space. When applied to thephenomena of Hering’s horopter and the locus of perceived egocentric equidistance, this model
provides a better qualitative explanation than the Luneburg theory. This model provides physicalexplanations for the di2erences among individual subject data, by accounting for the spatialdistribution of the visual experiences for learning the perception. When the perceptions areclassi4ed via psychophysical scaling, their class logically determines the learning signal in thismodel. As a result, the same ISLES model can predict each phenomenon when a subject learns
c 2002 Published by Elsevier Science B.V.
the physical loci under various conditions. Keywords: Binocular visual space; Horopter; ISLES model; Psychophysical scaling

1. Introduction
From a psychophysical standpoint, human binocular visual space is distorted anddoes not completely coincide with physical space. Even in darkness, humans can perceive the locations of points of light and their separation distance with binocular vision.
In such a situation, the subjective straight line to the objective point becomes the reference. However, the subjective straight line is observed to have a certain physical curve
which is generally not straight in the physical sense. This phenomenon is well known This paper was presented at CNS∗ 2001.

∗ Corresponding author. Tel.: +81-3-5841-6916; fax: +81-3-5841-8606.

E-mail address: maeda@star.t.u-tokyo.ac.jp (T. Maeda).

c 2002 Published by Elsevier Science B.V.
0925-2312/02/$ - see front matter PII: S 0 9 2 5 - 2 3 1 2 ( 0 2 ) 0 0 4 9 0 - 3

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Fig. 1. Hering’s horopter and the human oculomoter system. (a) Top view of typical horopter. (b) De4nitionof parameters.

as Hering’s front parallel horopter [3]. A horopter curve is a subjective frontal plane.
Fig. 1(a) shows a top-view of some typical front parallel horopter curves. In the 4gure,
L and R are the left and right eyeball positions of the observer, respectively. The shapeof the horopter curve depends on the distance from the observer. At a certain distance(Xa in this 4gure), it is practically straight. At closer distances, the horopter curves areconcave to the observer, while at greater distances they are convex [9]. Discussionsof binocular disparity are not necessary for this phenomenon. It can also be observedby alternately gazing at each point of light in darkness. In this case, the cues for theperception of the vertical are only the vergence angle , and the bipolar latitude .
With these cues, binocular visual space can be described as a perceptual space endowedwith a rich geometrical structure, which according to Luneburg [7] is a non-EuclideanRiemannian geometry of constant curvature. This hypothesis, together with certain psychophysical assumptions, provides a qualitative explanation of classical empirical phenomena. Luneburg’s metric of binocular visual space provides a good description forthe phenomena in the horizontal plane, but it does not give the reason as to why
perceptual space is distorted. A new hypothesis is proposed here, wherein perceptual
space is distorted as a result of some physiological constraint that renders the learning
for the perception incomplete, since the phenomenon is observed independent of thesubject’s mathematical knowledge of the geometry. In order to con4rm this hypothesis, the independent scalar learning elements summation (ISLES) model is proposed,
which is a neural network model for the developmental learning of perceptual space.

2. Background—geometry on binocular visual space
When a human subject gazes at a point of light, the location and orientation of
the eyeballs are identi4ed by the sensory signals of the vergence angle  and thebipolar latitude  (Fig. 1(b)). In the process of human visual space perception, atransformation is required to map  and  to physical world orthogonal coordinates x
and y that describe planes and lines. Under Euclidean geometry, this transformation is

T. Maeda et al. / Neurocomputing 44–46 (2002) 907 – 914909

as follows:
x=W(cos  + cos 2) ≡ X (; );sin y=W
sin 2 ≡ Y (; ):sin (1)On the other hand, in Luneburg’s metric, which is based on Riemannian geometry
and , the horopter is de4ned as a geodesic line. The line meets the -axis at a rightangle, and is asymptotical to one direction at in4nity. The geodesic line is describedas Eq. (2). K is the constant curvature of the space.  is the constant for scaling
the distance. These are de4ned as the individual constants of subjects in Luneburg’smetric. These constants are used to explain the di2erence among the individual data of
the horopters. Here, C is the invariant to de4ne a curve of horopter. When the curve
meets the -axis at 0 , C is de4ned as follows:
1
K 2
K= 2e− cos 2:(2)⇒ C=( + )−1=C ;0−−
=2esin
440
From the standpoint of psychological scaling, some invariant is necessary in order tode4ne a subjective line. This is the necessary condition for the nominal scale, which is
the 4rst class of the psychological scaling. The human perception process also needs
to have some function from the sensory cues to such an invariant, since the humanperception process constantly performs psychophysical measurement.

