A Differential Geometric Description of the Relationships among Perceptions

Journal of Mathematical Psychology 44, 241284 (2000) doi:10.1006jmps.1999.1240, available online at http:www.idealibrary.com on

A Differential Geometric Description of the Relationships among Perceptions David N. Levin University of Chicago

We present a differential geometric method for measuring and characterizing the perceptions of an observer of a continuum of stimuli. Because the method is not based on a model of perceptual mechanisms, it can potentially be applied to a wide variety of observers and to many types of visual and auditory stimuli. The observer is asked to identify which small transformation of one stimulus is perceived to be equivalent to a small transformation of a second stimulus, differing from the first stimulus by a third small transformation. The observer's identification of a number of such transformations can be used to calculate an affine connection on the stimulus manifold. This quantity describes how the observer encodes an evolving stimulus as a perceived sequence of ``reference'' transformations. This type of encoding is a multidimensional generalization of Fechner's method and reduces to his psychophysical scale when the stimulus manifold is one dimensional and the reference transformation is chosen to be a just noticeable difference. The intrinsic aspects of the nature of the observer's perceptions can be characterized by the curvature and torsion tensors derived from the connection. The multidimensional analogues of psychophysical scale functions exist if and only if these quantities vanish. Differences between the affine connections of two observers characterize differences between their perceptions of the same evolving stimuli. The affine connections of two observers can also be used to map a stimulus perceived by one observer onto another stimulus, perceived in the same way by the other observer. Unlike multidimensional scaling techniques, this method does not assume that the observer has a sense of distance (a metric) or that heshe can otherwise compare stimulus pairs that are oriented along perceptually different directions in the manifold. The method was used to measure the affine connections of observers of a dot moving on different background graphics; e.g., a blank screen, a grid, or two converging lines similar to those used to create the Ponzo illusion. The results comprise quantitative measurements of the background graphic's influence on each observer's perceptions of straightness, parallelism, and distance. The measurements demonstrate differences between the perceptions of the two observers.  2000 Academic Press Reprint requests should be addressed to David N. Levin, M.D., Ph.D., Department of Radiology, MC 2026, University of Chicago, 5841 S. Maryland Ave., Chicago, IL 60637.

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0022-249600 35.00 Copyright  2000 by Academic Press All rights of reproduction in any form reserved.

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I. INTRODUCTION

One of the most interesting issues in the study of perception is the extent to which two observers have the same experience when they see or hear the same physical stimulus. When exposed to a frequently encountered stimulus, most people believe that their experience is the same as that of other individuals with normal sensory organs. This apparent similarity of experience is remarkable, given the fact that there are likely to be significant differences between the neurosensory systems of any two persons. However, when an unusual stimulus is encountered, the perceptual experiences of individuals may differ noticeably. For example, consider the variety of reactions elicited by Rorschach diagrams or works of modern art and music. Evidently, different people employ different conventions for ``encoding'' uncommon stimuli as perceptions, i.e., for creating an internal representation of the world. In order to study these differences in a quantitative fashion, it is necessary to develop a general method of measuring and parameterizing the perceptions of an individual. It is not feasible to describe a person's perceptual experience in an absolute sense because this would be tantamount to a complete description of the brain state (i.e., the neuronal firing pattern) elicited by the stimulus in question. Furthermore, one does not even know which aspects of that brain state actually comprise conscious perceptual experience. However, it is possible to measure the perceptual relationships experienced by any observer (Rumelhart 6 Abrahamson, 1973). Specifically, one can determine if an observer perceives the relationship between two stimuli to be equivalent to the relationship perceived between another pair of stimuli. For example, one can ask if the observer perceives the relationship between two facial expressions of one individual to be the same as the relationship between two facial expressions of another individual. Likewise, one can determine if a listener perceives two musical notes to be related to one another in the same way as two other musical notes. For a continuum of stimuli, one can seek to determine a large number of such perceptual analogies. Specifically, we can ask the observer to identify which small transformation of one stimulus is perceived to be equivalent to a small transformation of a second stimulus, differing from the first stimulus by a third small transformation. Figure 1 illustrates such equivalence relations for a continuum of stimuli, consisting of faces parameterized by two variables controlling the configurations of the mouth and eyes. This notion of perceptually equivalent transformations is essentially a multidimensional generalization of Fechner's methodology (Fechner, 1860; Baird 6 Noma, 1978). He considered a one-dimensional continuum of stimuli with different intensities and assumed the perceptual equivalence of just noticeable differences (JNDs) at different intensity levels. These perceptual equivalence relations are the primitives that characterize the observer's perceptual experiences in a relative sense. Namely, they determine how the observer describes a succession of stimulus transformations in terms of his perceptions of ``reference'' stimulus transformations. This concept is a generalization of Fechner's insight that the JNDs determine the way an observer encodes the perception of a one-dimensional continuum of stimuli. Specifically, he showed that linear dependence of the JNDs on the physically defined intensity (i.e., Weber's law)

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FIG. 1. Consider a two-dimensional continuum of faces depending on two parameters, (x 1 , x 2 ), that control the configurations of the mouth and eyes in some way. Given two faces (A and C) that differ by a small transformation (C Ä A), one can determine which small transformation (A Ä B) of the face at A is perceived to be equivalent to a given small transformation (C Ä D) of the face at C.

implies that perceived intensity is proportional to the logarithm of the physical intensity. Figure 2a illustrates the role of perceptual equivalence relations for a two-dimensional continuum of stimuli, each of which consists of a dot at a position on a computer screen. Suppose that the observer watches a succession of stimuli, consisting of the dot moving along a trajectory. At the initial point on the trajectory (point P(0)), the movements of the dot along the vectors E(0) and N(0) are defined to be unit reference transformations in the east and north directions. Suppose that the observer perceives the movements, E(i) and N(i), at each point P(i) (i=1, ..., 7) on the trajectory to be equivalent to these reference movements. In other words, E(i) and N(i) are perceived to be the ``local'' east and north directions. These equivalence relations imply that the observer will perceive the evolution of stimuli along the trajectory to be a series of three northward movements, followed by a series of four eastward movements. Similarly, consider a succession of colors that is described by a trajectory through a two-dimensional manifold of colors, which correspond to various combinations of the intensities of the red and blue ``guns'' of a cathode ray tube. The observer can define two small reference transformations of the initial color that are perceived to make that color ``one shade redder'' and ``one shade bluer,'' respectively. At each point along the trajectory, suppose that the observer can find small transformations that are perceived to be equivalent to these reference transformations. Each small segment along the trajectory can then be expressed as a linear combination of these local ``red'' and ``blue'' transformations. At least in principle, by decomposing successive line segments along the trajectory in this manner, one can determine exactly how the observer encodes the trajectory as a succession of red and blue transformations. Now, reconsider the question of how to characterize the perceptual differences between two observers. The above examples suggest that if two observers perceive

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FIG. 2. (a) As a dot moves along the trajectory from P(0) to P(7), the first observer perceives local dot movements, E(i) and N(i), to be equivalent to the reference movements at P(0), namely E(0) and N(0). (b) At points P(3) through P(7), the second observer perceives a different set of local dot movements, E(i) and N(i) for i=3, ..., 7, to be equivalent to the reference movements at P(0).

the same equivalence relations, they will describe any succession of stimuli in the same way (as long as they agree to base their descriptions on the same reference transformations of the initial stimulus). However, if they perceive different equivalence relations, they will give different descriptions of some evolving stimuli. This is a multidimensional generalization of the following insight of Fechner. Suppose that one observer of a one-dimensional continuum of stimuli perceives JNDs that are linearly related to the physically defined intensity, but a second observer perceives JNDs that depend on the square root of the intensity. Then, assuming the perceptual equivalence of JNDs, the first observer will perceive stimulus strength to be related to the logarithm of the physical intensity, and the second observer will encode strength as proportional to the square root of the

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physically defined intensity. Figure 2 illustrates the two-dimensional version of this phenomenon. Suppose that a second observer of the moving dot perceives the movements E(i) and N(i) in Fig. 2b to be equivalent to the reference movements E(0) and N(0) at the initial point P(0). Heshe will describe the stimulus evolution as a series of three northward movements, followed by a series of movements in a direction somewhat south of east. The two observers give different descriptions because each observer disagrees with the other observer's choice of local directions along the second limb of the trajectory. Such differences would naturally develop if the two observers utilized different visual cues, different algorithms, andor different criteria for determining local directions in the stimulus space. However, neither observer is ``right'' or ``wrong,'' unless it is the convention to define local directions by a compass or some other objective means; for example, in this case, local directions could be defined relative to the local directions of the raster lines on the computer screen. This paper shows how to use the methods of differential geometry to describe an observer's perception of equivalency between transformations of different stimuli. As stated above, this work can be loosely regarded as a multidimensional generalization of Fechner's methodology. Specifically, the stimuli are taken to define the points of a continuous manifold. Then, transformations of stimuli induce line segments on the manifold. Therefore, an observer's perception of equivalence between transformations of different stimuli defines the equivalence between line segments at different points on the manifold. Mathematically, such a method of ``parallel transporting'' line segments across a manifold, without changing their perceived directions or lengths, serves to define an affine connection on the manifold. It describes the ``connection'' that the observer perceives between stimulus transformations at different points in the manifold. It should be emphasized that the proposed formalism merely provides a way of using differential geometric constructs to describe the perceptions of the observer, who is treated as a ``black box''; we do not seek to model the perceptual mechanisms of the observer. Therefore, the technique is potentially applicable to a variety of perceptual systems (humans or machines) and stimulus types (visual, auditory, etc.). It provides a systematic framework for measuring and characterizing certain types of perceptual experience. For example, certain intrinsic aspects of the nature of an observer's perceptions can be characterized by the curvature and torsion tensors that can be constructed from the measured affine connection. Furthermore, the perceptual experiences of two observers can be compared systematically by comparing their affine connections. As long as they use the same reference stimulus and reference transformations, their descriptions of all stimuli on a manifold are identical if and only if their affine connections are the same. This way of describing psychological spaces is different from that provided by multidimensional scaling or MDS (Shepard, 1964; Beals, Krantz, 6 Tversky, 1968; Rumelhart 6 Abrahamson, 1973; Baird 6 Noma, 1978; Lindman 6 Caelli, 1978; Holman, 1978; Carroll 6 Arabie, 1980; Townsend 6 Thomas, 1993). Both methods attempt to describe a psychological manifold that is populated by internal states of the observer (e.g., perceptions). However, in almost all versions of MDS, the observer is assumed to be able to compare stimulus pairs that are oriented along

