Psychological Review Copyright 1991 by the American PsychologicalAssociation~ Inc. 1991, Vol. 98, No. 3,430--453 0033-295X/91/$3.00 A Critical Review of Luneburg's Model With Regard to Global Structure of Visual Space Tarow Indow Department of Cognitive Sciences, School of Social Sciences University of California, Irvine Visual space (VS) is a coherent self-organized dynamic complex that is structured into objects, backgrounds, and the self. As a concrete example of geometrical properties in VS, experimental results on parallel and (equi) distance alleys in a frameless VS were reviewed, and Luneburg's interpretation on the discrepancy between these 2 alleys was sketched with emphasis on the 2 hypotheses involved: VS is a Riemannian space of constant curvature (RCC) and the a priori assumed correspondence between VS and the physical space in which stimulus points are pre- sented. Dissociating these 2 assumptions, the author tried to see to what extent the global structure of VS under natural conditions is in accordance with the hypothesis of RCC and to make explicit the logic underlying RCC. Several open questions about the geometry of VS per se have been enumerated. Visual space (VS) is the final product of the long series of processes from retina to brain, and phenomenologically it is articulated into individual objects, backgrounds, and the self (Figure 1). The self is a percept consisting of visual and pro- prioceptive experiences. Other visual percepts are due to stim- uli from the physical space (X). (Abbreviations such as VS and X are used throughout, and they are listed in the Appendix.) In contrast to most studies of visual perception, which are con- cerned with local phenomena in VS such as size or color con- stancy, stereopsis, and so forth, the main concern here is with the global structure of VS. The following features of VS will be discussed: VS. 1. VS is the most comprehensive percept that includes all individual visual objects appearing in front of the perceived self. Under ordinary conditions, in every direction we see some percept at a finite distance from the self, which means that VS is bounded in all directions. We never perceive anything to be at an infinite distance. The boundary of VS consists of mutually exclusive parts: individual objects, walls, terrain, or sky. No gap occurs in any part of the boundary. In other words, neither infinity nor emptiness exists in perception. Vacant space lies between the self and that part of the boundary in the direction of looking, but this intervening vacant space is not a percept in the same sense that the part of the boundary is. VS.2. VS has three major directions, which correspond to the three directions in X (e.g., the sky appears above the self on the ground, just as is true of their physical counterparts in X). This study was supported by National Science Foundation Grant IST80-23893. I express my heartfelt gratitude to R. Duncan Luce, a reviewer, who gave invaluable comments and suggestions to smooth out the style of English. Correspondence concerning this article should be addressed to Tarow Indow,Department of Cognitive Sciences, Universityof Califor- nia, Irvine, California 92717. It is through VS that we can guide our physical bodies to behave appropriately in X so as to reach or avoid physical objects. Hence, in the neighborhood of the self, at least, there is such a correspondence between structures of VS and X that enables us "to see things where they are" and "to behave properly" in terms of physical relationships in X. One cannot expect corre- spondence of this kind over the entire fields of VS and X. There is no physical entity that has the same form as the sky we per- ceive. VS.3. Percepts in VS, including the self, are hierarchically related with each other. Each percept is localized with respect to other percepts, which act as its framework. The self is not neces- sarily the ultimate framework for other percepts. It can be a percept that is localized, for example, with regard to the per- ceived wall of a room as the framework. If visible, the ground and the sky are the largest framework formed in VS. VS.4. We perceive geometrical properties in VS: curves, straight lines, intersections, betweenness, distances, con- gruences, parallelness, and so forth. Furthermore, we perceive the movement of objects, including the self. Some properties of perceived objects are magnitude-like and can be roughly or- dered in terms of their magnitude: area, angle, and so forth. This is particularly true with perceptual distance, such as the length of a perceptual straight line or the length of an interval marked by two end points. Of two perceptual distances, we can tell which is larger or that they are almost the same. Often, we can tell something more (e.g., quantitative relations between two perceptual distances, such as subjective ratio or difference). The perceptual distance of an arbitrary orientation is seen di- rectly, not derived from some calculation based on components separately perceived in the respective major directions of VS. VS.5. Under ordinary conditions, VS is structured in the way described above. However, how VS is structured depends on stimulus conditions in X. In complete darkness, VS has almost no structure, and we see only darkness of indeterminate depth. If the eyes are exposed to homogeneous light of a suffi- ciently low intensity, one feels as if being surrounded by a mist 430
Transcript
Psychological Review Copyright 1991 by the American Psychological Association~ Inc. 1991, Vol. 98, No. 3,430--453 0033-295X/91/$3.00
A Critical Review of Luneburg's Model With Regard to Global Structure of Visual Space
Tarow Indow Department of Cognitive Sciences, School of Social Sciences
University of California, Irvine
Visual space (VS) is a coherent self-organized dynamic complex that is structured into objects, backgrounds, and the self. As a concrete example of geometrical properties in VS, experimental results on parallel and (equi) distance alleys in a frameless VS were reviewed, and Luneburg's interpretation on the discrepancy between these 2 alleys was sketched with emphasis on the 2 hypotheses involved: VS is a Riemannian space of constant curvature (RCC) and the a priori assumed correspondence between VS and the physical space in which stimulus points are pre- sented. Dissociating these 2 assumptions, the author tried to see to what extent the global structure of VS under natural conditions is in accordance with the hypothesis of RCC and to make explicit the logic underlying RCC. Several open questions about the geometry of VS per se have been enumerated.
Visual space (VS) is the final product of the long series of processes from retina to brain, and phenomenologically it is articulated into individual objects, backgrounds, and the self (Figure 1 ). The self is a percept consisting of visual and pro- prioceptive experiences. Other visual percepts are due to stim- uli from the physical space (X). (Abbreviations such as VS and X are used throughout, and they are listed in the Appendix.) In contrast to most studies of visual perception, which are con- cerned with local phenomena in VS such as size or color con- stancy, stereopsis, and so forth, the main concern here is with the global structure of VS. The following features of VS will be discussed:
VS. 1. VS is the most comprehensive percept that includes all individual visual objects appearing in front of the perceived self. Under ordinary conditions, in every direction we see some percept at a finite distance from the self, which means that VS is bounded in all directions. We never perceive anything to be at an infinite distance. The boundary of VS consists of mutually exclusive parts: individual objects, walls, terrain, or sky. No gap occurs in any part of the boundary. In other words, neither infinity nor emptiness exists in perception. Vacant space lies between the self and that part of the boundary in the direction of looking, but this intervening vacant space is not a percept in the same sense that the part of the boundary is.
VS.2. VS has three major directions, which correspond to the three directions in X (e.g., the sky appears above the self on the ground, just as is true of their physical counterparts in X).
This study was supported by National Science Foundation Grant IST80-23893.
I express my heartfelt gratitude to R. Duncan Luce, a reviewer, who gave invaluable comments and suggestions to smooth out the style of English.
Correspondence concerning this article should be addressed to Tarow Indow, Department of Cognitive Sciences, University of Califor- nia, Irvine, California 92717.
It is through VS that we can guide our physical bodies to behave appropriately in X so as to reach or avoid physical objects. Hence, in the neighborhood of the self, at least, there is such a correspondence between structures of VS and X that enables us "to see things where they are" and "to behave properly" in terms of physical relationships in X. One cannot expect corre- spondence of this kind over the entire fields of VS and X. There is no physical entity that has the same form as the sky we per- ceive.
VS.3. Percepts in VS, including the self, are hierarchically related with each other. Each percept is localized with respect to other percepts, which act as its framework. The self is not neces- sarily the ultimate framework for other percepts. It can be a percept that is localized, for example, with regard to the per- ceived wall of a room as the framework. If visible, the ground and the sky are the largest framework formed in VS.
VS.4. We perceive geometrical properties in VS: curves, straight lines, intersections, betweenness, distances, con- gruences, parallelness, and so forth. Furthermore, we perceive the movement of objects, including the self. Some properties of perceived objects are magnitude-like and can be roughly or- dered in terms of their magnitude: area, angle, and so forth. This is particularly true with perceptual distance, such as the length of a perceptual straight line or the length of an interval marked by two end points. Of two perceptual distances, we can tell which is larger or that they are almost the same. Often, we can tell something more (e.g., quantitative relations between two perceptual distances, such as subjective ratio or difference). The perceptual distance of an arbitrary orientation is seen di- rectly, not derived from some calculation based on components separately perceived in the respective major directions of VS.
VS.5. Under ordinary conditions, VS is structured in the way described above. However, how VS is structured depends on stimulus conditions in X. In complete darkness, VS has almost no structure, and we see only darkness of indeterminate depth. If the eyes are exposed to homogeneous light of a suffi- ciently low intensity, one feels as if being surrounded by a mist
430
VISUAL SPACE 431
or fog of light, which is called the Ganzfeld phenomenon (Avant, 1965; Metzger, 1930). As the intensity of light is in- creased, the mist appears to retreat, finally consolidating into a surface in the structured VS. How far the boundary of VS ap- pears to the subject depends on the stimulus condition. In this sense, VS is the final product of a long series of processes, very dynamic in character.
The purpose of the present article is to discuss the geometri- cal structure of VS. When the subject is looking straight ahead in open space, the sky and ground or ocean always appear to converge at eye-level (Heelan, 1983; Sedgwick, 1980, 1982). On the retina, the boundary between the two images, sky and ground or ocean, passes through the fovea, and VS is structured so that a percept due to this stimulation occupies the same level with the eyes (Figure 1 ). As to retinal conditions, I will not go beyond this level in this article; there will be no discussion of retinal cues and physiological processes by which the articula- tion of VS is generated. The discussion will stay entirely at a phenomenological level. The survey of literature given here is not intended to be complete. Purely philosophical approaches to visual geometry (e.g., Angell, 1974; Craig, 1969; Hopkins, 1973) are not included. Also omitted are a number of geometri- cal approaches to patterns in a limited surface (Caelli, Hoff- man, & Lindman, 1978) and geometrical illusions related to the geometrical property of that local region (e.g., Dodwell, 1967; Hoffman, 1966, 1971, 1980; Watson, 1978). Also not cov- ered are the large number of studies on stereopsis in which solid patterns before or behind a base plane are studied. The article is limited to studies of geometrical patterns that extend over rela- tively large areas in VS.
Para l le l a n d D i s t a n c e Al leys
The relationship between distant stimuli and the eyes in X is given by geometrical optics, which is based upon Euclidean geometry. In most studies of visual perception Euclidean geom- etry is also used to describe, whenever necessary, the structure of the local phenomenon in VS. However, no a priori reason can be given that VS as a whole has to be structured as Euclid- ean. Indeed, it is perfectly possible that VS as a whole cannot be adequately described in terms of any conventional geometry.
Luneburg (1947, 1948, 1950) I was, perhaps, the first to dis- cuss the geometry of VS, though he was not explicit about how global it was. He concluded that VS is a hyperbolic space of constant curvature. It seems to me that two main motives led him, a geometer, to this problem: the well-known demonstra- tions of Ames and experiments on alleys and frontoparallel lines, which will be described in detail. At first, Luneburg tried to develop theoretical equations directly to X, and these were fitted to experimental data (1947). By the time of the last arti- cle (1950), which was published after his untimely death in 1949, equations were given first in a model space for Rieman- nian geometry and then mapped to X. This second approach is more flexible and will be followed here. To make the discussion concrete, let me begin by describing the alley experiment and Luneburg's theoretical interpretations in the Euclidean map (EM) of Riemannian spaces. The Luneburg model has been discussed by many others (e.g., Balslov & Leibovic, 1971; Blank,
J Forerunners to Luneburg are mentioned in Dr/~sler (1966 ).