3. Independent scalar learning elements summations model
A plane and a straight line are abstract concepts acquired developmentally. Therefore, the function for the invariant should be obtained from some learning mechanism
involving a visual space perception process. If the human learning mechanism could
completely learn the function from Eq. (1), then human sensory space would coincidewith physical space. However, the subjective straight line of a human operator di2ers
from a physical one, so the human learning mechanism is apparently incomplete inlearning such a function. So what kind of mechanism generates the characteristics forhuman visual space? The incompleteness should mostly comprise physiological factorsof neural networks of human brains, since everyone has the same tendency. We propose
here an assumption of the physiological learning rule that the physiological learning
mechanism cannot propagate error signals backward to any layer except the 4nal (output) layer. It is such a basic and natural constraint from a physiological viewpoint thatsuch learning mechanisms are actually found in a human brain [8]. In this case, thetraining signal for learning is not a vector but a scalar signal, because only one scalarsignal is made from a scalar evaluating function. We call this learning rule the scalarlearning rule. We provide a neural network model for the developmental learning ofperceptual space, which we call the ISLES model. (Fig. 2)
The ISLES model has similar constraints for learning as a biological neural networkfrom a scalar learning rule. This model does not apply the back-propagation method,

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Fig. 2. ISLES model. (a) ISLES model. (b) Scalar learning element.

but utilizes the Hebbian learning rule as a three-layer Perceptron. Further, the cellsof the hidden layer are simple cells with only one input signal, so the ability ofthis model to learn arbitrary functions is mathematically incomplete. When the model
has n independent input signals, it has n independent groups of simple cells. Eachgroup is called a scalar learning element. This model has n scalar learning elementsf̂ 1 (s1 )f̂ n (sn ) and only one summation unit for an output f̂:f̂(s1 ; s2 ; : : : ; si ; : : : ; sn ) =n

f̂ i (si ) + C;

(3)

i=1

where each f̂ i (x) is a nonlinear continuous scalar function acquired through training.
Each scalar function is made to learn its output with the error signal: Lf̂ i = Lf̂ ≡f − f̂. Here, f(s1 ; s2 ; : : : ; si ; : : : ; sn ) is a training function to be learned. These scalarlearning functions can be implemented by a neural network model with the constraint
mentioned above [5,6]. If the neural network model’s learning method is like thePerceptron or a method of steepest decent, then after suMcient training, each functionf̂ i (x) converges to each expectation as follows:2

lim E[Lf̂ ] = Cmin ⇒ lim E[f̂ − f] = Cmin;t→∞n⇒ lim E f̂ j − f + C  = Cmin

t→∞

t→∞

⇒ limt→∞j=1n

E[f̂ j ] = E[f] + C  :

(4)

j=1

Here, S is de4ned as the whole of the learning domain, and Si is de4ned as a subdomain where si = x. The output of f̂i (x) is determined with the subdomain Si , as

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follows:
limt→∞n
j=1ESi [f̂ j ] = ESi [f] + C  ⇒ lim f̂i (x) +t→∞n

Cj = ESi [f] + C  ;j=1
⇒ lim f̂i (x) = ESi [f] + Ct→∞(s)f(s) ds:⇔ ESi [f(s)] = Si(s) ds
Si(5)Every other function except f̂i (x) becomes constant because each function is dependent
on a single input signal that is independent of every other input. The expected value
ESi [f(s)] can be calculated directly by integrating over the region Si . Therefore, simulations calculated by this model do not need successive training when the distributionfunction (s) of learning points is known a priori. In the ISLES model, input signals
are “isled” with each other until they are summed up in the output layer. Therefore,
while this model cannot completely learn all the mathematical functions, it can makesome di2erences from ideal ones. When the di2erences are similar to those reported
in psychophysical experiments, the neural network model can be considered a good
approximation to the physiological process of spatial perception.

4. Simulation for the horopter curves calculated by ISLES model
ISLES model is incomplete in learning arbitrary functions. However, this incompleteness and the class of psychological scaling for learning made the curves of thefrontal parallel horopter. In order to learn a frontal parallel plane, the training function is not necessarily the X (; ) of Eq. (1), but it is necessarily invariant on thefrontal plane. In addition, the function must be monotonic to the depth x in order toconserve the order. This is the necessary condition for the ordinal scale, which is thesecond class of the psychological scaling, and is necessary and suMcient to measure
parallelity. Here, the value of the invariant is given as f0 , the output at the point P0
crossing the median plane on the same frontal parallel plane. As the de4nition of theinvariant, f0 and P0 correspond to C and 0 of Eq. (2) in Luneburg’s metric. In thiscase, the error signal for learning is de4ned as Lf̂ = f0 − f̂. The domain for learningin the ISLES model is de4ned as the domain where the cues  and  are e2ective
spatial cues in human binocular perception. In this study, the domain was de4ned as−=4 ¡  ¡ =4, 0:003 ¡  ¡ 0:3. This domain is equivalent to the domain of ordinal
human binocular visual experience. Luneburg’s individual constants K,  are explained
as the distribution of the learning points in ISLES model. In this result, the distribution was homogeneous along  and . After training, the experiments for measurementof horopter curves were simulated by the ISLES model. The results are shown inFig. 3(a).