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perceptually different directions in the manifold. For instance, heshe must be able to compare a mouth movement of a face to any combined moutheye movement. Furthermore, in metric versions of MDS, the observer is even assumed to have a consistent sense of the distances between points in this psychological space or at least a way of ranking various distances in this space, even if they are along different directions. In this paper, we make the weaker assumption that the observer imposes an affine connection on the space: given a transformation of a stimulus, heshe has a procedure for identifying perceptually equivalent transformations of neighboring stimuli. In other words, the observer has a way of determining whether transformations of neighboring stimuli have perceptually equivalent directions and, if so, whether they have the same magnitudes. However, we do not assume that the observer can compare pairs of stimulus transformations along perceptually different directions. For example, an observer of the stimuli in Fig. 1 is assumed to be able to find a combination of mouth and eye movements of one face that is perceptually equivalent to a given combination of movements of another face. However, unlike MDS, the observer need not know how to compare different types of facial transformations; i.e., heshe may not possess neural circuitry for comparing mouth movements with eye movements (or with combined moutheye movements). This is important because there is evidence that observers may not always have a consistent sense of stimulus changes along different directions (i.e., the proverbial ``apples vs oranges'' comparison; Shepard, 1964). There is a superficial resemblance between the methodology of this paper and mathematical formalisms that have been used to describe the geometry of vision (Watson, 1978; Hoffman, 1978; Yamazaki, 1987; Zhang 6 Wu, 1990; Naito 6 Cole, 1994). However, despite the technical similarity, there are important conceptual differences. The foregoing authors sought to describe the perception of a specific type of spatial manifold that was populated by points corresponding to the physical points in an image. They described the observer's sense of parallelism and distance by means of differential geometric constructs such as an affine connection and metric, respectively. In some cases, specific mechanisms were proposed to explain how these geometric entities were influenced by the image's contents. For example, Watson (1978) suggested that structures in the image acted as distant sources of ``force fields'' which determined a Riemannian metric on the visual space. In contrast, the present paper seeks to describe a different manifold: namely, the manifold of internal states in which each point represents the entire sensation elicited by a complete stimulus (e.g., an entire image or sound). In other words, geometry is used to describe the relationships perceived among entire stimuli, instead of describing the perceived relationships among the lines andor points within a single visual stimulus. This is a higher level description of perceptual phenomena than that of the aforementioned authors, and, at least in principle, it can be more readily applied to stimuli that are not spatial in nature. Furthermore, it is a more phenomenological approach in the sense that the affine connection is directly measured, instead of being derived from force fields other hypothetical mechanisms. The mathematical description of the method is presented in Section II. Section III discusses experimental techniques for using this approach to measure perceptual experience. The results of such experiments are described in Section IV. Section V

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is a discussion of the implications of this work. Some additional mathematical details are contained in the Appendix. It should be noted that all of the ideas in this paper, as well as preliminary experimental results, were previously presented at scientific meetings and published in abstract form (Levin, 1996a, 1996b, 1997, 1998a, 1998b). II. THEORY

A. Affine-Connected Perceptual Spaces We assume that we are dealing with a continuous two-dimensional manifold of stimuli, with each stimulus state being assigned a value of the coordinate x, having components x k where k=1, 2. Any convenient coordinate system can be used to identify the stimulus states; perceptual experience will be described in a manner that is independent of the definition of this coordinate system. Although the manifold has been assumed to be two-dimensional for the sake of simplicity, it is straightforward to generalize most of the following results to higher dimensional stimulus manifolds. Let h k be the components 1 of the small vector h (Fig 3), representing the displacement between the nearby points x and x+h; i.e, h represents the small transformation which changes the stimulus at x into the one at x+h. Let $x be the vector representing the small transformation between the stimulus at the point x and the one at x+$x. The fundamental assumption of this paper is this: there is a transformation h+$h of the stimulus at x+$x that the observer perceives to be equivalent to the transformation h of the stimulus at x. In geometric language, the observer perceives the transformations h and h+$h to have ``parallel directions'' and equal magnitudes, i.e., to be related by parallel transport. We expect that $h depends differentiably on x, h, and $x and, because null transformations at the two points should be perceived as equivalent, that $h Ä 0 as h Ä 0. Furthermore, continuity of perceptual experience dictates that $h Ä 0 as $x Ä 0. These two statements imply that $h is a bilinear function of h and $x, $h k =&

:

1 klm(x) h l $x m ,

(1)

l, m=1, 2

except for terms of order h($x) 2 and (h) 2 $x. This expression is parameterized by the quantity 1(x) that defines an affine connection on the manifold (Schrodinger, 1963; Weinberg, 1972). 1 describes how an arbitrary small transformation h at x can be moved along any direction $x without changing its perceived direction and length. It is important to note that the experimental methods for measuring the affine connection (Section III) require the observer to parallel transport small stimulus transformations across the manifold. Naturally, these transformations 1 Throughout this paper, raised indices are used to denote vector and tensor components that transform as contravariant quantities, i.e., that transform by the matrix dx$k dx l during the coordinate transformation x Ä x$. The components of tensors that transform covariantly (i.e., by the matrix dx k dx$l ) are given lowered indices.

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FIG. 3. The observer is assumed to perceive the equivalence between certain transformations of stimuli. The vector h represents a small transformation of the stimulus state having the coordinates x. The transformation h+$h of the stimulus at x+$x is perceived to be equivalent to h. The affine connection 1 relates these perceptually equivalent transformations.

must be at least as large as JNDs. Therefore, these experiments may not be described with perfect accuracy by the bilinear expression in Eq. (1), which really applies to the parallel transport of infinitesimal stimulus transformations. In this paper, we have ignored such discrepancies, which are expected to be of order h($x) 2 and (h) 2 $x. On the other hand, suppose that the observer is able to perceive any linear relationships that exist among infinitesimal stimulus transformations at each point. Then, all such linear relationships would have to be preserved by the parallel transport process that produces perceptually equivalent transformations at other points. This suggests that there are no corrections to Eq. (1) of order (h) 2 $x. 1 can be used to calculate how the observer will describe any evolving stimulus in relative terms; i.e., in terms of a reference stimulus and any two reference transformations. In this sense, the affine connection provides a relativistic method of encoding how an observer experiences an evolving stimulus. As an example, consider an observer of an evolving stimulus represented by the trajectory x(t), where t is a parameter with the range 0t1 (Fig. 4). Suppose that the observer describes hisher perceptual experience in terms of the reference stimulus at x(0) and in terms of two reference transformations at that point, h a , where a=1, 2. As in the examples in the Introduction, the observer will witness a series of infinitesimal transformations $x, and we assume that heshe can decompose each such increment into a linear sum of transformations, each of which is perceptually equivalent to a reference transformation. In mathematical terms, the observer describes the infinitesimal transformation $x at x(t) in terms of its components $s a along the vectors h a (t) that are perceptually equivalent to the reference transformations h a at x(0): $x k =h k1 (t) $s 1 +h k2 (t) $s 2 .

(2)

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FIG. 4. An evolving stimulus can be encoded in a coordinate-independent manner as a sequence of transformations that are combinations of local equivalents of reference transformations. Each increment $x of the stimulus trajectory x(t) can be decomposed into components $s a along transformations h a (t) of x(t), that are perceived to be equivalent to reference transformations h a of the initial stimulus x(0). The function s(t) describes the course of the evolving stimulus in terms of a series of transformations $s that are combinations of the reference transformations or their perceptual equivalents. The affinity 1 connects the coordinate-dependent and coordinate-independent descriptions.