The position of a stimulus point Q in X is given by either the Cartesian coordinates (x, y, z) or the bipolar coordinates (7, ~b, 0) as shown in Figure 2A. The origin 0 is the midpoint between the right and left eyes of the observer. In an alley experiment, all points Q~ are presented in the horizontal plane of eye level, HZ(E) (all z~ = 0, and equivalently Oi = 0), 2 and the space is made to be as frameless as possible. This is achieved by using for Q~ either small light points in darkness or small objects in an evenly illuminated surface with invisible edges. This condition is called the frameless VS. An example of the result is given in Figure 3, where the farthest pair of points of Q~ is fixed and the positions of other pairs are adjusted mainly in the direction of the y-axis by the subject according to the following two differ- ent criteria: (a) The two series of points of Q~ along the x-axis are adjusted to appear straight and parallel, which is called a paral- lel (P-alley; filled circles), and (b) each pair, Q~, is adjusted to appear in lateral separation equal to the separation of the fixed pairs, Q~, which is the conventional method and is called an (equi-) distance (D-alley; unfilled circles). Then, it is well known and replicated since Blumenfeld (1913) that the two alleys are not the same in X (Hardy, Rand, & Rittler, 1951; Indow, lnoue, & Matsushima, 1962a, 1962b, 1963; Shipley, 1957b; Squires, 1956; Zajaczkowska, 1956a, 1956b) and the D- alley tends to lie outside the P-alley. Hillebrand (1902) did a series of experiments on the P-alley for obtaining quantitative data on size constancy. In order to establish a quantitative rela- tionship between physical distance and physical size that yields perceptually the same size, he replaced the criterion of being equidistant by that of being "parallel." Hillebrand took it for granted that parallelness and equidistantness are synonymous. He did not experiment using D-alley instructions, but he ca- sually mentioned that each pair of points in his P-alley did not appear equally separated to the subject. The discrepancy can be regarded as a manifestation of the non-Euclidean property of VS. According to this interpretation, the fact that the D-alley lies outside the P-alley suggests that VS is hyperbolic. If VS is elliptic, a discrepancy of reversed direction would be expected. Schelling (1956) reached the same conclusion--that VS is hy- pe rbo l i c -on an entirely different basis.
Another experiment that has often been performed in a frameless VS and that seems to be related to the geometry of VS is as follows. Each series of points along the y-axis is adjusted to appear as a straight line that runs from left to right in parallel to the forehead of the subject. The point in the center, Q~ (filled square in Figure 3), is fixed. Often the series is called a (longitu- dinal) horopter (e.g., Luneburg, 1950; Ogle, 1964). The experi- ment can be done with a three-dimensional display of points and "apparent frontoparallel plane" (AFPP) would be a more accurate name (Foley, 1978,1980). The abbreviation HP is used for a frontoparallel plane, and H-curve for a frontoparallel line on HZ(E). In Figure 3, two Qs on both sides were adjusted in the x and y directions so that the three Qs appeared "frontopar- allel" and also satisfied the condition of either P- or D-alley It has long been known that the H-curve changes its shape accord-
ing to the distance of the fixed point, Q~. As seen in Figure 3, it is concave in X to the subject when Q~ is close, and convex when Q~ is far. This systematic behavior of H-curves can hold whether K (Gaussian curvature) is negative, zero, or positive; the only impact of K is on the position of the inflection point from concavity to convexity. Hence, as to the geometry of VS, H- curves are less diagnostic than the P- and D-alleys.
VS as a R i e m a n n i a n Space o f Cons tan t Curvature ( R C C )
Riemannian geometry is an extension of the geometry of curved surfaces. Associated with each point in this space is a parameter called the curvature, which may vary from point to point. Not unlike the way in which the derivative associated to each point of a curve determines a characteristic of the whole curve, the curvature determines a characteristic of the space. Luneburg set the postulate that VS is a Riemannian space (R) in which Gaussian curvature K is the same for every point in the space. Obviously, this assumption came in large part from the fact that, otherwise, the mathematics becomes intractable. But he also provided the following rationales for the postulate, which were later reiterated by Blank (1958a, 1959), a mathe- matician.
RCC.I. As stated in VS.3, a definite, direct impression ex- ists of the distance t~ 0 between any two points, Pv~ and P~j in VS, and furthermore one can judge whether ~ > ~k/or 8,j < 5kt or neither, in which case ~j = ~kt. Luneburg asserted that ~ is a ratio scale in the sense of measurement theory (Krantz, Luce, Suppes, & Tversky, 1971 ) by admitting perceptual concatena- tion: When Pvi, Poj, and P~k are collinear in VS in this order, then 60 + ~jk = ~k (extensive measurement). This claim will be discussed later. Blank followed the line of thought given by Busemann (1955, 1959). In order to make VS a metric space, two more conditions are necessary in addition to that ~ is a metric: finite compactness and convexity. In short, these condi- tions mean that one can think of a line segment between any two points in VS.
RCC.2. If VS, as a metric space, is assumed to be locally Euclidean, then VS becomes R. Assuming this property is in accordance with the general attitude taken in many studies of visual perception. It is true that figures we perceive on a sheet of paper do not exhibit any property inconsistent with Euclidean geometry. It is also true, however, that quantitative relationships between ~s are only partially explicit to our awareness. For ex- ample, looking at a right triangle, one sees that the oblique side is the longest but does not see the relationship between the three sides stated in the Pythagoras theorem.
RCC.3. One perceives plane surfaces at any place in VS with any orientation. Furthermore, a perceptually straight line, a geodesic in R connecting any two points in a plane, does not anywhere depart from the plane. We can visualize, in any place within VS, a large, fiat wall clock with the hand remaining on the plane no matter in which direction it points. It is im- portant to emphasize that we are thinking of a perceptually
2 In actual experiments, the eyes are often placed slightly higher than the plane for Qs to avoid occlusion.
VISUAL SPACE 433
Z
I I
PHYSICAL SPACE
;'0 '077 /
N
R
OBB'BSB~
Y
B
. / /~oooo.O°"°°°'°'
I LUNEBURG 'S MAPPING FUNCTIONS
EUCLIDEAN MAP C 1 ( pO =ze'~"
-- ~ B " ic circle
/ Be'.".. ° l l o t l n l l ~ ' l (~ =
w*°l 0 ..;. ........ 7<.......;.-,,,,,
. 'i--i" = 0
Figure 2. A: A stimulus point Q in the physical space X; Cartesian and bipolar coordinates. B: A point P in the Euclidean map EM; Cartesian and polar coordinates. (The whole visual space VS is represented within the sphere of radius max P0.)
flat plane, and it is a different question what physical surface, flat or appropriately curved, is required to give rise to such an appearance. In a space with this property (Desargue- sian), curvature cannot vary from point to point (Blank, 1958a, 1959). This space of constant curvature, according to the sign of K, is either one of three geometries: elliptic (K > 0), Eucli- dean (K = 0), or hyperbolic (K < 0).
In arguing that K is constant, Luneburg placed more empha- sis on the possibility of the free mobility of perceptual figures in VS than on its Desarguesian property. In the latter half of the 19th century, after geometries other than Euclidean had be-
come more than intellectual pastimes for mathematicians, Helmholtz and then Lie argued that, assuming differentiability, X must have constant curvature because of the existence and free mobility of solid bodies (Busemann, 1955; Freudenthal, 1965; Suppes, 1977). In VS, the free-mobility condition, how- ever, is more subtle. For example, in a recent review of the Luneburg model, a German mathematician wrote, "We do not see any good reason why constant curvature condition (CCC) a priori should be valid. Free mobility of rigid bodies cannot justify this assumption since the same body will have appar- ently different size in different positions" (Eschenburg, 1980, p.
13). Certainly, free mobility in X and that in VS are different and, to maintain the perceptual congruence, the physical ob- ject has to change its size according to its position and orienta- tion in X. However, it seems to me, what is required to ensure the homogeneity of VS is the possibility that a perceived figure can change its position in maintaining the perceptual con- gruence, and it is irrelevant what condition in X corresponds to this perceptual phenomenon. Although Luneburg (l 947) him- self gave somewhat confusing impressions about this point, he was very explicit in one place (Luneburg, 1947):
The visual shape and size of objects can be repeated in any posi- tion and orientation (in VS) . . . . We are convinced that any ob- ject can be moved as a visually rigid body to any desired position and orientation and the result of this movement is an object metri- cally congruent to the original object. We must, however, not as- sume that such a movement of an object is necessarily the same as a physical movement in a Euclidean space. (p. 48)
In fact, this possibility of VS is presupposed in the procedure to match in size a comparison object to a standard object that are
at different locations in VS. This is a standard procedure in size constancy experiments.
I f free mobility exists in VS, we can apply to VS the same logic as was applied by Helmholtz and Lie to X. However, their logic presupposes the space to be a manifold with differenti- ability properties. Without invoking differentiability, Buse- mann (l 955, chap. 2; 1959) showed that in Riemannian spaces, the Desargusian property holds only in the three cases K < 0, K - 0, and K > 0. If"two-point homogeneous" free mobility of small line segments, instead &figures, is used, the above-men- tioned three are only possible cases of so-called G-spaces of Busemann (wider than R) if G-space dimensionality is even or three (Busemann, 1955, chap. 6; Wang, 1951,1952). Hence, for the whole VS that is three-dimensional or perceptual planes in VS, ifRCC. 1 and RCC.2 are accepted and if two-point homoge- neity holds, we have to conclude that these are RCCs. How to interpret free mobility and two-point homogeneity in VS and in what part of VS it is expected to hold will be discussed in the last section.
If curvature varies from point to point in a space, all geometri- cal figures in the space, angle, or straight line that is called geodesic have to be defined, through the mathematics of varia- tion, from line elements associated with each point. The proce- dure is tedious. However, in an RCC, we can bypass this diffi- culty by using a mediating representation of RCC in Euclidean space (E). It is well known that a sphere surface or a saddle- shaped surface in three-dimensional E 3 can be a model for two- dimensional elliptic or hyperbolic surface R ~, respectively. There is another model to depict R ~ of constant K in E 3 (the same dimensionality). All these models were originally devel- oped in order to show noncontradiction of R when K # 0.
Euc l idean M a p o f VS ( E M )
As a Euclidean map ofVS, I shall refer to a model attributable to Poineare that differs from the more well-known Klein model. Standard textbooks on R and differential geometry do not often refer to these representations, and both models are usually discussed for only the two-dimensional ease (e.g., Greensberg, 1973; Smart, 1978). 3 Luneburg did not explain EM in detail. The author once gave an illustration of EM with derivations ( lndow, 1979), and it suffices here to enumerate its properties. Let us denote a perceived point in VS by P in EM. As shown in Figure 2B, the position of P is given in terms of Cartesian coordinates (4, ~?, ~') or polar coordinates (Po, ~, ~). Because we have no perceptual counterpart to the fact that the two eyes are involved in perceiving VS, the representation can be "cyclopeanY The origin of coordinates of EM, 0, is "Ieh- mire" ( DrOsler, 1966 ), from which directions in VS (left, right, above, etc.) are defined. When K = 0, VS and EM are identical. Hence, only the cases K > 0 and K < 0 are discussed later.
EM. 1. When K # 0, the entire VS as an R ~ can be repre- sented within a sphere in E 3, the center of which is 0. The sphere is called the basic sphere (BS) and, as shown in Figure 2, the intersection of the basic sphere and a plane with elevation angle
3 Recently I came to know that the m-dimensional Poincare model is discussed in Spivak (1979) and Berger (1987).
VISUAL SPACE 435
0 is called a basic circle (BC). We are only concerned with the front half of the sphere or circle. According to whether K > 0 or K < 0, what is represented by the basic sphere and the definition of metrics within the basic sphere are different.
EM.2. Denote by P0 or simply by O the Euclidean distance between two points P~ and P1 in EM. When one point is 0, it is the radial distance to P~ and denoted by po~ or simply by So. Line elements in the two spaces, ds in VS and do in EM, are related in the following way:
ds = G-2(po)ds,
in which
G(Oo) = (1 K \1/2 + 7 sg) , K ~ 0 . (1)
EM.3. The coefficient G-2(s0) in front of do is a function of the radial distance So to P only. Because G-2(o0) does not de- pend on the direction of do from P, any perceived angle defined by the two dss from a perceived point P~ in VS and the corre- sponding angle between two dos from P in EM are the same. Every angle in VS is representable in EM without distortion. A mapping preserving angles is called conformal, and EM is a conformal representation of VS (Loebell, 1950). Conformality holds between an angle from the perceived self in VS and the corresponding angle from 0 in EM, and hence P in Figure 2B represents a perceptual point P~ in VS appearing in the direc- tion ~ and 0 from the self.