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Y[mm]

Y[mm]

400

Model's Horopter
γ constant locus

400300300200200

1001000-100-200-300-400

The Locus of Egocentric
Equidistance by ISLES
Model
Vieth-Muller Circle

500500

1000

1500

2000

X
[mm]

0500

1000-1001500

2000

X
[mm]

-200-300-400-500-600

Fig. 3. Results of simulation by ISLES model. (a) Hering’s horopter. (b) Egocentric equidistance.

5. Discussion and additional simulation: egocentric equidistance
The shapes of the horopter curves are quite similar to those of humans that the ISLES
model provides a qualitative explanation for the phenomena of Hering’s horopter aswell as Luneburg’s model. Further, this model provides a physical explanation for individual data as the distribution of the learning points. In this result, the  of Luneburg’sindividual constants was slightly greater than the empirical human average, which implies that this distribution of learning points in the near domain is slightly greater thanaverage. This distribution on the ISLES model is concerned with the visual experiencesof the human subjects in physical space, while Luneburg’s individual constants K, can be de4ned only in Riemannian geometry. From this result, it seems that ISLES
model can provide a better explanation in the physical sense than the Luneburg theory.
Although the Luneburg theory provides a qualitative explanation of classical empiricalphenomena, the theory has been less successful in quantitatively predicting individual data [4], a fact that has sometimes been taken as evidence against the presumed
geometrical structure. Some of these shortcomings, however, may also be attributed
to psychophysical assumptions that do not depend on the geometry of visual space.
The locus of perceived egocentric equidistance, for example, has been found to deviate systematically from the Vieth–Muller circle postulated by Luneburg [2]. The locus
is always observed at the outside of the Vieth–Muller circle. However, the locus ofperceived egocentric equidistance is easily de4ned under the ISLES model. An interval scale is necessary for the perception of distances, which is the third class ofpsychological scaling. In order to learn egocentric distances, the training function f
has to conserve the interval. Here, the value of the invariant is given as the intervalLf0 , the di2erence of the outputs at the points whose interval on the physical egocentric distance is invariant LD0 . In this case, the error signal for learning is de4ned as
Lf̂=Lf0 −Lf0 , the di2erence between the Lf0 at a set of points and Lf0 at anotherset of points. The training function f must conserve the Lf0 at any LD0 . Therefore,
this function is necessarily equivalent to aD + b, where a and b are constants and D isx2 + y2 , the Euclidean norm of the learning point as the egocentric distance. AfterT. Maeda et al. / Neurocomputing 44–46 (2002) 907 – 914

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learning the front parallel horopter as well, the loci of perceived egocentric equidistance were simulated by the ISLES model. The results are shown in Fig. 3(b). The
loci by the ISLES model were always observed to be outside the Vieth–Muller circles.
This result means that the same ISLES model can generate both Hering’s frontal parallel horopter and the locus of egocentric equidistance when it learns the physical lociunder each psychological scaling. This con4rms that the ISLES model can provide a
better qualitative explanation than the Luneburg theory. Lately, a computational modelof depth perception based on headcentric disparity was proposed by Erkens and Ee [1].
This model is based on three types of retinal disparity, and integrates the contradictions
among those disparities into the error of the oculomotor signals, in order to compensate
each other’s disparity. The approach of that model may ultimately be closely related
to our study of the ISLES model, although the main depth cues for the two models
di2er.
6. Conclusions
In this paper, we provided the ISLES model, a neural network model for the developmental learning of perceptual space. This model was applied to the phenomenaof Hering’s front parallel horopter and the locus of perceived egocentric equidistance,and provided a better qualitative explanation than the Luneburg theory, especially forthe perception of egocentric equidistance. In addition, this model provides the physicalexplanation for the di2erences among individual data as the spatial distribution of thevisual experiences for learning the perception. Furthermore, the perceptions are classi4ed by psychological scaling, and the class of scale determines the learning signallogically. As a result, the same ISLES model can predict each phenomenon when themodel learns the physical loci under each psychological scaling. This result suggeststhat the class of psychological scaling in perception is signi4cant not only for psychophysical measurement but also for the human learning process, since the perception
system of a human brain is in itself a psychophysical measurement system.
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