Integration of the $s leads to a function s(t) that describes the way the observer interprets or encodes the evolution of the stimulus in terms of the ``standard'' stimulus changes represented by the reference transformations h a . For example, Fig. 2a shows that the first observer of the dot perceives it to move three units in the local north direction followed by four units of movement along the local east direction. Therefore, in this case, s(t) describes an s trajectory with a three unit limb parallel to the s 2 axis, connected to a four unit limb parallel to the s 1 axis. It is important to note that s(t) describes perceptions in a manner that is independent of the x-coordinate system. This is because s(t) describes the stimulus solely in terms of other perceptual experiences of the same observer, i.e., in terms of transformations perceived to be equivalent to reference transformations of the reference stimulus. It does not describe the stimulus evolution in terms of x or some physically defined stimulus coordinates. This coordinate independence is a mathematical consequence of the fact that s is defined to be a scalar by Eq. (2). For a one-dimensional continuum of stimuli, this approach is closely related to Fechner's method (Fechner, 1860). To see this, let x denote the physically defined stimulus intensity (e.g., luminance of a patch of light), and let h(x) be the JND at each intensity level x. Fechner assumed the perceptual equivalence of the intervals h(x) at different values of x. This implies that h(x+$x)=h(x)+$h(x) and h(x) must be related by parallel transport along the segment $x. This relationship is described by a one-dimensional version of Eq. (1), $h(x)=&1(x) h(x) $x, where 1(x) is the affine connection on the manifold. Weber's discovery that the JNDs were approximately linear over a reasonable range of intensities implies h(x)tc(x+x 0 ),

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where c and x 0 are empirical constants. It follows from the above two equations that Weber's law is equivalent to the statement 1(x)t &1(x+x 0 ). Fechner's psychophysical law is obtained by substituting the linear form of the JNDs into the one-dimensional version of Eq. (2) and then integrating the result in order to derive x 1 +x 0 1 s(x 1 )&s(x 2 )t ln , c x 2 +x 0

\

+

where s(x) denotes the perceived strength of the stimulus with physical intensity x. In other words, the results of Weber and Fechner can be reformulated in terms of a one-dimensional version of the geometric framework of this paper. Weber's law follows from substituting a 1x form of the affine connection in Eq. (1) and then integrating it to find h(x). Fechner's psychophysical scale is the s description of stimuli that follows from choosing a JND to be the reference transformation; this scale can be found by substituting Weber's law in Eq. (2) and integrating over $s. Therefore, Fechner's ``global'' psychophysical scale is derived from the ``locally'' measured affine connection by a procedure involving two integrations. Notice that Fechner constructed his scale by assuming the perceptual equivalence of JNDs, a measure of discriminability. In contrast, this paper shows how a description of an observer's perceptions can be derived from measurements of any set of perceptually equivalent stimulus transformations. It is not necessary to consider JNDs or to postulate their perceptual equivalence. Furthermore, there are other subtle but important differences between a more rigorous formulation of Fechner's method (Luce 6 Edwards, 1958) and the one-dimensional version of the formalism in this paper. In the former, one deals with stimulus ``quanta'' (single JNDs) and concatenates them additively in order to obtain a psychophysical scale. In contrast, our formalism utilizes small stimulus increments ($x and h(x)) along a stimulus continuum and is only exact if higher order terms (terms of order h($x) 2 and (h) 2 $x) are taken into account. Even though these increments must be at least as large as JNDs in actual experiments, we have ignored these corrections in our analysis of experimental data. Fechner also took this liberty, but he was subsequently criticized for it (Luce 6 Edwards, 1958). As shown in Fig. 5a, this technique can be used to compare the perceptual experiences of two observers. If they describe the phenomenon with identical functions s(t), they will perceive the evolving stimulus to be the result of the same sequence of transformations that are perceptually equivalent to the same reference transformations. If these functions are not identical, the two observers will describe the evolving stimulus differently. This means that they must have utilized different transformations as the local equivalents of the reference transformations. In other words, the two observers used different perceptual equivalency relations, corresponding to different affine connections. For example, each observer of the moving dot in the Introduction encoded the stimulus as an s-trajectory by utilizing E(0) and N(0) as the first and

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FIG. 5. (a) The perceptual experiences of two observers of the same stimulus x(t) can be compared by comparing their coordinate-independent descriptions, s(t) and s$(t), derived from the common reference transformations h a and each individual's affine connection (1 and 1$). (b) The affine connections of two observers can be used to calculate evolving stimuli, x(t) and x$(t), that they perceive to result from the same sequence of transformations s(t).

second reference transformations, respectively. The first observer (Fig. 2a) perceived an s-trajectory with two limbs: an upward vertical segment of three units along the s 2 axis, followed by a rightward horizontal segment of four units, parallel to the s 1 axis. The second observer (Fig. 2b) encoded the stimulus as an s-trajectory with an upward vertical segment of three units along the s 2 axis, followed by a second segment that was directed to the right and slightly downward in the s-plane. These differences in the s-trajectories resulted from different perceptions of what directions were equivalent to east and north along the second limb of the trajectory. In principle, the coordinate-independent description s(t) of a trajectory x(t) can be calculated exactly in terms of the affine connection, taken together with the reference transformations (Fig. 4). Once the 1 of an observer has been measured, Eq. (1) can be used to calculate the h a (t), the transformations perceived to be equivalent to the reference transformations at each point along the trajectory x(t). Then, each incremental stimulus transformation can be decomposed in order to derive s(t) as in Eq. (2). As shown in the Appendix, the function h a (t) must be found by solving

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an integral equation in which 1 is the kernel. Although this equation cannot be solved exactly, it can be used to develop a perturbative solution for h a (t) that is valid if the affine connection is sufficiently small. The first three terms in the corresponding perturbative solution for s(t) are derived in the Appendix: s a (t)= : h ka[x k(t)&x k(0)]& k=1, 2

1 klm h ka

: k, l, m=1, 2

|

t

[x l (u)&x l (t)]

0

dx m du du

1 klm + : 1 jlm 1 kji h ka x i i, k, l, m=1, 2 j=1, 2 t dx _ [x l (u)&x l (t)][x i (u)&x i (0)] m du. du 0 &

_

:

&

|

(3)

Here, the affine connection and its derivatives are evaluated at the origin of the trajectory x(0), and h ka is the inverse of the matrix defined by the reference transformations, i.e., h l 1 h k1 +h l 2 h k2 =$ kl , where $ kl denotes the Kronecker delta. The expression in Eq. (3) does not account for terms which involve higher order derivatives andor higher order products of 1. Given an observer's affine connection and hisher choices of the reference stimulus and reference transformations, one can calculate the stimulus trajectory x(t) corresponding to a given coordinate-independent description s(t). In other words, the relationship in Eq. (3) can be inverted (Fig. 4). The perturbative form of this inverse relationship is x k(t)=x k(0)+ : h ka s a (t)+ a=1, 2

+

: i, l, m, a, b, c=1, 2

|

: l, m, a, b=1, 2

_

k lm

[s a (u)&s a (t)] s b(u) 0

|

t

[s a (u)&s a (t)] 0

ds b du du

1 & : (1 kjm 1 jli +1 klj 1 jmi ) h la h ib h m c x i j=1, 2

t

_

1 klm h la h m b

&

ds c du. du

(4)

Here, the affine connection and its derivatives are evaluated at the origin of the trajectory x(0). As before, we have dropped terms that involve higher order derivatives andor higher order products of 1. As shown in Fig. 5b, this equation can be used to find different stimuli that two observers, having different affine connections, will describe as the same sequence of transformations, applied to the same initial stimulus state. Taken together, Eqs. (3)(4) can be used to map a stimulus perceived by one observer onto another stimulus that is perceived in the same way by another observer. This process of ``translating'' a stimulus so that its perceptual meaning is preserved is analogous to the translation of a sentence into another language without altering its content. There is a particularly simple interpretation of a trajectory x(t) corresponding to an s(t) that is a straight line. This type of evolving stimulus is perceived to be the result of repeated applications of the same transformation. For instance, the trajectory x(t), generated by substituting the straight line s a (t)=t$ 1a into Eq. (4), consists

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of repeated transformations that are perceptually equivalent to h 1 . For example, this might correspond to a musical chord being transformed repeatedly to a higher and higher pitch. Such trajectories, which are the analogs of ``straight lines'' in the observer's perceptual space, are the geodesics of differential geometry. B. Flat Perceptual Spaces without Torsion Consider a perceptual space that has the following intrinsic property: every stimulus trajectory forming a simple closed loop (a round trip) corresponds to a coordinate-independent description s(t) that also forms a simple closed loop. This implies that the observer perceives no net change in s when there has been no net change in the physical state of the stimulus. This condition can be restated in another way (Fig. 6): any two trajectories, x(t) and x$(t), with the same endpoints (stimuli A and B) correspond to two coordinate-independent descriptions, s(t) and s$(t), with identical endpoints. In other words, if a stimulus evolves in two different ways from state A to state B, the net transformations perceived by the observer are identical. The perceptual experiences of normal observers commonly have this type of self-consistency, at least to a good approximation. For example, most observers perceive the same net change whether a face (1) first undergoes a right eye movement and then a left eye movement or (2) first undergoes the same physical movement of the left eye, followed by the same physical movement of the right eye. Manifolds with this property have the following feature: because the net change in s between any two stimuli is independent of the path between them, each stimulus state can be assigned a value of s with respect to some fixed reference stimulus. In other words, the transformations perceived to relate each state to the reference state can be used to establish a ``natural'' coordinate system with the reference stimulus at the origin. Then, the coordinates of two stimuli (their s-space coordinates) can be used to characterize all possible sequences of transformations leading from one to the other. For example, suppose that the observer sees a face with a closed right eye transform itself into a face with a closed left eye by undergoing left eye closure,

FIG. 6. A manifold is flat and torsionless if and only if every loop-like stimulus trajectory corresponds to a loop-like coordinate-independent description. In this case, any two trajectories, x(t) and x$(t), with the same endpoints (A and B) correspond to two coordinate-independent descriptions, s(t) and s$(t), that also have common endpoints.