EM.4. When K ~ 0, it is impossible to map straight lines in VS into EM without distortion. A nonradial perceptual straight line in VS is represented in EM in the following way. When K < 0, a geodesic is represented by a circle that meets the BS orthogo- nally. When K > 0, it is a circle that meets with the BS on antipodal points (two intersections of a diameter with BS). These are termed geodesic circles. All geodesic circles intersect within or at the basic sphere when K > 0, and hence there are no nonintersecting coplanar geodesics in elliptic geometry. There are an infinite number of such geodesics in hyperbolic geome- try. In either geometry, all perceptually straight lines in VS origi- nating from the self are represented in EM by radial lines from 0. The perceptual distance along a given line in VS cannot be represented by the Euclidean length of the corresponding geo- desic circle in EM. In other words, VS and EM are not isomet- ric. However, the length of a perceptual distance in VS Can be readily obtained from the corresponding o in EM.
EM.5. A perceptual distance of 6 o between two points in VS that are represented by Pi and Pj in EM is related to o0 in the following way:
f 2 I ~ 1 1 1 7 ~ sin- -T 'soG- (o0,)G- (o0j), K > 0
6ij 2 I V - K l 1 0, (2) /
l - - ' - - ~ s inh- ' - ~ Oi.iG- (Ooi)G- (So1), K<
where G is defined in Equation 1. As a special case, radial dis- tance is given by
r 2 J tan -To0 K>o
6° = ] 2 h ' ~ (3) --rSo K<0
EM.6. The radius of BS is So = 2r in terms of the curvature radius r and So = 2 / ~ for K > 0 and So = 2 / - ~ for K < 0 in terms of the curvature KofVS. From Equation 3, when K > 0, 60 is intrinsically limited because tan -1 oe = ~r/2, which implies that the elliptic space is closed. When K < 0, BS is the set of points P that appear at infinity because 6o = oe for o0 = 2 / - ~ , (tanh -1 1 = oo). As indicated in VS. 1, nothing in VS appears at infinity. Hence, when K < 0, VS must be represented as a lim- ited area. When K > 0, VS must also be represented as a limited area within the basic sphere, because otherwise nonintersecting coplanar lines such as the P-alley cannot be represented in EM.
EM.7. Let us represent the whole VS within a sphere in EM and denote its radius by max So, which is shown by the dotted circle in Figure 4. This sphere must be inside the basic sphere and max o0 < 2/ fK- or 2 / - ~ . Then it is convenient to select the metric unit of EM so that max So = 2. In terms of this unit, Equation 3 y i e ld s - 1 < K < 1. Let us call the boundary of VS the effective limit of VS. Its radius, max 60, is 2.
Al leys a n d F ron topa ra l l e l Curves in E M
If discussion is limited to the region in EM that represents VS, Po < max p0 (within broken circles in Figures 2B and 4) for both K > 0 and K < 0, then, for any pair of points, there are an infinite number of geodesic circles that pass through these points and do not intersect within this region. Hence, in order to represent a P-alley passing through a fixed pair of points Pi that are on the eye-level plane, HZ(E) , and symmetrical to the ~-axis (~, _+ ~, 0), Luneburg adopted such a pair of geodesic circles that pass through P1 and are orthogonal to the ~-axis. This choice captures the property that in VS the P-alley consists of two straight lines parallel to the invisible median line repre- sented by the (-axis, which is orthogonal with the ~/-axis (P- curves in Figure 4).
When a point P~ is given on the (-axis (~ , 0, 0), the fronto- parallel curve /~ passing through this point represents the straight line in VS that is parallel to the invisible line passing through the self(i.e., the n-axis). Hence, /~ is the geodesic circle orthogonal to the (-axis at P, (H-curves in Figure 4).
Then, the D-alley passing through Pi should be the locus of such a point on/~-curves that maintains constant perceptual distance from P when P, moves along the (-axis. This is also given by a circle that passes the intersection of the (-axis and BC when K < 0 and that is orthogonal to BC when K > 0 (D-curves in Figure 4). But the D-curve is not a geodesic circle, and some subjects really notice that, after having constructed a D-alley, its two sides do not appear completely straight. It is of interest to see that, as shown in Figure 4, the P-curve when K < 0 and the D-curve when K > 0 are given by the same equation as are the D-curve when K < 0 and the P-curve when K > 0.
Theoretical curves for the P- and D-alleys passing through a pair of points P~ are different in EM when K 4= 0. Hence, if the
436 TAROW INDOW
P EM 3 /%\ 2
HZ(E) o \ ,,L, K.=O
zon, / 1 I t I L -l l
max PO = 2
-I < K < I
- 1:0 (P0) 2 (n / (P0/ 2 C ( ) n D
0 (o) ÷% - 1 : 0 ~ - % + i = 0
r= i/
Figure 4. P- and D-alleys and H-curves in the Euclidean map EM for the horizontal plane of eye-level HZ(E). (The right half is for K < 0 and left half is for K> 0. Equations for alleys are included.)
K < 0
K > 0
correspondence between EM and the physical space X is mon- otone, we have a discrepancy in the same direction in X, and the experimental result shown in Figure 3 corresponds to the case K < 0 (the left half of Figure 4).
It has often been said that the definition of the D-alley is straightforward but that Luneburg's choice of the equation for the P-alley is questionable (e.g., Shipley, 1957b; Squires, 1956). However, the impression the subject has about the P-alley, being "neither converging nor diverging" in the direction correspond- ing to the ~-axis, is really captured by Luneburg's choice of such geodesic circles that are orthogonal to the n-axis. Furthermore, the D-alley is defined on the basis of/-~-curves that are actually P-curves with respect to the n-axis.
Luneburg ' s M a p p i n g Func t ions
What we obtain by the alley or frontoparallel curve experi- merit is a configuration of stimulus points {Qi } in x adjusted by the subject. Hence, the theoretical equations in EM have to be mapped back into X. Luneburg assumed the following map- ping functions (Figure 2):
Po = g ( 7 ) = 2 e - ~ , ~o = @, 0 = 0. (4) Then, we have the theoretical curves in X, which are deter- mined by two parameters: the curvature K and a in the first mapping function. Because
~o ~o ~o dr 400 dr aG-2(po)Po (5)
and G2(p0) given in Equation 1 is always positive, a can be interpreted as the sensitivity for depth perception. When a stim- ulus point Q is moved away from the subject in X, and the convergence angle 3' decreases, the perceptual distance ao in- creases. The larger the value of a, the larger the rate given in Equation 5 for a given value ofpo. The second and third equa- tions of Equation 4 imply that the direction that P is located with regard to 0 in EM, and hence the direction that the point is perceived in VS (conformality in EM.3), is the same as the direction of the stimulus point Q with regard to the observer in X.
When the theoretical curves in EM are mapped to X using Equation 4, the curves thus defined in X describe the configura- tions of stimulus points {Qi } for the P- and D-alleys or H-curves fairly well, provided that the parameters K a n d a are appropri- ately estimated. Luneburg proposed a procedure to estimate these parameters through two additional experiments, and Za- jaczkowska (1956a, 1956b) used this procedure. However, in my experience, the procedure has not yielded satisfactory re- sults (Indow et al., 1962a, 1962b, 1963). On the other hand, when the values are estimated directly from {Q~} for alleys,
VISUAL SPACE 437
H-curves, or both, the fit is always of the order shown in Figure 3. A comprehensive table of values of K and a obtained in this way up to 1979 was presented in Indow and Watanabe (1984a) and in all cases K < 0, which corresponds to the fact that the D-alley lies outside the P-alley. There are studies in which some subjects exhibit the opposite result, which implies that K > 0 (e.g., Battro & Pierro Netto, 1976; Hardy et al., 1951; Shipley, 1957b; Squires, 1956). The entire discussion of the present arti- cle holds independent of the sign of K.
It has become clear that, although the results are always suc- cessfully described by the theoretical equations, K and a are not "universal individual constants" for each subject that are valid under all observing conditions and for all configurations {Q~ }. Experiments have been performed with the same subject in the dark and or in illuminated frameless spaces with various fixed pair QI. Although I had expected that K would be closer to 0 under the illuminated condition than in the dark, that was not necessarily the case. The difference between the two conditions is more distinctly reflected in values of a, which is larger under the illuminated condition. The subject is more sensitive to change in 3" in an illuminated space. Furthermore, in the dark, Kand cr are conditional on the size of{Q1 }. In Figure 5, values of K and tr are plotted against the radial Euclidean distance to QI in X, el = (x~: + y12) ~/2, for 3 subjects who constructed P- and D-alleys over a few years under the same condition (dark and on HZ(E) ) with various fixed farthest points QI. Clearly, a is an increasing function of el and IKI is larger for QI at about 16 m (in a gymnasium) than for Qi of less than 5 m in the laboratory. Dependency of K upon {Qi } has also been reported by others (Ehrenstein, 1977; Hagino & Yoshioka, 1976; Higashiyama, 1981, 1984).
If the effective limit of each frameless VS is defined by max 60 corresponding to max P0 = 2 in EM, though invisible, its posi- tion is determined by K and a in the sense of saying how far the limit is compared with the farthest point P~ visible in VS. Fig- ure 6 gives ratios max 6o/6o~ under various conditions, where max 60 is a function of Kalone and 6Ol depends on both Kand a (Indow, 1984). Although Kand ~ change their values according to conditions, this ratio seems to behave more systematically, in frameless VS at least. In the dark, the subject does not feel that VS extends far beyond the farthest point P~l, whereas the sub- ject sees something behind Po~ in an illuminated space and the ratio has to be larger. In other words, the effective limit of VS depends on where the farthest percept is localized. According to Gogel (1972), when a light point Q is presented with no distance cue at all (monocular observation through a small arti- ficial pupil) , P~ appears to be at 3.0 ft (0.91 m) with a standard deviation of 4.8 ft (1.46 m). In an outdoor experiment, Gilinsky (1951, 1955 ) estimated the parameter corresponding to max 60 in the horizontal direction to be 300 ft (90 m). Allegedly, Bour- don estimated, as early as 1897, that the night sky appeared to most people to be 80-150 m away. In these cases, perceptual distances, 6o, are given in terms of physical unit. We have to interpret that the perceptual distance is equivalent in length to the appearance of that physical distance under the ordinary condition. The perceptual distance to the horizon, max 6 o , when looking at the ocean, may be determined by the texture gradient in the retinal image of the water surface (Figure 1 ). In this article, suffice it to mention that VS is dynamically
bounded, and the discussion of how 60 to the furthest percept is related to the physical condition is not included.
The mapping functions (Equation 4) are egocentric and the three variables are assumed to have their effects independently. In the first equation, convergence angle 3, is effective only when Q is in the neighborhood of the subject. There are many studies on how the perceptual radial distance 60 is related to 3" or more generally to radial distance to Q in X, e0. The present discus- sion is limited to implications following from the mapping functions (Equation 4). As discussed in EM.3, the last two equations of Equation 4 imply that ~v = ~b, ~ = O where ~v, 0o represent the corresponding directional angles in VS: We see P~ in VS in the same direction from the self as Q is from the subject in X. These two assumptions imply something more about which angles in X are preserved in VS. Suppose do is a line element from P in EM. Denote by S(P) the small plane orthogo- hal to the vector OP at P (Figure 7), and denote by d~ the projection ofdp on S(P) . Then, do and the corresponding ds in EM are related as
ds = G-2(po)[d2po + d~] 1/2
d~ = p0[d2tp + cos2~pd21, q] 1/2. (6)
We can think of similar small plane S in VS and that in X. Denote by ds and de small vectors on those Ss in VS and in X that correspond to d~. Suppose two line elements ds and d~ from P~ in VS with angle ~0~ that are represented in EM by dp and d~o with angle co, and denote by ~ the angle between d~ and d'-fi on ~'(P). Because of EM.3, o~ = o~ and ~ = c0~, where ~ represents the angle in VS that corresponds to ~ in EM. It is determined by mapping functions how o~ and ~ from P in EM are related to o~ x and ~x from Q in X. If ~ = $, 0 =/9, from Equation 6
d p = po[d2~b + cosE~bd2O] 1/2, (7)
which implies that ~o = cox and hence any angle ~ defined on S(Q) in X is preserved as o~o in VS. Notice that the retinal image of a stimulus pattern around Q is approximately proportional to the pattern on S(Q ). In other words, a stimulus pattern gener- ates such a percept in VS that is conformal with its retinal image. The mapping functions in Equation 4 as a whole imply more. Because
dpo = (dpo/d3")d'y,
i fp 0 as a function of 3" satisfies the condition that dpo/d3, = Cpo, where c is a constant, then Equation 6 can be written as
ds = poG2(po)[c2d23" + d2~ + cos2f~d20] 1/2, (8)
and the first expression in Equation 4 satisfies the condition: dp0/dv = co o. Hence, Equation 8 holds under the mapping func- tions in Equation 4, which means that the angular relationship in VS remains the same for transformation 3/= ~/+ c,, where c~ is a constant (Eschenburg, 1980). This is in agreement with the fact that the transformations (i.e., 3 /= 3' + c,, q~ = ~, O' = O), sometimes called iseikonic, can be used in designing a family of Ames rooms that is a sequence of distorted rooms in X that appear the same to binocular observation from a fixed position (Hardy et al., 1953; Ittelson, 1960; Luneburg, 1947).