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followed by right eye opening. On this kind of manifold, the observer can be certain that the same physical state will be reached if the right eye is perceived to open, followed by closure of the left eye. Thus, this natural coordinate system makes it easy for the observer to ``navigate'' among stimuli ``without getting lost.'' The function s(x) is the multidimensional generalization of Fechner's psychophysical scale function. In the language of differential geometry, such a manifold must be intrinsically flat and have a symmetric affine connection (see Appendix or Schrodinger, 1963); namely, B klmn(x)=0 V k(x)=0,

(5)

where B klmn is the RiemannChristoffel curvature tensor B klmn =&

1 klm 1 kln + + : (1 k 1 i &1 kin 1 ilm ) x n x m i=1, 2 im ln

(6)

and V k is the torsion, equal to the antisymmetric part of the affine connection V k =1 k12 &1 k21 .

(7)

Manifolds with flat torsionless affinities have the property that there are special ``geodesic'' coordinate systems in which the affine connection vanishes everywhere and in which all geodesic trajectories are straight lines. These are the above-mentioned s coordinate systems in which the coordinates of each stimulus state directly describe perceived transformations leading to that state (i.e., correspond to values of psychophysical scale functions). Flat manifolds have another important property: the transformation of stimulus B that is perceived to be equivalent to a given transformation of A is independent of the configuration of the trajectory taken from A to B. This corresponds to the condition h(1)=h(1) in Fig. 7. Thus, the observer always equates the same transformation at B to a given transformation at A, no matter what sequence of transformations led to B. In other words, the observer can navigate among the stimuli of the manifold without losing his ``bearings'' or becoming perceptually disoriented. For example, this would be the case if equivalent transformations at each point on the manifold were defined solely in terms of information that is intrinsic to the stimulus at that location, i.e. without using ``historical'' information dependent on the path leading to that stimulus. In the example of the moving dot in the Introduction, this could be done if a small compass symbol were printed at each location on the screen. Flat perceptual spaces can be characterized by functions that are simpler than the affine connection. Given any set of reference transformations h a of a reference stimulus, the observer can identify perceptually equivalent transformations of other stimuli with various coordinates (x). Because the transformations identified at each

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FIG. 7. For a curved manifold, the transformation at B that is perceptually equivalent to a reference transformation h at A may depend on the path taken from A to B. For example, the equivalent transformation at B is h(1) or h(1) for the right and left paths, respectively. For flat manifolds, there is only one equivalent transformation: h(1)=h(1).

point do not depend on the path leading there, they are well-defined functions of the point's coordinates; i.e. they can be written as h a (x). It can then be shown (Schrodinger, 1963) that the affine connection can be written in terms of these functions and their inverses: 1 klm(x)=&h l 1(x)

h k1 h k &h l 2(x) 2 . x m x m

(8)

In other words, the eight components of the affinity can be expressed in terms of two functions, each of which has two components. In practice, if the values of h a (x) are measured at a dense enough collection of points on the manifold, h a (x) at other points might be estimated by interpolation (e.g., by fitting the measured values to a parametric form such as a Taylor series or by using a suitably trained neural network). For example, as shown earlier in this section, Weber's law can be regarded as the statement that, in the one-dimensional case, the x-dependence of the perceptually constant vector h(x) is well approximated over a surprisingly wide range by the first two terms in its Taylor series. C. Perceptual Spaces with Curvature andor Torsion If B klmn does not vanish everywhere, the manifold is said to have intrinsic curvature. This means that the observer's perceptual system has the following intrinsic property: some loop-like trajectories x(t) correspond to coordinate-independent descriptions s(t) that are open curves. Therefore, if the stimulus evolves from a point B on the loop to a point A, the net transformation perceived by the observer

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depends on which segment of the loop was the path of evolution (Fig. 8). This means that each stimulus is not associated with a well-defined net perceived transformation relating it to a fixed reference stimulus. In other words, the s values perceived by the observer cannot be used to establish a coordinate system on the manifold. This implies that the observer does not impose a well-defined psychophysical scale function on the manifold. A confusing situation may develop if the observer is not cognizant of the manifold's curvature and tries to use the s values as coordinates (i.e., as values of scale functions): the perception of a stimulus (A or A in the right panel of Fig. 8) may depend on the configuration of the trajectory leading to it from another stimulus (B). Equivalently, if the same physical stimulus A is observed before and after observations of other stimuli (e.g., a round trip in x-space, A Ä B Ä A in Fig. 8), the observer may perceive that there has been a net change in A, namely, A Ä A in s-space. Heshe may not even recognize this stimulus state. For example, this would happen if the observer was only conscious of the s-representation (A ) of the stimulus and compared it to a memory of its previous encoding (A). For instance, suppose that an observer travels on a sphere and uses the usual curved affine connection that is induced by the Euclidean metric of the embedding 3D space. This means that segments of great circles are perceived to be geodesics or straight lines. Consider a trajectory consisting of the following movements: an initial movement along a great circle by one quarter of the circumference, followed by a leftward movement along the locally orthogonal great circle by one-quarter of the circumference, followed by a similar leftward movement along the locally orthogonal great circle. This describes a round-trip journey that takes the traveler back to the starting point on the sphere. However the corresponding trajectory in s-space has an ``open leg,'' because it consists of three straight lines that are connected by two right angle vertices. Therefore, if the traveler does not recognize that his perceptual space is curved and tries to use the s values as coordinates (i.e., as psychophysical scale values) on the manifold, heshe may perceive the journey to have an open leg; i.e., he may not even recognize that the starting point has been revisited. Of course, if the curvature is explicitly

FIG. 8. If the affine connection has curvature andor torsion, some loop-like trajectories in x-space will have descriptions in s-space that do not return to the initial point. Therefore, if the stimulus evolves from any point on the trajectory (point B) to the initial point on the trajectory (point A in x-space), the net transformation perceived by the observer (A &B vs A&B in s-space) depends on which limb of the trajectory was followed.

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taken into account, it is possible to navigate accurately, i.e., to have consistent, reproducible perceptions of evolving stimuli. Curved perceptual spaces have another potentially problematical property (Schrodinger 1963): the transformation of stimulus B, which is perceived to be equivalent to a given transformation of A, depends on the configuration of the trajectory taken from A to B. This corresponds to the condition h(1){h(1) in Fig. 7. In contrast to flat perceptual spaces, there is no unambiguous, path-independent choice of an equivalent transformation. Therefore, the observer's perceptual reference frame at a given stimulus depends on the prior evolution of the stimulus state. This will not confuse an observer who is aware of the curvature of hisher perceptual space and accounts for it. However, inconsistent perceptions will occur if the observer erroneously assumes that the perceptual space is flat. For example, if the ``flat-minded'' traveler on a sphere revisits a point at which directional conventions were originally established, heshe may fail to identify those directions correctly. Perceptual problems can also arise if the curvature tensor vanishes, but the torsion does not, i.e. if the first part of Eq. (5) is true, but the second part is not. In this case, it can be shown (Schrodinger, 1963) that the observer's perception of the net transformation between two stimuli is dependent on the trajectory of stimulus evolution between those points (Fig. 8). Thus, just as in curved spaces, the observer's perceptions of transformations cannot be used to establish a coordinate system (i.e., psychophysical scale functions) on the manifold. The naive observer, who is unaware of the torsion of the connection and tries to use the s values as coordinates (i.e., as psychophysical scale values), will perceive confusing relationships between different stimuli, as in the case of a curved space. On the other hand, because the curvature is zero, the observer's perceptions do suffice to define equivalent ``local'' transformations in an unambiguous fashion. Therefore, even the naive observer will be able to recognize previously encountered directions upon revisiting a stimulus; i.e., the observer will not suffer from the sense of ``disorientation'' possible in curved spaces. III. EXPERIMENTAL METHODS

A. Overview The affine connection can be written as the sum of parts that are symmetric and antisymmetric in the lower two indices. Knowledge of the symmetric part is equivalent to knowing (Schrodinger, 1963) (1) the shapes of all geodesic trajectories and (2) the ``metric'' along each geodesic curve (i.e, a method of subdividing each geodesic curve into small segments that are carried into one another by ``parallel transport'' along the curve). The first part of the experiment was designed to determine the symmetric parts of the observer's affine connection by measuring the configurations of trajectories of this type. The observer used a computer mouse to modify computer-generated stimuli. He was presented with an initial stimulus and was asked to perform a series of perceptually equivalent transformations of his choice, each of which operated on the result of the previous transformation. In the

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experiments in this paper, each stimulus consisted of a background graphic, with a superposed dot at a location controlled by the position of a computer mouse. The observer was asked to move the dot in any direction and then to perform a series of perceptually equivalent transformations, each of which began with the result of the previous transformation. For instance, heshe might have initially chosen to move the dot a certain distance in a certain direction, and then heshe performed a series of dot movements, each of which translated the dot by a perceptually equivalent distance in a perceptually equivalent direction. If the stimulus had consisted of faces as in Fig. 1, the observer would have been asked to deform the face's expression in any way and then to perform a succession of perceptually equivalent facial transformations. For example, the observer could deform the face by a series of incremental transformations, each of which was perceived to close the eyes by an additional 100 and simultaneously increase the mouth's upward concavity by an additional 100. In effect, the observer started at an initial position in stimulus space and ``drove'' through the stimulus space in a ``straight'' line, leaving ``milestones'' at constant ``distance'' intervals. The resulting trajectory constituted a geodesic curve that was subdivided into segments related by parallel transport. The configurations of a number of such curves were used to deduce the symmetric part of the observer's affine connection. The antisymmetric parts of the affine connection (i.e., the components of the torsion) were determined from the configurations of pairs of stimulus trajectories that connected the same two points in s-space. Therefore, in the second part of the experiment, the observer was implicitly asked to ``navigate'' along such trajectories. First, he was asked to review any two of the geodesics (say, 1 and 2) that he had already drawn at a given initial point. Next, he was asked to transform the stimulus at the terminus of geodesic 1 along a geodesic perceived to be equivalent to geodesic 2. Later, the observer again reviewed geodesics 1 and 2 and he was asked to transform the terminal stimulus of geodesic 2 along a geodesic perceived to be equivalent to geodesic 1. Suppose that initial movements along geodesics 1 and 2 were taken to be reference transformations. Then, the concatenation of geodesic 1 and the perceptual equivalent of geodesic 2 comprised an s-space trajectory having its first limb along the s 1 axis and its second limb parallel to the s 2 axis. Likewise, the concatenation of geodesic 2 and the perceptual equivalent of geodesic 1 had its first limb along the s 2 axis and its second limb parallel to the s 1 axis. Each of these two-limbed trajectories formed two adjacent sides of a rectangle in s-space; each extended from the origin to the diagonally opposite corner of the rectangle. For the dot stimuli described in these experiments, the observer might have chosen geodesics 1 and 2 that moved the dot ``N 1 units along direction 1'' and ``N 2 units along direction 2,'' respectively. Then, he would have been asked to move the dot at the end of geodesic 1 by ``N 2 units along direction 2.'' Later, he would have been asked to move the dot at the terminus of geodesic 2 by ``N 1 units along direction 1.'' Suppose that the reference transformations were defined to move the dot from its initial position ``one unit along direction 1'' and ``one unit along direction 2,'' respectively. Then, geodesic 1, together with the concatenated geodesic, formed two adjacent sides of a rectangle in s-space, having one corner at the origin and a diagonally opposite corner at coordinates (N 1 , N 2 ). Likewise, geodesic 2, together

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with the geodesic concatenated with it, comprised a two-limbed trajectory forming the other two sides of the same rectangle. The antisymmetric parts of the observer's affine connection were derived from the configurations of these two trajectories (together with similar pairs of trajectories that formed other rectangles in s-space). The generality of this experimental paradigm is illustrated by the following example. If the stimulus space had consisted of the faces in Fig. 1, the observer would have been asked to repeatedly transform an initial face in any way (i.e., to define a geodesic in face space) and then to repeatedly transform the same initial face in a second way (i.e., to define a second geodesic). Then, heshe would have been asked to transform the face at the end of the first geodesic in a way that was perceptually equivalent to the transformation represented by the second geodesic. Later, the observer would have been asked to transform the face at the end of the second geodesic in a manner that was perceptually equivalent to the transformation represented by the first geodesic. The experimental results were recorded numerically, e.g., as values and variances of the affine connection. However, the results were also displayed in a more intuitive fashion by graphing the geodesics of the measured connection that were ``parallel'' to convenient reference transformations at the origin of stimulus space. These families of geodesics provided ``maps'' that showed how the observer perceived stimuli to be related by sequences of reference transformations. In other words, these maps displayed the relative ``positions'' that the observer assigned to various stimuli, when he used the chosen reference transformations as ``meter sticks''. Differences in the perceptions of two observers were detected by demonstrating statistical differences in the numerical values of their affine connections. The perceptions of two individuals were also compared by inspecting the above-mentioned maps, generated by their affine connections. If two observers had different perceptions, these maps could be used to ``warp'' a stimulus trajectory observed by observer 1 so that observer 2 perceived the stimuli along the ``warped'' trajectory in the same way that observer 1 perceived the stimuli along the original trajectory. In other words, stimuli could be ``translated'' from the perceptual framework of one observer to that of the other so that perceptual ``meaning'' was preserved. Notice that the above-described experimental method does not attempt to fully account for the stochastic nature of perceptual phenomena. For example, the observer was not asked to draw the same geodesic on multiple occasions so that the statistically average trajectory could be determined. In this respect, the methodology differs from current implementations of Fechner's technique, in which JNDs are defined statistically (Luce 6 Galanter, 1963). B. Experimental Apparatus In the experiments described below, the stimuli consisted of a dot at various positions on a computer screen, which was otherwise blank or which displayed a constant background graphic. The observer used a computer mouse to change the stimulus, i.e., to move the dot across the screen. The dot could move over a visual angle of approximately 19% and was observed freely (i.e, without gaze fixation). As explained above, during each part of the experiment, the observer was presented

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with an initial stimulus (the dot at a certain position) and was asked to transform it along a geodesic trajectory of his choice, i.e., to move the dot along a line that he perceived to be ``straight''. After he made an attempt to do this, the computer displayed the resulting trajectory as a ``movie'' loop that repeatedly showed the dot moving from its initial location to its final location. If the observer felt that the trajectory was not actually ``straight'', he rejected it with a keystroke and attempted to draw another geodesic. Once he was satisfied that he had drawn a ``straight'' line, he could use keystrokes to modify the local speed of the dot at each point along the trajectory until it was perceived to be moving at a ``constant speed''; i.e., trajectory segments traversed in equal time intervals were perceived to represent equivalent transformations. Once this was accomplished, the trajectory was ``accepted'' with another keystroke. Each accepted trajectory was recorded by the computer as a sequence of pixel locations. The computer also recorded the ``dwell time'' associated with each location, which represented the length of time that the dot paused at that location. Because the trajectory was perceived to be a geodesic that was traversed at a ``constant speed'', short segments that were traversed in the same time intervals represented perceptually equivalent transformations; i.e., they were perceived to be in the same direction and to have the same length. Several steps were taken to control extraneous conditions (i.e., conditions other than the state of the stimulus) that might influence the observer's performance. First of all, the observer viewed the computer screen through a short cylindrical tube that enclosed his head at one end and circumscribed the video display at the other end. This tube was lined with black felt, and the experiments were conducted in a darkened room so that the observer was not influenced by extraneous visual cues. Second, steps were taken to prevent the observer's perception of ``straightness'' and ``speed'' from being influenced by his perception of the mouse's trajectory across the mouse pad. Specifically, movements of the mouse were mapped onto movements of the dot by means of quadratic functions. This made it necessary for the observer to move the mouse along a segment of a rotated conic section (e.g., an ellipse, hyperbola, or parabola) in order to create a dot movement that was straight in the Euclidean sense (except for possible deformations due to screen curvature, miscalibration of the CRT beam, etc.). The quadratic function was changed without notice at frequent intervals during the experiment. This prevented the observer from basing hisher perception of ``straightness'' and ``speed'' on proprioceptive information about movements of his hand. Instead, the observer was forced to rely on hisher perception of the visual stimuli on the screen. C. Measurement Procedure The first observer was a middle-aged male who wore glasses with continuous bifocal lenses that corrected his vision for myopia, presbyopia, and astigmatism. The second observer was a young male (17 years old) who wore contact lenses that corrected his vision for myopia. The first observer performed three experiments, using the following background graphics: (1) a blank screen, (2) a 15_15 grid with uniform spacing, which filled the visible screen (Fig. 9a), and (3) two converging lines similar to those used to create the Ponzo illusion (Fig. 9b; Spillmann 6

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FIG. 9. (a) One set of stimuli consisted of images of a dot at various positions on a grid. (b) Another set of stimuli consisted of images of a dot at various positions on a Ponzo background (two converging lines).

Werner, 1990). The second observer performed two experiments, utilizing the grid and the Ponzo diagram as background graphics. Each experiment 2 was performed in five sessions, which took approximately 30 min each. During each session, the observer was shown a stimulus consisting of the dot at a certain point on the screen, and he was asked to draw two or more ``constant speed'' geodesic trajectories of his choice, each of which began at that stimulus. This procedure was repeated with the dot presented at four other positions that were scattered across the visible screen. The positions of the presented dot were varied from session to session and from experiment to experiment. Next, the observer was presented with a pair of the geodesic trajectories that he had already drawn from the same initial position. Specifically, the computer first displayed a ``movie'' of the dot traversing one of these geodesics (say, 1), and then it displayed a movie of the second geodesic (say, 2). The second movie ended with 2

The program for collecting experimental measurements was written in C (Think C, Symantec, Inc.) and ran on a Macintosh platform (Apple Computer, Inc., Cupertino, CA). The program could easily be used with auditory stimuli or with other visual stimuli (e.g., colors, animated diagrams, color or grayscale images) that the observer could control by changing two parameters with a mouse (Levin, 1997, 1998a).

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the dot at the terminus of the second geodesic. Starting at this point, the observer was then asked to move the dot along a geodesic that was perceptually equivalent to the first geodesic. He did this by means of the same mouse and keyboard controls mentioned above. In other words, he created a movie that showed the dot moving away from the terminus of geodesic 2, in a direction perceived to be ``parallel'' to trajectory 1 and with a ``speed'' perceived to be the same as that of trajectory 1. Much later in the same session, he reviewed geodesics 1 and 2, and he was asked to perform the reverse operation, i.e., to begin at the terminus of geodesic 1 and draw a geodesic with a direction and ``speed'' perceived to be equivalent to those of trajectory 2. A typical session would produce five pairs of freely chosen geodesics (one pair at each of five presented dot positions) as well as five pairs of parallel geodesics. Therefore, an experiment with five sessions produced approximately 100 geodesics consisting of 50 initial geodesics and 50 geodesics drawn parallel to them. A typical geodesic consisted of 100200 points that were traversed during a 3 s movie. Figure 10 shows the 97 trajectories that were drawn by the first observer in the presence of the Ponzo graphic.

FIG. 10. Each panel shows 48 or 49 of the 97 trajectories drawn by the first observer in the presence of the Ponzo graphic. The narrow solid lines are the experimental trajectories, and the heavy dashed lines are the corresponding geodesics of the fitted affine connection.