438 TAROW INDOW
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- .7
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2.5 5 10 16 m
e(O,Q1): Distance to the farthest point QI
Figure 5. K and a as functions of distances to the farthest point Qi in the physical space X for 3 subjects who constructed P- and D-alleys with QI at various positions in the dark horizontal plane of eye-level HZ(E). (An open circle indicates that there is more than one point. Oblique straight lines are only for separating three sets of points.)
C o m m e n t s on Luneburg ' s Model o f VS
Luneburg's model consists of two assumptions: (a) VS is an RCC, which enables us to use EM: and (b) the correspondence between X and EM is given by Equation 4. Before focusing on the global structure of VS, it may be appropriate to enumerate comments about these assumptions.
L 1. If two assumptions are taken together, the correspon- dence between VS and X must be isotropic; the same relation- ship holds for any direction. Because Kis constant, P0 does not change according to radial direction from 0 in EM and P0 is
assumed to be independent of@ and 0 in Equation 4. However, there are a number of observational phenomena that exhibit the existence of the anisotropy in VS. For example, when two line segments of the same length are presented in X at the same egocentric distance, one horizontally and the other vertically, they do not appear to be of the same length in VS (e.g., Kiannapus, 1955). At a more global level, the sky does not ap- pear to be a sphere of constant radius. These facts contradict the conclusion of the isotropy of VS when the two assumptions are taken together. Furthermore, the phenomenon of size con- stancy (Kirkpatrick & Ittelson, 1953) needs explanation if it is
VISUAL SPACE
(9 ® ® Light points in dark (Lab.) Black points on illuminated table Ught points in dark (Gym.),
surrounded by white curtain,
1976 3Ss 1963 3 S s 1977 3Ss
® ® Light points in front of dimly illuminated checkerboard pattern
1983 2 Ss
439
x 1 =417.5, Yl = _ 4 O c m
max <501501 1.28
Geometric mean
x 1 = 355, Yl " + 30 cm
1.70
broad x 1 - 1610, Yl " + 150 crn
narrow x I - 1610, y 1 - + 75cm
x 1 = 409.5, Yl = + 40 cm
Two moving points
no background 1.52 1.20 broad 1.38 near (at 424.5 cm ) 1.58 1.28 narrow 1.36 far (at 676.5 cm ) 1.60 1.28
@
1 .28
I /
0 Figure 6.
® ® ® ® 1.70 1.52 ~'1.60
, /
/ ; / back- /
T" ' i / / / Pl ground ' P1
/ / I ,/
0 0 0 0
1,20"-1.28
Geometric means of max ~o/~o~ that are estimated from K and o under various conditions.
to be understood within the framework of Luneburg's model. Luneburg tried to account for size constancy on the basis of RCC and the mapping functions in Equation 4. However, the frameless VS, in which the functions of Equation 4 are most likely to hold, is a space where the least degree of size constancy is expected. Because the two assumptions are separable, we can assume more flexible correspondence between EM and X while retaining the RCC assumption. For more structured VS, it will be more natural to replace the mapping functions by some other form that is suitable for size constancy. The assump- tion that such simple mapping functions as Equation 4 hold for all stimulus conditions is equivalent to the constancy hypothe- sis, which assumes a rigid and context-free correspondence be- tween local stimulus condition and sensation (Indow, 1974a). Foley (1966, 1977, 1978, 1985) presented a number of pieces of experimental evidence showing that the first equation g(~/) does not hold, even in a frameless space. On the other hand, an experiment of Lukas (1983) supports g('r) of Equation 4 in a certain region of EM.
L.2. A reason why alley and frontoparallel curve experi- ments have been carried out only in frameless VS is that no one expected the simple egocentric mapping functions of Equation 4 to hold in VS with a framework. However, one such experi- ment was performed in a natural condition. Battro and Pierro Netto (1976) constructed P- and D-alleys in a large garden and
also in a polo field using wooden stakes as Q~, which were placed by the experimenter according to directions of the sub- ject. Of the largest {Qi}, the farthest fixed pair Q1 was (240 m, _+24 m). In total 56 subjects participated. Almost a third of cases exhibit settings of overconstancy in which all Q,- of both alleys have ly~l > 13'11, which they called "divergent." There was also another nonregular pattern, and they discarded all these cases. Theoretical curves for alleys were fitted to the remaining 45% of"regular" cases. There must have been a number of subjects who failed to understand the instructions and constructed the alleys "physically" parallel or equidistant. One more surprising thing is that the theoretical curves used were based on mapping functions in Equation 4. Clearly, "r is not meaningful in this situation. The authors reported only values of K obtained, not values ofo: K > 0 for 52 cases and K < 0 for 38 cases. As the experiment was carried out in the daytime, a number of objects must have been visible beyond the farthest pair of wooden stakes Qi (trees, terrain, etc.). The limit of VS, max ~0, must have been beyond these objects. Hence, no matter what map- ping function holds, the ratio we considered in Figure 6 has to be large and {Q~ } for these alleys has to be represented by a small {Pi } in the area closer to 0 in EM. In other words, the geometrical structure of {Pi } cannot be very different from Euclidean. Although values of K obtained are problematic be- cause they are based on the first equation of Equation 4, the
440 TAROW INDOW
co : angle between dp and d'9 : angle between d~ and d ' . 5 ~
Figure Z Decomposition in Euclidean model of a line element do into dOo and d~. [~¢(P) is a small plane orthogonal to the direction at Pfrom which line elements originate. Angle ~o between two line elements dp and d'p is projected on S(P) as ~.]
ambiguous results with regard to K may be attributed to the fact that the alleys occupied only a small part of VS.
L.3. As pointed out in L. 1, the mapping functions of Equa- tion 4 are not flexible enough to cope with the variety of obser- vational facts. If we assume more flexibility in mapping func- tions, then we have to take into account the possibility that the discrepancy between P- and D-alleys in X is attributed to a possible change of correspondence between VS and X accord- ing to observing condition. Suppose that P- and D-alleys are actually the same in VS, and the subject sees one and the same pattern. When constructing the P-alley, it is natural for the sub- ject to scan two series of Qs (i.e., along the x-axis in Figure 3). When constructing the D-alley, the main direction of eye move- ment would be along the y-axis. Hence, if the appearance in VS of {Q~ } in X changes according to the scanning direction, two different sets of{Q~} will be obtained in X. However, the same discrepancy was obtained when P- and D-alleys were con- structed by two moving Qs instead of a stationary {Qi }. Scan- ning directions can be controlled by the direction of movement and, even when two Qs moved in the opposite direction in the two series along the x-axis (e.g., toward the subject on the left and away from the subject on the right), the same discrepancy was obtained (Indow & Watanabe, 1984a). Namely, the discrep- ancy is perceptually genuine, so that being parallel and being equidistant are perceptually not the same. This is the main thrust for the first assumption that VS is R with K 4: 0.
L.4. When told that VS may not be Euclidean, people tend to ask "why?" I have never heard the same question raised to the idea that VS is E, even though E is R with K = 0. However, there is no a priori reason that makes it natural for VS to be Euclidean and unnatural to be hyperbolic or something else. A naive ques- tion is often raised: "If VS is an R with K 4= 0, why do we not perceive it curved?" If K 4: 0, our VS may look curved to a creature in a space of higher dimension who can observe our VS
from outside, but not to those living in that space. What is meant by saying that VS is hyperbolic, for example, is that per- ceived straight lines and angles behave in the same way as geode- sics and angles of that geometry behave. The assertion implies nothing les~ and nothing more.
Specifying the geometry of VS is a step in the level ofpheno- menology, not the ultimate answer. The real question is why and how the brain generates VS as we see it. It will be an effective step toward understanding this problem, however, i f a variety of phenomena are formulated in terms of a single geometry. The discrepancy between P- and D-alleys is only a symptom for VS to be R with K 4: 0. In order to state that VS is structured according to a particular geometry, it must be shown that geode- sics and angles behave in all respects in the way predicted by that geometry. Because all of the experiments discussed so far were concerned with {Q~ } in the horizontal plane of about eye- level, HZ(E) , it is essential to extend the same approach to other subspaces in VS and to three-dimensional VS as a whole.
D i r ec t C o n s t r u c t i o n o f Conf igura t ion {Pi } in E M
In the preceding sections, {P~ } in EM was defined by map- ping {Qi } in X into EM through Luneburg's mapping functions (Equation 4). If data on perceptual distances 60 between two stimulus points Qi and Qi are available, then {P~ } can be de- fined directly without using any a priori assumed mapping functions.
Denote by d o numerical values representing perceptual dis- tances 6 U in VS. Suppose that we have d o as data; then d o can be converted to Euclidean distances P0 in EM through the in- verted functions of Equations 2 and 3. Denote by (Pu) the ma- trix ofps thus obtained for a given configuration of points {Pv~ } in VS, which includes the self as the origin 0 = Pro (i = O, 1, 2, . . . . n). Then, by applying a multidimensional scaling method (MDS) to (P0), we can obtain such a configuration {Pi} in EM that gives the most satisfactory correspondence between data d~ and dk that are obtained through Equations 2 and 3 from {P0}, interpoint Euclidean distances of{Pi } constructed. It is not necessary to have d o for all possible combinations of points of {P~ }, and hence (Pu) can have vacant cells. This is a method of MDS to construct {P~ } in an RCC from Riemann- ian metrics d o (Indow, 1974b, 1975, 1982). In the stage of con- verting d o to Oo, we need a constant c = ~zK_/__2u where u is the unit of numerical values of d~, because ( ~ f / 2 ) 6 is necessary to use the inverse functions of Equations 2 and 3, and d is assumed to be proportional to &
d = u 6 , 6>_0, u > 0 , (9)
where, although latent, 6 is assumed to behave like a quantita- tive variable. This step is not involved in traditional methods of MDS in which {P~ } is constructed in E where K = 0. Estimating the optimum value of c, ~, is equivalent to obtaining informa- tion on K.
The following method was used to obtain the data (do) in our experiments (Indow, 1975, 1977, 1982). The subject was asked to assign orally numerical values, ri, jk, to perceptual ratios 60/6~k between the two perceptual distances to Pvk and Pvj from a point Pv~. From the data matrix (r~, jk ) where i and j , k are system- atically varied, it is possible to define the matrix (do) where all
VISUAL SPACE 441
ds are given by an arbitrary common unit u (Indow & Ida, 1975). This is a finer grained procedure than direct magnitude estimation on ~0" We can apply a metric MDS program to (cd o) where c is the constant defined above. Then, by systematically varying values of c, we find the value ~ that results in the best fit in the following sense.
In general, the fit can be evaluated by the agreement between the data d o and d 0 from {Pi } obtained from (cd o). If{Pvi} has an intrinsic geometrical structure, that information can be used for the evaluation of goodness of fit. Suppose that P- or D-alleys and H-curves at various distances were determined for the sub- ject under a given condition. Then, intersections of the P- or D-alley with H-curves can be used as {Q~ }. Figure 3 shows an example of such {Q;}, and each of the three subjects made dis- tance judgments with {Qi } in darkness (n = 28) and in a frame- less illuminated space (n = 26). For each value ofc selected, the theoretical pattern {P~lc} based on c under the three geometries (K < 0, K = 0, K > 0) was constructed, and deviations of the data {Pi } from {Pi Ic} were obtained. In all cases, the optimal value of 6 that minimizes the root mean square (RMS) of the deviations was obtained when Equations 2 and 3 for K 4:0 are used. Using this ~, we can define in EM the optimum configura- tion {/5 } and the theoretical curves in EM (curves on the left in Figure 4) for this {/3,- }. Although the fit is quantitatively not very satisfactory, one fact became very clear: VS under discussion is not Euclidean. It can be shown in the following way. Directly from the data (d~k), without using the inverse functions of Equations 2 and 3, we can construct a configuration {Pj} in E (K = 0) by a traditional MDS and define interpoint distances djk. Insofar as the reproducibility o f d by d is concerned, there is almost no difference between this {Pj} and the previous {/~i} that is based on 6. However, if VS were Euclidean, this {P~} would be its quantitative representation and this {P~ } should exhibit the property according to which the subject constructed {Q~ } (e.g., the two series of Ps for P-alleys should be straight and parallel). This was clearly not the case (Figures 12 and 13 in Indow, 1982). It was also shown that, when radial distances P0 in {/3 } are plotted against their values of'y, points for/5 with various values o f ~ are scattered along a single curve g(~,), but the form ofg(~) is not exactly the first equation of Equation 4. If this empirically defined g(~,) has the asymptote, max Po, then we can determine the value of K in the way stated in EM.7 by defining max P0 = 2. It was not easy to pinpoint max ~ by extrapolation. If necessary, however, the value of Kcan be given by the other unit. For example, K = -(2c32 with the unit that u = l .