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D. Calculation of the Affine Connection As mentioned above, the symmetric part of the affine connection was determined by the configurations of all of the geodesic trajectories. To see this, recall from IIA that a geodesic is represented by a straight line in s-space, e.g., s a (t)=_ a f (t), where _ a determines the line's direction and where f (t) is a function of elapsed time t with f(0)=0. For the ``constant speed'' geodesics considered here, f (t) can be taken to be equal to t. Substituting this into Eq. (4) gives x k(t)=x k(0)+W kt+Y kt 2,

(9)

where x k(0) are the coordinates of the stimulus at the beginning of the geodesic, W k = a=1, 2 _ a h ka , h a are the reference transformations of the stimulus at x(0), Y k =& 12  l, m=1, 2 W lW m1 klm , and 1 is evaluated at x(0). We have only included terms up to (t 2 ) in Eq. (9) because we expect the geodesics to be smooth enough to be fitted by the conic sections parameterized by this approximation. Because the dropped terms involve either derivatives of 1 or are quadratic in 1, this is equivalent to assuming that the affine connection is small and nearly constant. Each geodesic in a typical experiment was separately fitted by Eq. (9). In other words, for each of the 100 geodesics in an experiment, we found the values of x k(0), W k, and Y k that best fitted its 100200 points according to a least squares criterion. 3 The resulting values of Y k and W k were then substituted into the above expression for Y k to produce approximately 100 pairs of linear constraints on the symmetric parts of 1 : 1 k11 , 1 k22 and 1 k12 +1 k21 . Linear regression was used to find the values of these components that best fitted the 100 paired constraints. In this last step, we implicitly assumed that the affinity had similar values at the initial points of all geodesics; i.e., we again assumed that the affine connection was essentially constant across the stimulus manifold. This assumption, which was made for simplicity, is not consistent with the results of Weber and Fechner for some 1D stimulus manifolds, as mentioned in IIA. This assumption could be circumvented by expanding the affine connection as a Taylor series around one point in the manifold. Then, the linear constraints on the affinity's symmetric parts could be used to determine its derivatives, in addition to its value, at the center of convergence of the Taylor expansion. However, it would be best to have even more experimental data to determine this additional information about the affine connection. The antisymmetric part of the affine connection was determined from the configurations of the pairs of geodesics that were freely drawn at each presented stimulus, together with the configurations of the two geodesics drawn ``parallel'' to them. To see this, consider any two trajectories in x-space that connect the origin of s-space to an identical point in s-space. Let the difference between the terminal points of these x-space trajectories be 2x, which may or may not be zero. If we subtract Eq. (4) for one of these trajectories from Eq. (4) for the other one, we find 3 The data analysis in (Wolfram Research, Inc., 8500150 took less than 5 of the processor time was

IIIDF was performed by a program that was written in Mathematica Urbana, IL) and that ran on a Macintosh platform. A Power Macintosh min to analyze the data from a typical experiment with 100 geodesics. Most consumed by three plotting routines.

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FIG. 11. Perceptual maps consisting of families of ``parallel'' geodesics, derived from the affine connection measured in each experiment. The left panels of (c) and (e) show that the two Ponzo observers would describe slightly different physical trajectories (origin Ä A Ä B and origin Ä E Ä F ) in the same way (``two rightward moves, followed by two upward moves''). (a) Observer 1blank background. (b) Observer 1grid background. (c) Observer 1Ponzo background. (d) Observer 2grid background. (e) Observer 2Ponzo background.

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the following relationship between 2x and the antisymmetric part of the affine connection, 2x k =(h 11 h 22 &h 12 h 21 ) A s(1 k12 &1 k21 ),

(10)

where h a are the reference vectors and where A s is the area in s-space enclosed by the two curves. As in the derivation of Eq. (9), we have omitted terms that are higher order in 1 or that involve derivatives of 1. As outlined in IIIA, Eq. (10) was applied to each pair of geodesics that had been freely drawn by the observer. First, we defined the reference transformations to be along their initial directions. Then, either one of these geodesics, together with the geodesic drawn ``parallel'' to the other freely drawn geodesic of the pair, formed two adjacent sides of an s-space rectangle. These two-limbed trajectories connected the corner of the rectangle at the origin of s-space to the diagonally opposite corner. Therefore, the x-space difference between the terminal points of the two ``parallel'' geodesics satisfied Eq. (10). Here, A s was the area of the rectangle in s-space, and the reference transformations h a were equal to the initial directions of the freely drawn geodesics (i.e., equal to the vectors W in Eq. (9) that were fitted to each of these geodesics). Because of the relationship W k = a=1, 2 _ a h ka in Eq. (9), this choice of reference transformations implied (_ 1 , _ 2 )=(1, 0) and (_ 1 , _ 2 )=(0, 1) for the first and second freely drawn geodesics, respectively. Because s a (t)=_ a t, it followed that the length of the s-space trajectory of each freely drawn geodesic was equal to the time it took to traverse that geodesic. Therefore, A s could be set equal to the product of the traversal times of these two trajectories. By substituting the measured values of 2x k , W k and A s into Eq. (10), we derived one pair of linear constraints on the antisymmetric part of the affine connection. A typical experiment produced 25 sets of freely drawn geodesics and their ``parallel'' counterparts; therefore, each experiment resulted in 25 pairs of constraints. These were subjected to linear regression in order to find the values and variances of the antisymmetric part of the affine connection. E. Display of Results The results of a single experiment included the values and variances of the eight components of the affine connection obtained by the above-described regression procedure. The symmetric part of the affine connection was derived from two independent regressions: one regression on the three quantities 1 k11 , 1 k22 , and 1 k12 +1 k21 for k=1 and the other regression on the same three quantities for k=2. The corresponding 3_3 variance matrices were used to compute ellipsoidal joint confidence volumes in the three-dimensional spaces of each of these triplets. The antisymmetric part of the affine connection was derived from two other regressions: one on 1 k12 &1 k21 for k=1 and the other on the same quantity for k=2. The resulting variances were used to derive confidence intervals for each of these quantities. The following method (left panels of Fig. 11) was used to provide a more intuitive understanding of the affine connection derived from each set of experimental data. First, the measured affine connection was used to create the geodesic that passed through the x 1 unit vector at the origin of x-space. This curve was

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described by Eq. (9) with x(0)=(0, 0) and W=(1, 0). This geodesic was then divided into segments that were carried into one another by parallel transport along the curve. Therefore, these segments represented stimulus transformations (called h 1 transformations) that would be perceived to be equivalent to one another by an observer with the measured affine connection. The endpoints of these segments were found by evaluating x(t) at small equal increments of t. We then constructed vectors at each of these endpoints that were perceptually equivalent to the unit vector along the x 2 -axis at the origin. This was done by using the measured affine connection to parallel transport that unit vector (Eq. (1)) from the origin to the endpoint of each segment. Next, we constructed a geodesic through each of these endpoints that was directed along the local perceptual equivalent of the x 2 unit vector. This geodesic was described by Eq. (9) with x(0) equal to the coordinates of the segment's endpoint and with W equal to the parallel transported version of the x 2 unit vector. Each of these geodesics was then divided into segments that were carried into one another by parallel transport along the curve, as well as by parallel transport along the first-drawn geodesic. Therefore, these segments represented stimulus transformations (called h 2 transformations) that would be perceived to be equivalent to one another by an observer with the measured affine connection. The endpoints of these segments were found by evaluating Eq. (9) at small equal increments of t; these points were marked with dots as fiducials. The resulting family of geodesics constituted a map of how the stimulus manifold was perceived by the observer with the measured affine connection. Specifically, it showed the observer's perception of what transformations were required to create any stimulus, i.e., the number of h 1 transformations, followed by the number of h 2 transformations, that would be required to transform the stimulus at the origin into the stimulus of interest. In other words, the map could be used to derive an s-space description of a trajectory that extended from the stimulus at the origin to any other stimulus of interest. For example, consider the observer, whose map is shown in the left panel of Fig. 11c. He perceived the stimulus (dot) at location B to be reached by two h 1 transformations (two ``rightward moves''), followed by two h 2 transformations (two ``upward moves''). Notice that this type of description of a stimulus is independent of the definition of the x-coordinate system; instead, it depends on the observer's notion of parallel transport and on hisher choice of reference stimulus and reference transformations. Also note that if the observer's affine connection has curvature or torsion, the s values of a stimulus are path-dependent (Fig. 8) and cannot be used to define a manifold coordinate system (i.e., the multidimensional analogue of a psychophysical scale function). The middle panel of Fig. 11c is another map of the same observer's perception. It shows how to create any stimulus by first performing h 2 transformations, followed by h 1 transformations. As mentioned in IIB, if the stimulus manifold has no curvature and torsion, two trajectories connecting the same two points in s-space correspond to x-space trajectories that also connect the same two points (Fig. 6). In other words, the same stimulus would be produced if one began with the stimulus at the origin and performed the same number of h 1 and h 2 operations in either order. However, if the affine connection has a nonvanishing curvature or torsion, different stimuli may be produced by performing the h 1 and h 2 operations