When the data d o are plotted against their theoretical values do, which are defined through Equations 2 and 3 from {P~ } in EM, the scatter of points is very small, and d is either propor- tional to d or is a slightly accelerated function ofd . Although it is true that the hyperbolic geometry accounts for the data (d o ) better than any other RCC, it is an open question whether the not very satisfactory fit is due to the scaling procedure for ob- taining (d~), which heavily depends upon verbal reports, or to the inadequacy of the postulate that VS is RCC. Many investi- gators have tried asking the subject to assess ratios of perceived distances (e.g., Baird, 1970a; DaSilva, 1983; Foley, 1980). Ac- cording to my experience, ratios of scaled distance do/d~k and verbal reports r~,~k agree quite well. However, this consistency
holds even if ratio judgments are made in such a way that r~,jk = (60/Sik)a,fl ~ 1 (Indow, 1968; Krantz, 1972; Shepard, 1978). If ~4: l, {Pi} based on (do) derived from the basic data (ridk) would give a distorted image of the perceptual pattern {Pv~} in VS.
In the second procedure, direct mapping through Rieman- nian powered distance (DMRPD), instead of Equation 9, d is assumed to be a power function of ~ in order to take care of possible human bias in assigning numerical values r~, jk (Indow, 1983):
d = a~ ~, a > O, fl > O. (10)
Initial values of/~, K, and the initial configuration {P~ } are as- signed, and the program modifies these by the method of steep- est descent so as to obtain {/~ } in EM that gives those ds that are most closely related to data ds in the following sense: When d o are plotted against d o, the relationship is a power function with the exponent/~. An example of {/5 } constructed by DMRPD for {Q~ } similar to Figure 3 is given in Figure 8. Because max Po is not given, K is defined by the unit P0~ = 1. As shown in A in Figure 9, d is again a slightly accelerated function of d. The degree of scatter of points (Kruskal, 1966a, 1966b) is given by
d 0 is given by Equations 2 and 3 for K ~ 0 from interpoint Euclidean distance/~0 of {/5i }.
Using DMRPD, I tried to obtain {/6 } representing the config- uration of stars in the night sky. Ten stars of about the same brightness and the subject (0 = P0) were embedded as {/5/} in three-dimensional EM in three different ways: dwas defined by Equations 2 and 3 for K < 0 and for K > 0 and {P~ } was directly constructed in E from d (K = 0). The data (do), taken from Indow (1968), were scaled by the previously described proce- dure: assigned r~.jk to the perceptual ratio ~O/~k. In this case, as shown in Figure 10, embedding according to each of the three geometries gave almost the same {P~ }. Furthermore, the three embeddings show almost no difference in the trend and the goodness of fit in the d - d plot (two examples are given in B and C of Figure 9). In a word, we cannot distinguish the geometry of the perceived night sky on the basis of(d0) alone. We do not have any theoretical pattern as in Figure 8, and RMS is not definable.
Global Structure o f Three-Dimens iona l VS
The perceptual shape of sky, the vault, has been studied by meteorologists and psychologists. As early as 1728, Rob Smith bisected the sky arc between perceived horizon and zenith. If the sky appears as a sphere, it is expected to be 45 °, but he obtained 23 ° from the horizon, suggesting that the sky is flat- tened in the direction of zenith (Filehne, 1912). Meteorologists estimated from the half-arc angle the ratio of~o to the horizon to 60 to the zenith. These attempts are always made under the assumption that VS is Euclidean. The result in Figure 9 suggests that which geometry is to be used is not crucial. Of course, the
442 TAROW INDOW
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Figure 8. An example of a configuration in EM of intersections {/~ } of P- and D-alleys with H-curves on HZ (E) that has been constructed from distance judgments. (RMS = root mean squares of deviation of P~ from the respective theoretical curves. K is defined with the unit that Po~ = 1. The inset shows the right side of the configuration when constructed in E. If VS is Euclidean, both sets of points, P- and D-alleys, should appear on the line passing P~ and perpendicular to the n axis. DMRPD = direct mapping through
VISUAL SPACE 443
half-arc angle changes according to condition: daytime sky, clear or cloudy, or night sky, moonlit or moonless (Miller & Neuberger, 1945; Neuberger, 1951 ): the more cloudy, the more flattened. In my experiment discussed above, 60 means the length of the line segment connecting two stars, P; and Pj, not the length of arc along the surface of sky. It is easier for the subject to interpolate between stars in this way, and this is the definition of distance required in MDS or DMRPD. On the other hand, in bisecting the sky, it is easier to see the length of an arc than to imagine a line segment connecting an invisible bi- secting point in the sky with the horizon or the zenith. Psycholo- gists are interested in the shape of sky as a way to account for the moon illusion (e.g., Baird & Wagner, 1982; McCready, 1986; Reed, 1984; Rock & Kaufman, 1962). The moon illusion is, however, beyond the scope of the present article.
The configuration {P~ } of stars in Figure 10 extends most in the direction of 71 and least in the direction of ~ (i.e., the night sky under discussion is not flattened in the way described above). The experiment was performed on a long beach, and subjects were seated facing in the direction of the ocean hori- zon, where nothing was visible. In this direction, the situation was almost the same as in the Ganzfeld experiment (VS.5). Subjects were allowed to move their heads to see stars in various directions, and the silhouette of the terrain was visible on the left and right peripheries. Under these conditions, all subjects agreed that the sky appeared flattened in the direction of the horizon more than in the direction of the zenith. This appear- ance is captured in Figure 10. Baird and Wagner (1982) showed that perceptual distances 6o to the horizon sky varies according to what the subject sees on the ground in that direction. To my knowledge, it is not well understood how the perceptual dis- tance to the boundary of VS, max 60, is determined by the physical condition in that direction. As shown in Figure l, there must be the maximum effective physical distance eo in X that corresponds to max 6 o in VS. In the case of the sky, all stars are physically well beyond this limit, and hence they are perceived as lying on the vault. If an aircraft is far away, its movement and its vapor trail, if visible, are perceived as curved on the vault. According to Fieandt (1966), aerial observers in antiaircraft batteries during World War II had trouble because an enemy plane passing horizontally looked as if its course was curved. Concerning vapor trails of jet planes, Gombrich (1974) wrote: "I have come to appreciate the reasons why some students of art, including the great Panofsky, asserted with such conviction that we really see straight lines as curved" (p. 86). If straight lines mean physical ones in X, the statement is understandable. Otherwise, the statement is a contradiction by itself. Nobody will deny the fact that we can see in VS a straight line, if its length is limited. Of course, its counterpart in X needs not to be straight.
Galanter and Galanter (1973), using aircraft and boats at various distances, Co, in X, showed that do is a power function of
eo when do is given by magnitude estimation and the exponent changes from 0.80 (in the direction of zenith) to 1.25 (in the direction of horizon) according to the elevation angle of sight. The data show no sign of the existence of an asymptote for d o , even though e0 covered a range of about 10 km. Power functions between do by magnitude estimation and eo in the outdoor hori- zontal direction, more limited in the range, have been reported by a number of investigators (DaSilva, 1983; DaSilva & Da- Silva, 1982; DaSilva & Dos Santos, 1982; Kiinnapus, 1960; Teghtsoonian & Teghtsoonian, 1970).
In our daily life, we are very unlikely to come across alleys such as represented by P- and D-alleys. Two parallel straight lines in X appear to converge to the vanishing point on the horizon in VS (Sedgwick, 1980, 1982). What we see as being parallel is, in most cases, on a plane in front of us. To our casual observation, frames of a window look straight and parallel. De- note by HP (P) the frontoparallel plane passing through a given point P. Theoretical equations in EM that represent H P ( P ) and alleys on H P ( P ) , parallel or equidistant in the horizontal or vertical direction, were developed (Indow, 1979, 1988). If this subspace in VS, H P ( P ) , is structured according to R ofK:~ 0, these P- and D-alleys should exhibit a discrepancy. Two experi- ments were performed on these alleys on HP (P) around a stim- ulus point Q(x, 0, 0), where x was either 98, 186, 276, or 320 cm. The subject adjusted {Qi} so that all points appear to be frontoparallel and form either two or three horizontal series of points, one above the other. Once they were adjusted to appear straight and parallel, and once all corresponding sets of points in the series were vertically adjusted to have the same percep- tual distance. The results were analyzed in two ways: fitting theoretical curves to the configuration {P~ } on HP (P) obtained from {Qi } by using Luneburg's mapping functions and the anal- ysis of judgments on interpoint distances of the (Q~ } by DMRPD (Indow & Watanabe, 1984b, 1988). Both analyses showed that no other geometry than Euclidean is necessary for these patterns in HP(P) . An example of{Q~ } is given in Figure 11, where no systematic discrepancy is observed between {Qs } for the P-alley (o) and {Q~} for the D-alley (x). According to calculation using the mapping functions (Equation 4), when such values o f K a n d tr are used that are typical for alleys on the H Z (E), the discrepancy between these alleys on H P (P) is ex- pected to be of such magnitude that not only would it be easily detectable by experiment, but it is hard to believe that no phe- nomena related to it have been noticed in our daily life (Indow, 1988).
The majority of studies on visual perception are concerned with patterns on a plane like HP (P), and it is taken for granted that VS is structured in that way: A perceptual plane is per- ceived at a certain distance 60 from the self. Furthermore, when- ever necessary, HP(P) is regarded as Euclidean. When HP(P) is small, it is natural and legitimate because R is always locally Euclidean. In the experiment shown in Figure 11, HP(P) cov-
P-alley and front•parallel H-curves on Horizontal Plane of Eye - level HZ . /
DMRPD-I DARK, T.I. ~, HALF OF POINTS PLO'F'FED)
1976 A
K = - 0.80 A
(POl = 1.o)
p = 1.27 ". • • • e
stress (Q1)=0.09 STARS
(OISO BEACH) T.I.
1962 "" DMRPD-1
.04
plane HP
~: interpoint distances in {Pj)
Figure 9. Scaled perceptual distance d and interpoint distance d of{P/} constructed in EM. (A: For {Pi } in Figure 8. B: For two cases of{Pi } in Figure 10. C: An example for {Pi } on a front•parallel plane HP(P) in Figure 11. DMRPD = direct mapping through Riemannian powered distance.)
ered a large portion of VS of the subject whose head is fixed and still the HP(P) is consistent with E. Of course, if l i P ( P ) were clearly different from Euclidean, it is unlikely that human be- ings would have created Euclidean geometry. The surface of sky
can be said to be an extension of l i P ( P ) where every Pis at max 60, and the result in Figure 10 may also imply that no geometry other than Euclidean is necessary to the structure of that con- cave surface facing toward the self.
Experiments of P- and D-alleys on the horizontal plane of eye-level, HZ (E), gave the results consistent with the hypothe- sis that this subspace in VS is structured in terms of RCC with K ~ 0. However, K and a vary according to {Qi} being pre- sented on HZ(E) , which was regarded by Heelan (1983) as evidence for failure of Luneburg's (Heelan, 1983, p. 49) model. As stated in the RCC section, the free mobility of perceptual figures in a given VS is the basis for the assumption RCC that VS is of constant curvature K. No free mobility can be defined between two VSs on different occasions, one with a {Qi} and another with a different {Qi }, and hence we have no reason to claim that the horizontal plane in VS is structured in the same way irrespective of {Qi} presented on HZ(E) . The real ques-
tion is whether we can regard that the horizontal plane with a given {Pi } is really an RCC with K of some fixed value. So far, only experiments using alleys have been discussed. There are a few other experiments that are relevant to the assumption of RCC.
Blank (1961 ) performed an ingenious experiment with small light points {Q1, Q2, 03} on HZ(E) that appear as an isosceles triangle (Figure 12, plotted by the author). The subject was asked first to bisect the sides PvI-P~2 by Q4 and the side Pv:Pv3 by Q5, and then to adjust Q4' and Q5' along the base of the triangle so that 624, = 645 and 635, = 645. If VS with this {Pi } is Euclidean, Q4' and Q5' should coincide and both be at the mid- point of Q2-Q3. If hyperbolic, Q4' is on the right and Q5' is on the left of the midpoint so that we have a gap between two segments, Q2-Q4' and Q3-Q5'. If elliptic, the relation between
446 TAROW INDOW
U
M
L
!~ 1 Dark 2 T .W,
stationary -]5ocm ~ o P-alley
-lOOcm " s o ~ 1 ~ ~ ~ ~ 3 x D-alley (standard)
- ~ Z = 50
-lOOcm _ _ ~ lOOcm
-50cm ~ . ~ L...,._ 150cm .[.~ -"
y v ooo~
IbOcm
Figure 11. An example of{Q~}s in the physical space X for P- and D-alleys on a frontoparallel plane HP(M3).
Q4' and Q5' is reversed and the two segments should overlap. For 6 out of 7 subjects, gaps between Q4' and Q5' were found and the mean gap of all subjects was 12.8 cm. However, we see an unusual fact in the result of this experiment. All subjects set Q4 and Q5 closer to the apex QI than the midpoints of sides, and furthermore, with regard to this setting, subjects were clearly divided into two groups as shown by unfilled and filled symbols in Figure 12. As shown in the bottom part of Figure 12, it is hard to say whether the size of the gap between Q4' and Q5' is different between these two groups. It is an open question, however, whether the same gap would occur if Q4 and Q5 were set at a more natural position closer to the base and hence the interval between Q4 and Q5 were larger.
Hagino and Yoshioka (1976) presented small light points QI at various positions on the x-axis and asked the 5 subjects to set Qs around each Q1 to form a perceptual circle on HZ(E) . They analyzed the result according to Luneburg's model. When K and cr were estimated, K turned out to be negative in almost all cases, but these values varied according to Q1 and the radius of circle. This is not surprising given the view expressed in the present article. However, with {Qi } of each circle, they admitted that the result is not quite consistent with the model (RCC) and the mapping functions of Equation 4. Hence, neither study can be taken as decisive additional empirical support that the per- ceived HZ(E) is RCC of K < 0.
Experiments on P- and D-alleys in the horizontal direction on frontoparallel planes, HP(P) , gave the results consistent
with the hypothesis that each subspace H P ( P ) is E, an RCC of K = 0. If the two hypotheses, the perceived HZ(E) as RCC of K + 0 and HP (P) as E, are taken together, there are two possibil- ities. One is that VS in a given condition consists of two sub- spaces, HZ along the line of sight and HP perpendicular to it, and each is structured according to a different RCC. Then, the three-dimensional VS as a whole is not an RCC. In this case, RCC. 1 to RCC.4 are valid in each subspace taken separately, but do not hold between two subspaces or in the whole VS. The other possibility is that either one or both of these subspaces are actually not structured in terms of RCC. Then, there must be at least one condition in the set RCC. 1 to RCC.4 that does not hold even in a subspace.
1. In the preceding discussion, VS is assumed to exist, under a given condition, as a geometrical entity with stable and coherent structure in which perceptual distance ~ and angles, although both are latent, are assumed as quantitative variables. We can see only a part of VS on a single glance, and perception of the whole VS is a result of multiple glances. In spite of that, the environment is perceived in more or less the same way all the time. This is especially true for VS with frames such as the walls in a room or the terrain in an open field. The appearance of the whole VS remains almost the same independently of movement of the eyes and head. This is VS by which our behav- ior is guided. Perhaps human beings have had VS structured in this way since remote ages. However, our awareness of the surroundings in a single moment is limited, and it is possible to
V I S U A L SPACE 447
x in.
i00
8o
6o
4o
,l A D
m
L 70 cm
!
2O
275 cm
0
I I Q2 Q5' Q4'
I I
o 2O
. . . . . . . O-- • . . . . IX O
Q3 ~
[y] i I
1o @6.4 cm o
Y
in . x in. 29
70 em J.
Figure 12. The result in the physical space, X, of Blank's (1961) ex- periment using an isosceles triangle. (Results of 7 subjects are given by different symbols.)
think that VS changes its geometrical property from moment to moment according to whatever aspect of VS is receiving at- tention. For example, in the P- and D-alley experiments, lateral distances between pairs of points may be geometrically differ- ent entities according to which alley is being constructed, not because of the difference of scanning direction as discussed in L.3 but because of the difference in focus of attention. If that is the case, we cannot assume one and the same metric space for both alleys as we did in the preceding sections (Yamazaki, 1987).
2. We can cast doubts on the assumption RCC. 1: "whether it is at all justified to consider VS as a metric space with the fixed metric" (Balslov & Leibovic, 1971 ). The same skepticism has been expressed by many (e.g., Foster, 1975 ). DrOsler (1979) proposed an approach that, without using the concept of 6, gives experimentally testable conditions sufficient for the exis- tence of ~ and for specifying the uniqueness of 6 within eight geometries, including the three RCCs discussed here. Yama- zaki (1987 ) developed a formulation of the P-alley that does not
presuppose the concept of& There are two problems concern- ing a: One is the nature of a as a latent variable and the other is, if 6 can be regarded as a quantity-like variable, how to obtain data on &
Gogel (1977a), who invented the adjustable pivot method, which does not rely upon verbal reports about 60, wrote: "It is clear that spatial perceptions are metric. One object is perceived to be twice the size of another and at three times its distance." '%ll are examples of metric perception. Although numbers are often used to describe these perceptions, the perceptions do not depend on the observer having a concept of number; ' "the abil- ity of a rat to modify the force of his jump as a function of the distance between one stand and another implies the rat has a metric perception of distance" (Gogel, 1977b, p. 135). The state- ments imply the following three points: (a) Gogel regarded 6 as a quantity, although it is latent; (b) the quantity a can guide physi- cal movements of the organism in X; and (c) because scaling through verbal reports may be susceptible to contextual effects, if at all possible, we should use a more "objective" procedure. This view may be shared by many investigators of visual per- ception.
Suppose that P~, Pv2, and Pa are collinear in VS in this order and, when focused separately, each perceptual distance fi is felt like a quantity. Still, there is no guarantee that the three quanti- ties are related so as to satisfy the additivity conditions; a12 + 623 = ~13. If the condition is satisfied, we can regard collinearity to be equivalent to concatenation in measurement theory (Krantz et al., 1971 ) - - t h e operation common in physical mea- surements but not possible in scaling of most sensory attributes such as brightness and loudness. When data (du) are given, then it is possible to examine the additivity of ds for Ps collinear in VS. In the experiments discussed earlier, the following points are collinear in VS: Ps on a P-alley in HZ(E) and Ps in a hori- zontal P-alley on an HP (P). When d e is plotted against the sum diE + d23 + • • " + dj _ l.j on log-log coordinates for Pl, P2 . . . . . P , collinear in this order, we can expect the straight line of unit slope if the additivity holds. Figure 13 gives such plots on the basis o f the data o f Indow (1982; an illuminated HZ (E)) and o f Indow and Watanabe 0988; an HP(P) in the dark). In each collinear series, two sets o fds are plotted in Figure 13, one from an end as Pt and the other from the other end as Pl (o and x). All ds were scaled by ratio assessment as described in the section on direct mapping. Eight plots in Figure 13 are well fitted by straight lines of unit slope and, for one case, the best fit is given by the straight line of the slope of 0.9. However, the fit of the unit slope line is not unacceptable at all. Hence, we can con- clude that ds satisfy the additivity condition. Then, it is very unlikely that such data ds are obtained when the additivity does not hold for the latent variables as themselves. It implies that ~ is equal to 1 in Equation 10. We cannot take this situation for granted. For example, such additivity fails to hold for ds repre- senting perceptual color differences for colors collinear in the color space constructed by MDS or DMRPD (Indow, 1987).
3. Desarguesian property (RCC.3) was tested by Foley (1964a, 1964b) with triangles in three-dimensional clark VS, and the answer was affirmative. He also tested the possibility of congruence between two triangles in a dark HZ(E) and ob- tained a negative answer (Foley, 1972). The subject constructed
448 TAROW INDOW
5
4
dti 2
d j5
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5
4
3
dd 2
d15
Para l le l A l l eys , Eye- leve l , I l l u m i n a t e d Room 1982
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/ / - ° x / I I T.I. o ~ L.S. , ~ T.F. / =~i ,,
~ / /,," , y , ~ ! X / / ~ / ~101 / O// ~ ~ l " I / 0 s
/ . ~ . . ' x - , s =.9 / . . . . , ¢ ~,/ / 13=1.0 / / / ( " v" / P - ] ' " / i l
l - t 6 O j = 2 ~ 6 ~-~.di, i+ 1 X j = 5 ~ 1 T~.di_l, i
i=1 i= j+ l
Horizontal Hh - curves on Frontoparallel Plane HP
/ /
0 UI"->U5 Ox / 0 M1-->MS / 0 L1-->L5 ~ U x U 5 - > U / x M S - - > / ~ g X L5---> L / ~ M
/ / L ,o/ / /
, , , , / / , , , # , , , , ,
• 8 1 2 3 1 / / 2 3 1 2 3 4 5
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1 2 3 1 2 3 1 2 3 4 5
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Figure 13. Additivity of scaled perceptual distances in two different configurations.
1 2 3 4 5
0 0--.--0 0 0
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VISUAL SPACE 449
(1) A VS (11)
A
Figure 14. Perceptual pattern in Foley's experiment (1972).
such {Q~} with a fixed point A that appears as shown by the pattern (I) in Figure 14, where O represents the observer. First, B was adjusted so that AABO appeared as an isosceles right triangle: fiAB = flOW and AB_LOB. Then, C was set in such a way that ABOC appeared as an isosceles right triangle: fiBO = ~OC' and CO±OB. The angle @A in X to which 9A in VS corresponds was fixed at 9.9 in degree and the position of Q~ was varied in two ways to change the size of the pattern. Using 24 students from the Massachusetts Institute of Technology as subjects, Fo- ley came to two conclusions. Even if it is assumed that angles appear larger in VS than in X (1.1 times according to his esti- mation), each isosceles right triangle did not have two acute angles of 45 °, and hence VS with this {Pi } is not Euclidean. There is no doubt that ~BC is larger than ~OA (about 1.2 times), and hence AABO and ABOC are not congruent. The second conclusion is contradictory to RCC.3, and Foley denies the pos- sibility of congruence in the horizontal plane in VS.
Several comments may be made about this experiment, but two will suffice. One is the difficulty or ambiguity that exists in constructing a right angle in VS when viewed in an oblique way. When a sketch (II) of Figure 14 is presented, people are apt to say tha t / -ABO =/--BOC = 90 ° and that even when it is stressed, what matters are angles on the sheet of paper. A similar ambigu- ity may be involved in this experiment. The other is that attend- ing to equality and perpendicularity of two sides is a bottom-up approach. It is not clear whether the observer sees two con- gruent triangles when the observer completed the settings. It would be interesting to try a top-down approach such that two perceptually congruent right triangles are constructed first and then to check whether 6AB = 6OC' or/--ABO =/__BOC, and so forth. It would be highly desirable to perform similar experi- ments in VS having a frame and using three-dimensional fig- ures. It is a routine procedure in size constancy experiments to establish perceptual matching between two figures at different distances or different directions from the self. The Ames dem- onstration of distorted rooms is a succession matching between the appearances of two physically different three-dimensional patterns. This is a top-down approach. It is an open question, however, what the subject would say if asked to compare corre- sponding parts in the two patterns--whether all corresponding angles appear the same and all corresponding sides appear to be of the same length.