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in different orders. A small effect of this type can be detected in the maps in the left and middle panels of Fig. 11c. For example, the observer produced one stimulus (B in the left panel) by performing two rightward moves, followed by two upward moves; he produced a slightly different stimulus (D in the middle panel) by performing two upward moves, followed by two rightward moves. The right panel of Fig. 11c is a third map of the same observer's perception. It shows how to create any stimulus by first performing h 1 operations, followed by a series of operations perceptually equivalent to movements along a 45% line at the origin. F. Statistical Analysis of Results We tabulated the values and variance matrices of the affine connection measured during each experiment performed by one observer with one background graphic. We then tested the hypothesis that the affine connection was zero. Specifically, we used an F-test to determine if a vanishing affine connection was within various joint confidence regions for the parameters 1 k11 , 1 k22 , and 1 k12 +1 k21 and for the parameters 1 k12 &1 k21 that were derived by linear regression. In other words, we tested the hypothesis that the observer's affine connection was the same as that of a ``Euclidean'' observer who behaves in the following way: (1) he perceives a dot to be moving in a ``straight'' line if and only if it moves along a line that is straight according to the Euclidean geometry of x-space; (2) two ``straight'' line trajectories of the dot are perceived to be ``parallel'' if and only if the trajectories are parallel in the Euclidean sense; (3) segments of any given ``straight'' line are perceived to have equal ``length'' if and only if they have equal length in a Euclidean sense. We also tested the hypothesis that the observer's affine connection was the same as that of a ``pseudo-Euclidean'' observer. The latter is an observer whose notions of ``straight'' and ``parallel'' coincide with those of Euclidean geometry, but whose notions of ``length'' along each geodesic differ from those of Euclidean geometry. It can be shown (Schrodinger, 1963) that a pseudo-Euclidean observer must have a symmetric affine connection of the form 1 klm =$ kl U m +$ km U l ,

(11)

where $ kl is the Kronecker delta and where U k denotes two parameters that determine the observer's notion of ``length'' along the geodesics. Equation (11) defines a plane in the six-dimensional space of the six symmetric parts of the affine connection. We performed an F-test to determine whether any of the pseudo-Euclidean affinities in this plane were within the 95 0 joint confidence volume of the measured affine connection. In this way, we tested the hypothesis that the experimental observer only differed from the Euclidean observer with respect to his notions of ``length'' along geodesics. We also compared the affine connections measured for one observer of the dot moving against two background graphics. First, we tested the hypothesis that the two affine connections were the same; i.e., we tested the hypothesis that the differences in the background graphics did not affect the observer's perception of the

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dot's motion. We also tested the hypothesis that the two affinities only differed by terms of the form in Eq. (11). This tested the hypothesis that the background graphics only influenced the observer's perception of ``length'' along the geodesics, not his perceptions of ``straightness'' and ``parallelism''. Both of these hypotheses were evaluated by using F-tests to compute joint confidence regions for the difference between the two affine connections. Here, we assumed that there were no covariances between the two connections. Finally, we compared the affine connections measured for the two observers of the dot moving against the same background graphic. First, we tested the hypothesis that the affine connections of the two observers were the same. Then, we tested the hypothesis that their affine connections only differed by terms like those in Eq. (11). This tested the hypothesis that the two observers only differed in their perceptions of ``length'' along the geodesics, not in their perceptions of ``straightness'' and ``parallelism''. As above, both of these hypotheses were evaluated by using F-tests to compute joint confidence regions for the difference between the two affine connections. Here, we assumed that there were no covariances between the two connections. IV. EXPERIMENTAL RESULTS

A. Individual Experiments with One Observer and One Background Graphic 1. Overview. Table 1 lists the affine connections measured in the five experiments. Each component of the affinity is listed along with its standard deviation, computed from a diagonal component of the variance matrix. Note that some TABLE 1 The Values and Standard Deviations of the Affine Connections Measured in Each Experiment 1 111

1 1 1 2(1 12 +1 21 )

1 2 2 2(1 12 +1 21 )

1 222

1 1 1 2(1 12 &1 21 )

1 2 2 2(1 12 &1 21 )

Obs 1 Blank

0.31 \0.20

0.42 \0.18

0.17 0.20 \0.25 \0.20

0.53 \0.17

&0.15 \0.24

&0.03 \0.16

0.03 \0.09

Obs 1 Grid

&0.62 \0.23

0.07 \0.18

0.20 0.24 \0.29 \0.20

&0.13 \0.15

&0.55 \0.25

&0.05 \0.05

&0.02 \0.03

Obs 1 Ponzo

&1.20 \0.20

0.54 \0.24

0.02 0.19 \0.30 \0.15

&0.35 \0.17

&0.34 \0.22

&0.13 \0.10

0.20 \0.13

Obs 2 Grid

0.16 \0.18

&0.08 \0.16

&0.08 &0.01 \0.22 \0.14

&0.01 \0.12

0.04 \0.18

&0.05 \0.05

0.01 \0.06

Obs 2 Ponzo

&0.15 \0.18

0.15 \0.22

0.10 0.26 \0.22 \0.21

&0.74 \0.26

&1.24 \0.25

&0.24 \0.19

0.12 \0.16

Experiment

1 122

1 211

Note. The standard deviations of the symmetric parts of the affine connection cannot be interpreted ``jointly,'' because these components of the connection were derived by simultaneous regression on three variables. Bold type denotes components whose joint confidence volume excluded a vanishing (i.e., Euclidean) affine connection at the 950 confidence level.

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symmetric components of some of the measured affinities are printed in bold type. This means that the above-described F-tests forced us to reject the hypothesis that those components were all zero; i.e., we rejected the hypothesis that the observer's perception was the same as that of a Euclidean observer. Specifically, the 95 0 confidence volume in the three-dimensional space of those three parts of the affinity did not include the origin. On the other hand, it turned out that all of the antisymmetric parts of the measured affinities were not significantly different from zero. In other words, in each case, we could not reject the hypothesis that the affinity was torsionless. As mentioned in IIIF, we also tested the hypothesis that the symmetric part of the measured affinity was the same as that of a pseudo-Euclidean observer. Computation showed that we could reject this hypothesis in only one experiment (observer 2Ponzo background); i.e., for all of the other experiments, the 95 0 confidence region for the symmetric parts of the affinity included pseudo-Euclidean affinities. Figure 11 shows the perceptual maps corresponding to each of the five measured affine connections. These figures demonstrate how each observer's notions of ``straight,'' ``parallel,'' and ``equal length'' along a geodesic differed from those of a Euclidean observer, whose map would consist of a Cartesian grid. Of course, the statistical significance of these apparent differences is determined by the tests mentioned in the preceding paragraph. 2. Observer 1blank background. In this case, both triplets comprising the symmetric parts of the affine connection were different from those of a Euclidean observer at the 95 099 0 level of confidence. The left and middle panels of Fig. 11a show dot movements that the observer perceived to be parallel to movements along the axes of x-space. A Euclidean observer would perceive these to be slightly curved and to be converging slightly toward the upper and right halves of the plane. Furthermore, from the Euclidean standpoint, the experimental observer equated diagonal movements in the lower left quadrant to shorter diagonal movements in the upper right quadrant (right panel of Fig. 11a). 3. Observer 1grid background. In this experiment, the symmetric affine connection was not different from that of a Euclidean observer at the 95 0 level of confidence. The left and middle panels of Fig. 11b show dot movements that the observer perceived to be parallel to movements along the axes of x-space. There are slight differences between these maps and the maps of a Euclidean observer (a uniform grid), but these differences are not statistically significant. 4. Observer 1Ponzo background. In this case, the symmetric parts of 1 1lm were different from zero at a confidence level greater than 99 0 (F=13.0), while the symmetric parts of 1 2lm differed from zero at the 950 confidence level. The left panel of Fig. 11c shows moderate convergence and slight curvature of the geodesic movements perceived to be parallel to the x 2 -axis. The middle panel shows a noticeable tendency for short dot movements in the left half-plane to be parallel transported into longer segments in the right half-plane. 5. Observer 2grid background. In this experiment, we could not reject the hypothesis that the measured affine connection was equal to that of a Euclidean

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observer. Figure 11d shows that the corresponding maps resemble the uniform grids that comprise the perceptual maps of a Euclidean observer. 6. Observer 2Ponzo background. In this case, the symmetric parts of 1 2lm differed from zero at a confidence level greater than 99 0 (F=9.0). We could not reject the hypothesis that the symmetric parts of 1 1lm were equal to zero. Therefore, there were significant differences between the perceptions of this observer and those of a Euclidean observer. Furthermore, this observer's affine connection differed from the affine connection of all possible pseudo-Euclidean observers at the 95 0 confidence level. Figure 11e shows that this observer's geodesics tended to converge as they moved toward the upper and left halves of the stimulus space. The right and left panels also demonstrate a tendency for geodesics to curve and to have a ``shrinking length scale'' as they enter the lower half-plane. B. Comparison of Experiments with One Observer and Different Background Graphics 1. Observer 1blank vs grid backgrounds. There were differences at the 99 0 and 950 levels between the symmetric parts of 1 1lm and 1 2lm , respectively, for these two experiments. However, we could not reject the hypothesis that these differences were solely due to different length scales along the geodesics; i.e., we could not reject the possibility that the two affine connections only differed by terms like those in Eq. (11). There was no significant difference between the antisymmetric parts of these two affine connections at the 95 0 level. 2. Observer 1blank vs Ponzo backgrounds. There was a difference at the 99 0 level between the symmetric parts of both 1 llm and 1 2lm for these two experiments (F=10.6 and 4.6, respectively). However, we could not reject the hypothesis that these differences were solely due to different ``length'' scales along the geodesics. There was no significant difference between the antisymmetric parts of these two measured affine connections at the 95 0 level. 3. Observer 1grid vs Ponzo backgrounds. There was no significant difference between either the symmetric or antisymmetric parts of these two affine connections at the 95 0 level. 4. Observer 2grid vs Ponzo backgrounds. The symmetric parts of 1 2lm for these two experiments differed at the 99 0 level of confidence (F=7.1), but the symmetric parts of 1 1lm did not differ at the 95 0 level. We could not reject the hypothesis that these differences were solely due to differences in ``length'' scales along geodesics, rather than the ``straightness'' and ``parallelism'' of geodesics. The antisymmetric parts of these two affine connections did not differ at the 95 0 level.