4. Free mobility in VS is continuous maintenance of the perceptual congruence of a figure. It is irrelevant how its coun- terpart in X changes its shape according to its position to meet this condition. As described in L.3, it is possible in HZ(E) to construct a D-alley by two moving Qs. There is free mobility along the x-axis o f a frontoparallel line segment, and hence the perceived HZ (E) has the property of"two-point homogenity" stated in RCC.3. However, congruence of angle is not included in this experiment, and the conclusion should be reserved until we have a direct demonstration in this subspace of free mobility of two-dimensional figures, not of a line segment. On the other hand, in a frontoparallel plane H P ( P ) , it seems to me that the possibility of free mobility of two-dimensional figures, or that of congruence between two figures at least, is taken for granted in our daily life. This subspace of VS has one more important characteristic. It is also taken for granted in photographs of a fiat figure on H P ( P ) that we see the same figure in these pic- tures despite the fact that sizes are different from the original. It implies that, in addition to congruence, the relationship of simi- larity is possible on HP (P). Mathematically, similarity of any two figures at different positions is possible only in RCC of K = 0 (i.e., E, not in other RCCs). This logic is of interest in the view of the experimental results stated earlier that P- and D-alleys from left to right on an HP(P) coincide and this plane in VS is Euclidean. If the perceived HZ(E) is RCC o f K 4 : 0 and H P ( P ) is RCC of K = 0, then logically it is impossible to have free mobility between these two subspaces of VS. It would be very important to carry out careful studies to make explicit quantita- tive relationships in all respects between perceptually con- gruent figures in various orientations in VS and their pictures presented on an HP (P) or in other orientations with or without change of size (similarity and congruence).
Conc lus ion
A tentative formulation as to frameless VS is as follows. The perceived horizontal plane of eye-level, HZ(E) , is structured according to Riemannian geometry of nonzero curvature K, whereas frontoparallel planes, HP (P), or surfaces perpendicu- lar to the line of sight are structured according to Euclidean geometry, K = 0. The formulation may "explain" why P- and
450 TAROW INDOW
D-alleys are not the same on H Z ( E ) , why similarity holds in H P (P) , and so forth. This is a formulation at a phenomenologi- cal level, and this formulation itself needs an "explanation" of why H Z ( E ) and H P ( P ) are structured in these ways. I f a vari- ety of visual phenomena are describable in this way, however, it should facilitate finding a more fundamenta l "explanation" This is a top-down approach that contrasts with bot tom-up approaches in which models for VS are constructed under the constraints of either geometrical optics on the retina or physio- logical findings on cells in the brain (e.g., Fry, 1950; Gtinther, 1961; Hoffman, 1968, 1977; Leibovic et al., 1970). To decide whether a geometric approach is productive and, if it is, to p in down the most appropriate geometry needs profound theoreti- cal as well as experimental considerations (Baird, 1970a, 1970b; Dodwell, 1983; Heelan, 1983; Hoffman, 1980; Robert, 1970; Robert & Suppes, 1967; Suppes, 1977). VS is dynamic, not a solid empty container into which various percepts are put with- out affecting its contours and intrinsic structure. Hence, the model proposed by Luneburg, which consists of two assump- tions, RCC and the mapping functions of Equation 4, is too rigid even for two-dimensional subspaces in VS such as H P ( P ) or the perceived H Z ( E ) . However, the two assumptions are separable. It is particularly important to extend the geometrical approach to VS under natural condit ions and to see how con- text affects the geometry and the mapping functions. Several open questions about the geometry of VS per se have been enumerated. These questions may be ins t rumental in designing new experiments.
References
Angell, R. B. (1974). The geometry of visibles. Nous, 8, 87-117. Avant, L. L. (1965 ). Vision in the Ganzfeld. Psychological Bulletin, 64,
246-258. Baird, J. C. (1970a). Psychophysical analysis of visual space. New York:
Pergamon Press. Baird, J. C. (Ed.), (1970b). Human space perception: Proceedings of
the Dartmouth Conference. Psychonomic Monograph Supplements, 3(13, Whole No. 45).
Baird, J. C., & Wagner, M. (1982). The moon illusion: I. How high is the sky? Journal of Experimental Psychology: General, 111, 296-303.
Balslov, E., & Leibovic, K. N. (1971 ). Theoretical analysis of binocular space perception. Journal of Theoretical Biology, 31, 77-100.
Battro, A. M., & Pierro Netto, S. (1976). Riemannian geometries of variable curvature in visual space: Visual alleys, horopters, and trian- gles in big open fields. Perception, 5, 9-23.
Berger, M. (1987). Geometry I and 11. New York: Springer-Vedag. Blank, A. A. (1953). The Luneburg theory of binocular visual space.
Journal of Optical Society of America, 43, 717-727. Blank, A. A. (1957). The geometry of vision. British JournalofPhysio-
logical Optics, 14, 1-30. Blank, A. A. (1958a). Axiomatics of binocular vision. The foundation
of metric geometry in relation to space perception. Journal of Opti- cal Society of America, 48, 328-334.
Blank, A. A. (1958b). Analysis of experiments in binocular space per- ception. Journal of Optical Society of America, 48, 911-925.
Blank, A. A. (1959 ). The Luneburg theory of binocular space percep- tion. In S. Koch (Ed.), Psychology: A study of a science (pp. 395- 426). New York: McGraw-Hill.
Blank, A. A. (1961). Curvature of binocular visual space. Journal of Optical Society of America, 51, 335-339.
Blank, A. A. (1978). Metric geometry in human binocular perception: Theory and fact. In E. L. J. Leewenberg & H. E J. M. Buffart. (Eds.), Formal theories of visual perception. (pp. 83-102 ). New York: Wiley.
Blumenfeld, W. (1913). Untersuchungen tiber die scheinbare Gr0sse in Schraume [Studies on apparent sizes in visual space ]. Zeitshrift far Psychologie, 65, 241-404.
Brock, E W. (1960). Parallel alleys as clues to the constitution of visual space. American Journal of Optometry, 37, 395--402.
Busemann, H. (1955). The geometry of geodesics. New York: Aca- demic Press.
Busemann, H. (1959). Axioms for geodesics and their implications. In L. Hopkin, P. Suppes, & A. Tarski (Eds.), The axiomatic method with special reference to geometry and physics (pp. 146-159). Amster- dam: North-Holland.
Caelli, T., Hoffman, W., & Lindman, H. (1978). Subjective Lorentz transformations and the perception of motion. Journal of Optical Society of America, 68, 402-411.
Craig, E. J. (1969). Phenomenal geometry. British Journal of Philo- sophy of Science, 70, 121-131.
DaSilva, J. D. (1983 ). Ratio estimation of distance in a large open field. Scandinavian Journal of Psychology, 34, 343-345.
DaSilva, J. D., & DaSilva, C. B. (1982). Scaling apparent distance in a large open field: Some new data. Perceptual and Motor Skills, 56, 135-138.
DaSilva, J. D., & Dos Santos, R. A. (1982). Scaling apparent distance in a large open field: Reference of a standard does not increase the exponent of the power function. Perceptual and Motor Skills, 55, 267-274.
Dodwell, P. C. (1967). A model for adaptation to distortions of visual field. In W. Wathen-Dunn (Ed.), Models for the perception of speech and visual form (pp. 262-272). Cambridge, MA: MIT Press.
Dodwell, E C. (1982). Geometrical approaches to visual processing. In D. J. Ingel, A. M. Goodale, & R. J. W. Mansfield (Eds.), Analysis of visual behavior (pp. 801-825 ). Cambridge, MA: MIT Press.
Dodwell, P. C. (1983 ). The Lie transformation group model of visual perception. Perception & Psychophysics, 34, 1-16.
Dr0sler, J. (1966). Das beid~ingige Raumsehen [The binocular space perception ]. In W. Metzger, Handbuch der Psychologie Band I (pp. 590-615). G/~ttingen, Federal Republic of Germany: Hogrefe.
Dr0sler, J. (1979). Foundations of multi-dimensional metric scaling in Cayley-Klein geometries. British Journal of Mathematical and Sta- tistical Psychology, 32, 185-211.
Ehrenstein, W. H. (1977). Geometry in visual space--some method- dependent (arti)facts. Perception, 6, 657-660.
Eschenburg, J. H. (1980). Is binocular visual space constantly curved? Journal of Mathematical Biology, 9, 3-72.
Fieandt, von, K. (1966). The world of perception. Homewood, IL: Dor- sey Press.
Filehne, W. (1912). Die mathematische Ableitung der Form des scheinbaren Himmelsgewolbes [Mathematical derivation of the form of perceived sky ]. Archiv far Physiologie, 5, 1-32.
Foley, J. M. (1964a). Desarguesian property in visual space. Journal of Optical Society of America, 54, 684-692.
Foley, J. M. (1964b). Visual space: A test of the constant curvature hypothesis. Psychonomic Science, 1, 9-10.
Foley, J. M. (1966). Locus of perceived equidistance as a function of viewing distance. Journal of Optical Society of America, 56, 822- 827.
Foley, J. M. (1972 ). The size-distance relation and intrinsic geometry of visual space: Implications for processing. Vision Research, 13, 323-332.
VISUAL SPACE 451
Foley, J. M. (1977). Effect of distance information and range on two indices of visually perceived distance. Perception, 6, 449-460.
Foley, J. M. (1978). Primary distance perception. In R. Held, H. W. Leibowitz, & H. L. Teuber (Eds.), Handbook of sensory physiology: Vol VIII. Perception (pp. 181-213). New York: Springer.
Foley, J. M. (1980). Binocular distance perception. Psychological Re- view, 87, 411-434.
Foley, J. M. (1985). Binocular distance perception. Egocentric dis- tance task. Journal of Experimental Psychology: Human Perception and Performance, 11, 133-149.
Foster, D. H. (1975). An approach to the analysis of the underlying structure of visual space using a generalized notion of visual pattern recognition. Biological Cybernetics, 17, 77-79.
Freudenthal, H. (1965). Lie groups in the foundations of geometry. Advances in Mathematics, 1, 145-190.
Fry, G. A. (1950). Visual perception of space. American Journal of Optometry and Archives of American Academy of Optometry, 11, 531 - 553.
Galanter, E., & Galanter, P. (1973 ). Range estimates of distant visual stimuli. Perception & Psychophysics, 14, 301-306.
Gilinsky, A. S. (1951). Perceived size and distance in visual space. Psychological Review, 58, 460-482.
Gilinsky, A. S. (1955 ). The effect of attitude upon the perception. Amer- ican Journal of Psychology, 68, 173-192.
Gogel, W. C. (1972). Scalar perception with binocular cues of distance. American Journal of Psychology, 85, 477-697.
Gogel, W. C. (1977a). An indirect measure of perceived distance from oculomotor cues. Perception & Psychophysics, 21, 3-11.
Gogel, W. C. (1977b). The metric of visual space. In W. Epstein (Ed.), Stability and constancy in visual perception: Mechanism and pro- cesses (pp. 129-181 ). New York: Wiley.
Gombrich, E. H. (1974). The sky is the limit: The vault of heaven and pictorial vision. In R. B. MacLeod & L. E Herbert, Jr. (Eds.), Percep- tion: Essays in honor of James J Gibson (pp. 84-94). Ithaca, NY." Cornell University Press.
Greensberg, M. J. (1973). Euclidean and non-Euclidean geometries: Development and history. San Francisco: Freeman.
Giinther, von N. (1961). Versuch zur Aufstellung einer exakten Theorie des Sehraumes [A study to describe an exact theory of visual space ]. Optik, 18, 3-21.
Hagino, G., & Yoshioka, I. (1976). A new method for determining the personal constraints in the Luneburg theory of binocular visual space. Perception & Psychophysics, 19, 499-509.
Hardy, L. H., Rand, G., & Rittler, M. C. (1951 ). Investigation of visual space: The Blumenfeld Alley. Archives of Ophthalmology, 45, 53-63.
Hardy, L. H., Rand, G., Rittler, M. C., & Boeder, P. (1953). Thegeome- try of binocular space perception. New York: Knapp Memorial Labo- ratories, Institute o f Ophthalmology, Columbia University College of Physicians and Surgeons.
Heelan, E A. (1983). Space-perception and philosophy of science. Berkeley: University of California Press.
H igashiyama, A. (1981 ). Variation of curvature in binocular visual space estimated by the triangle method. Vision Research, 21, 925- 933.
Higashiyama, A. (1984). Curvature of binocular visual space: A modi- fied method of right angle. Vision Research, 24, 1713-1718.
Hillebrand, E (1902). Theorie der scheinbaren GrOsse bei binocu- larem Sehen [Theory of apparent sizes in binocular vision]. Denkschrisfien der Wiener Akademie, Mathematisch-Naturwissen- schaft Klasse, 72, 255-307.
Hoffman, W. C. (1966). The Lie algebra of visual perception. Journal of Mathematical Psychology, 3, 65-98.
Hoffman, W. C. (1968). The neuron as a Lie group germ and Lie prod- uct. Quarterly Journal of Applied Mathematics, 25, 423-440.
Hoffman, W. C. (1971 ). Visual illusions of angle as an application of Lie transformation groups. SIAM Review, 13, 169-184.
Hoffman, W. C. (1977). The Lie transformation group approach to visual neuropsychology. In E. L. J. Leenwenberg & H. Buffart (Eds.), Formal theories of visual perception (pp. 66-77). New York: Wiley.
Hoffman, W. C. (1980). Subjective geometry and geometric psychol- ogy Mathematical Modelling, 1, 349-367.
Hopkins, J. (1973). Visual geometry. Philosophical Review, 82, 3-34. Indow, T. (1968 ). Multidimensional mapping of visual space with real
and simulated stars. Perception & Psychophysics, 3, 45-64. Indow, T. (1974a). On geometry of frameless binocular perceptual
space. Psychologia, 17, 50-63. Indow, T. (1974b). Applications of multidimensional scaling in percep-
tion. In E. C. Carterette & M. P. Friedman (Eds.), Handbook of per- ception (Vol 2, pp. 493-525). San Diego, CA: Academic Press.
Indow, T. (1975, August). An application of MDS to study of binocular visual space. US.-Japan Seminar: Theory, methods and applications ofmultidimensionalscalingandrelatedtechniques. University of Cali- fornia, San Diego.
Indow, T. (1977). A mathematical analysis of binocular visual space under illumination with no a priori assumption on mapping functions (9th Report). Yokohama, Japan: Psychological Laboratory, Hiyoshi Campus, Keio University.
Indow, T. (1979). Alleys in visual space. Journal of Mathematical Psy- chology, 19, 221-258.
Indow, T. (1982). An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics. Journal of Mathematical Psychology, 26, 204- 236.
Indow, T. (1983, August). Multidimensional mapping in Riemannian space and its application. Paper presented at the Psychometric Soci- ety Annual Meeting, Los Angeles.
Indow, T. (1984, June). On finiteness of visual space. Paper presented at the 17th Mathematical Psychology Meeting, Chicago.
Indow, T. (1987, June). A critical review of basic judgments for multidi- mensional mapping of colors and points in visual space. Paper pre- sented at the International Conference in honor ofG. Th. Fechner: Developments in psychophysical theory, Bonn, Federal Republic of Germany.
Indow, T. (1988). Alleys on apparent frontoparallel plane. Journal of Mathematical Psychology, 32, 259-284.
Indow, T. (1990). On geometrical analysis o f global structure o f visual space. In G. Geisler (Ed.), Psychophysical explorations of mental structures (pp. 172-180). G6ttingen, Federal Republic of Germany: Hogrefe & Huber.
Indow, T., & Ida, M. (1975). On scaling from incomplete paired com- parison matrix. Japanese Psychological Research, 17, 98-105.
Indow, T., Inoue, E., & Matsushima, K. (1962a). An experimental study of the Luneburg theory of binocular space perception (1), The 3- and 4-point experiments. Japanese Psychological Research, 4, 6-16.
Indow, T., Inoue, E., & Matsushima, K. (1962b). An experimental study of the Luneburg theory of binocular space perception (2), The alley experiments. Japanese Psychological Research, 4, 17-24.
Indow, T., Inoue, E., & Matsushima, K. (1963). An experimental study of the Luneburg theory of binocular space perception (3), The ex- periments in a spacious field. Japanese Psychological Research, 5, 1-27.
Indow, T., & Watanabe, T. (1984a). Parallel- and distance-alleys with moving points in the horizontal plane. Perception & Psychophysics, 35, 144-154.
Indow, T., & Watanabe, T. (1984b). Parallel- and distance-alleys on horopter plane in the dark. Perception, 13, 165-182.
452 TAROW INDOW
Indow, T., & Watanabe, T. (1988). Alleys on an extensive apparent frontoparallel plane: A second experiment. Perception, 17, 647-666.
Itteison, W. H. (1960). Visual space perception. New York: Springer. Kirkpatrick, E R, & Ittelson, W. H. (1953). The size-distance invar-
iance hypothesis. Psychological Review, 60, 223-23 I. Krantz, D. H. (1972). A theory of magnitude estimation and cross-mo-
dality matching. Journal of Mathematical Psychology, 9, 168-199. Krantz, D. H., Luce, R. D., Suppes, E, & Tversky, A. (1971). Founda-
tions of measurement, Vol. 1. Additive and polynomial representations. San Diego, CA: Academic Press.
Kruskal, J. B. (1966a). Multidimensional scaling by optimizing good- ness of fit to a nonmetric hypothesis. Psychometrika, 29, 1-24.
Kruskal, J. B. (1966b). Nonmetric multidimensional scaling: A nu- merical method. Psychometrika, 29, 115-129.
Kilnnapus, T. M. (1955). An analysis of the vertical-horizontal illu- sion. Journal of Experimental Psychology, 49, 134-140.
Kiinnapus, T. M. (1960). Scales for subjective distance. Scandinavian Journal of Psychology, 1, 187-192.
Kuroda, T. (1971 ). Distance constancy: Functional relationships be- tween apparent distance and physical distance. Psychologische For- schung, 34, 199-219.
Leibovic, K. N., Balslov, E., & Mathieson, T. A. (1970). Binocular space and its representation by cellular assemblies in the brain. Jour- nal of Theorectical Biology, 28, 513-529.
Lindman, H., & Caelli, T. (1978). Constant curvature Riemannian scaling. Journal of Mathematical Psychology, 17, 89-109.
Loebell, E (1950). "Landkarten" der nichtenklidischen Ebene [Maps of non-Euclidean plane ]. Jahresbericht derdeutschen Mathematiker Vereiniging, 54, 4-23.
Lukas, J. (1983). Visuelle Frontalparallelen: Ein Entscheidungs exper- iment zu den Theorie von Blank, Foley and Luneburg. [Visual front parallels: A decisive experiment to the theory of Blank, Foley, and Luneburg ]. Zeitschrift far Experimentelle und Angewandte Psycho- Iogie, 30, 610-627.
Luneburg, R. K. (1947). Mathematical analysis of binocular vision. Princeton, N J: Princeton University Press.
Luneburg, R. K. (1948). Metric methods in binocular visual percep- tion. In Studies and essays." Courant Anniversary Volume (pp. 215- 240). New York: Wiley (Interscience).
Luneburg, R. K. (1950). The metric of binocular visual space. Journal of Optical Society of America, 50, 637-642.
Metzger, W. (1930). Optische Untersuchungen am Gansfeld II. Zur Ph~inomenologie des homogenen Ganzfeld [ Visual studies on total field II. On phenomenology of homogeneous total field ]. Psycholo- gische Forschung, 13, 6-29.
Miller, A., & Neuberger, H. (1945). Investigations into the apparent shape of the sky. Bulletin of American Meteorological Society, 26, 212-216.
Neuberger, H. (1951 ). General meteorological optics. In T. H. Malone (Ed,), Compendium of meteorology (pp. 61-78 ). Boston: American Meteorological Society.
Ogle, K. N. (1964). Researches in binocular vision. New York: Hafner. Reed, C. F. (1984). Terrestrial passage theory of the moon illusion.
Journal of Experimental Psychology: General 113, 489-500.
Robert, E S. (1970). Notes on perceptual geometries (Report No. P-4423). Los Angeles: Rand Corporation.
Robert, E S., & Suppes, P. (1967). Some problems in the geometry of visual perception. Synthese, 17, 171-201.
Rock, I., & Kaufman, L. (1962). The moon illusion II. Science, 136, 1023-1031.
Schelling, H. von. (1956). Concept of distance in attine geometry and its applications in theories of vision. Journal of Optical Society of America, 46, 309-315.
Sedgwick, H. A. (1980). The geometry of spatial layout in pictorial representation. In M. A. Hagen (Ed.), The perception of pictures (Vol. 1, pp. 33-90). San Diego, CA: Academic Press.
Sedgwick, H. A. (1982). Visual modes of spatial orientation. In M. Potegal (Ed.), Spatial abilities: Development and physiological foun- dations (pp. 3-33 ). San Diego, CA: Academic Press.
Shepard, R. N. (1978 ). On the status of"direct" psychophysical mea- surement. In C. W. Savage (Ed.), Minnesota studies in the philosophy of science (Vol. IX, pp. 441--490). Minneapolis: University of Min- nesota Press.
Shipley, T. (1957a). Convergence function in binocular visual space. I. A note on theory. Journal of Optical Society of America, 47, 795-803.
Shipley, T. (1957b). Convergence function in binocular visual space. II. Experimental report. Journal of Optical Society of America, 47, 804- 821.
Smart, J. R. (1978). Modern geometries. Monterey, CA: Brooks/Cole. Spivak, M. (1979). A comprehensive introduction to differential geome-
try (Vol. 5 ). Berkeley, CA: Publish or Perish, Inc. Squires, P. C. (1956). Luneburg theory of visual geodesics in binocular
space perception. Archives of Ophthalmology, 56, 288-297. Suppes, P. (1977 ). Is visual space Euclidean? Synthese, 35, 397-421. Suppes, P., Krantz, D. M., Luce, R. D., & Tversky, A. (1989). Founda-
tions of measurement, Volume II. Geometrical threshold, and proba- bilistic representations. San Diego, CA: Academic Press.
Teghtsoonian, R. L., & Teghtsoonian, M. (1970). Scaling apparent distance in natural outdoor settings. Psychonomic Science, 21, 215- 216.
Wang, H. C. (1951 ). Two theorems on metric spaces. Pacific Journal of Mathematics, 1, 473-480.
Wang, H. C. (1952 ). Two-point homogeneous spaces. Annals of Mathe- matics, 55, 177-191.
Watson, A. S. (1978). A Riemann geometric explanation of the visual illusions and figural after-effects. In E. L. J. Leewenberg & H. E J. M. Bullart (Eds.), Formal theories of visual perception (pp. 83-102). New York: Wiley.
Yamazaki, T. (1987). Non-Riemannian approach to geometry of vi- sual space: An approach of aflinely connected geometry to visual alleys and horopter. Journal of Mathematical Psychology, 31, 270- 298.
Zajaczkowska, A. (1956a). Experimental determination of Luneburg's constants cr and K. Quarterly Journal of Experimental Psychology, 8, 66-78.
Zajaczkowska, A. (1956b). Experimental test of Luneburg's theory. Horopter and alley experiments. Journal of Optical Society of Amer- ica, 46, 514-527.
VISUAL SPACE 453
VS X R E K r
ff
EM BS, BC P-alley D-alley H-curve HP
HZ 0 Pv P
Append ix
Abbrevia t ions a nd Symbols
visual space 6 physical space p Riemannian space e Euclidean space RCC (Gaussian) curvature, K = _+ 1 ]r 2 MDS curvature radius RMS a parameter in the Luneburg mapping functions d Euclidean map for R of constant K c basic sphere, basic circle in EM parallel alley x0 (equi) distance alley frontoparallel curve ( ) frontoparallel plane in VS, HP(P) when it is defined { }
by P on the ~-axis horizontal plane in VS, H Z (E) when it is on eye-level stimulus point in X (x, y, z) or (3, ~, 0) perceptual point in VS point representing Pv in EM, (~, n, g') or (Po, ~o, 0)
perceptual distance in VS (geodesics in R) Euclidean distance representing 6 in EM Euclidean distance in X the assumption that VS is an R of constant K multidimensional scaling root mean square scaled values of 6 as inputs to MDS a parameter to convert d to p theoretical values obtained by MDS (x for E d, o, c) respective radial distances from the self or the origin
(x for 6, d, e, p) matrix configuration of points
Received March 7, 1988 Revision received July 2, 1990 Accepted December 28, 1990 •
C a l l for N o m i n a t i o n s for Neuropsychology
The APA Publications and Communications (P&C) Board has opened nominations for the editorship of Neuropsychology for the years 1993-1998. B. P. Uzzell is the incumbent editor of this newly acquired APA journal in the area of experimental and applied neuropsychology, which will begin publication under APA in 1993.
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