C. Comparison of Experiments with Two Observers of the Same Background Graphic 1. Grid background. There were no differences at the 95 0 level between either the symmetric or antisymmetric parts of the affine connections of the two observers when they observed dot movements against a grid background.

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2. Ponzo background. The symmetric parts of 1 1lm for these two experiments differed at the 99 0 level of confidence (F=5.6), and the symmetric parts of 1 2lm differed at the 95 0 level. However, once again, we could not reject the hypothesis that these differences were solely due to differences in ``length'' scales along geodesics, rather than the ``straightness'' and ``parallelism'' of geodesics. The antisymmetric parts of these two affine connections did not differ at the 95 0 level.

V. DISCUSSION

It is not surprising that the affine connections of both observers in the grid experiments were statistically indistinguishable from the vanishing affinity of a Euclidean observer. This is because the observers tended to measure the directions and lengths of dot movements with respect to the grid and to use those measurements to define perceptual equivalence according to Euclidean geometry. Thus, the presence of the grid tended to bias each observer's perception so that it was the same as that of a Euclidean observer. This bias would be present even if the dot trajectories were warped by computer-related effects (e.g., screen curvature or warping of the screen display due to miscalibration of the cathode ray tube or due to ambient magnetic fields) and by observer-dependent factors (e.g., distorted optical path between the screen and the retina or distorted processing of visual information). This is because the grid would be warped by the same effects. The measured affine connection of the blank screen observer differed from that of a Euclidean observer at the 95 0 level. It also differed at the 95 0 level from the same observer's affine connection for dot movements against the grid background. Because the grid was not present to influence the observer's perception, these differences could be due to some of the computer-dependent or observer-dependent factors mentioned above. Additional experiments would have to be performed to determine at what stage the differences occurred. The affine connections that were most strikingly different from that of a Euclidean observer (differing at the 99 0 level) were those of both observers of the dot motion on the Ponzo background. Some of these differences represented perceptual distortions that were consistent with the Ponzo illusion itself. For example, consider the convergence of geodesics perceived to be ``parallel'' to the x 2 -axis (left panels of Figs. 11c and 11e). This effect suggests a tendency to succumb to the Ponzo illusion, i.e., to overestimate the length of a horizontal line in the upper half plane, relative to the perceived length of a physically identical horizontal line in the lower half plane. However, the meaning of the other perceptual distortions associated with the Ponzo experiments is unclear. At least some of these effects are produced by the nature of the Ponzo diagram itself. This is suggested by the fact that the affinity of observer 1 was significantly different (at the 99 0 level) from his affine connection for a blank screen. Likewise, the Ponzo affinity of observer 2 was significantly different (at the 990 level) from his grid affinity. However, the exact nature of these perceptual distortions was observerdependent. This is suggested by the fact that there was a highly significant difference (at the 99 0 level) between the affine connections of the two observers of the Ponzo diagram.

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The main purpose of this work is not to perform definitive experiments on the perception of specific visual stimuli, 4 which have been studied by earlier investigators (for example, see Mandriota, Mintz, 6 Notterman, 1962). Rather, its purpose was to introduce a new method for the mathematical description of perceptual experiences. This work is based on the following assumption: if two stimuli differ by a small transformation, an observer can tell whether a small transformation of one stimulus is the ``same or different'' as a small transformation of the other stimulus. Differential geometry provides the natural mathematical language for describing such equivalence relations between transformations. Specifically, these equivalence relations endow the manifold with structure, represented by an affine connection. The affinity provides a way of describing the essential perceptual experience of the observer without modeling the mechanism of perception; i.e., without specifying how the observer derives the affinity from sensory data. Therefore, it can potentially be applied to diverse types of stimuli and to a variety of observers (machines, humans, even other creatures). This theory is relativistic in the sense that it describes how an observer perceives stimuli in terms of the observer's perceptions of other stimuli (a standard reference stimulus and reference transformations as in Fig. 4). It is not surprising that this is best accomplished with differential geometry, which is the mathematics used in the theory of general relativity. This is because both general relativity and the formalism in this paper seek to describe continua (spacetime and the stimulus manifold, respectively) in a coordinate-independent manner, and differential geometry is the natural mathematical language for accomplishing this. As shown in Fig. 11, the measured affine connection of any observer can be used to create maps that illustrate how heshe perceives stimuli in the manifold. It should be emphasized that these maps are not intrinsic to the manifold of physical stimuli, but are imposed upon the stimulus manifold by the observer's mind. In effect, these are maps of an inner psychological space, populated by the observer's brain states that are elicited by the stimuli. As stated above, one of our main assumptions is that the observer is endowed with some sort of mechanism for performing parallel transport on the stimulus manifold. In other words, the observer is able to find perceptually equivalent transformations of different stimuli on the manifold. This means that the observer is 4 These experiments could be improved in several ways. (1) It would be best to eliminate as much extraneous visual information as possible by having the stimuli viewed with a truly immersive display (e.g., a head-mounted display). (2) In order to characterize the exact physical nature of the presented stimuli, it would be helpful to use a carefully calibrated cathode ray tube with a screen of known curvature. (3) It would be useful to provide each observer with a standard clock (e.g., a metronome-like time signal) so that the observer's perception of time does not influence hisher perception of ``speed.'' (4) Another experimental issue concerns the assumption that the affine connection was small and nearly constant across the stimulus manifold, i.e., the assumption that the observer's perceptions did not differ very much from those of a Euclidean observer. This assumption was justified by the smooth nature of the trajectories in these experiments, and it was supported by the fact that the experimental geodesics were well fitted by such a constant connection. In principle, it is straight forward to circumvent this assumption by retaining higher order terms in the derivation of Eqs. (9)(10) and by fitting the values of Y k to a Taylor series for the affine connection's variation across the manifold. However, as mentioned in the Discussion, more abundant experimental data would be needed in order to determine the additional parameters (i.e., derivatives of the affine connection).

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assumed to be able to determine if two stimulus transformations are the same or different. This can be done by comparing like attributes of the two transformations, i.e., by determining if the first transformation changes each stimulus attribute (e.g., pitch and loudness) in the same way that the second transformation changes the same attribute. To appreciate the weakness of this assumption, consider what would happen if it were not true. The observer of an evolving stimulus would not be able to ``connect'' or relate his perception of how the stimulus is changing at one moment to his perceptions of how it changed at any earlier time. As the stimulus evolved, heshe would experience a confusing stream of ``jumbled'' or unrelated perceptions. 5 Because the vast majority of individuals do not experience this type of perceptual chaos, it is reasonable to assume that their perceptual experience is described by a parallel transport process. As shown in IIA, a sufficiently ``differentiable'' mechanism of parallel transport affects infinitesimal transformations in a bilinear fashion that defines an affine connection. We attempted to measure this affine connection by fitting it to experimentally determined geodesics. In order to do this, we implicitly assumed that the affine connection accurately described the parallel transport of noninfinitesimal segments of these geodesics (i.e., small segments that were at least as large as JNDs). In effect, we ignored possible nonlinear corrections to Eq. (1). As mentioned in Section IIIA, the affine connection is sufficient to define relative distances along each geodesic (Schrodinger, 1963). This limited notion of distance can be used to compare the magnitudes of a series of transformations along perceptually equivalent directions (e.g., to compare a series of qualitatively similar facial movements of one face). In order to compare distances along different directions, 6 an observer must be able to impose a Riemannian metric g kl on the manifold. In this paper, we did not assume the existence of a metric, which implies that the observer has an innate sense of the absolute distance between stimuli. In fact, one can show that some affinely connected manifolds do not even admit a metric. To see this, note that a metric must be consistent with the observer's sense of parallelism in the following way: the metric at any two points must preserve the length of a geodesic segment that is ``parallel transported'' between the two points. This leads to a mathematical consistency condition between the observer's metric and hisher affine connection (see Eqs. (9.11)(9.12) in Schrodinger, 1963). For a given affine connection, there may be no Riemannian metric that satisfies this constraint. On the other hand, for other affine connections (e.g., flat, symmetric connections), an infinite number of metrics may be mathematically possible. However, any given observer may or may not have the natural sense of distance 5

This phenomenon is loosely reminiscent of the pathological experiences of a few individuals, who became sighted for the first time as adults (Sacks, 1995). Presumably, these persons never ``learned to see'' at an earlier age; i.e., in the language of this paper, they never learned how to perform a parallel transport operation on the manifold of visual stimuli. Certain types of abstract music and art may create similar inchoate sensations in novice observers. These stimuli may take them into a kind of perceptual terra incognita in which there are no familiar landmarks that can be used to establish a parallel transport process. The assumption that the observer has a mechanism for performing parallel transport is equivalent to the assumption that the stimuli do not evoke this type of perceptual confusion. 6 There is some experimental evidence that observers can make this type of comparison in some situations (Marks, 1974; Stevens, 1975; Baird 6 Noma, 1978).

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that corresponds to it. For example, consider a set of stimuli consisting of two tones having pitches ( p 1 , p 2 ), which are played one after the other at a fixed time interval. Suppose that the observer knows the equally tempered scale and has a pathindependent sense of whether any transformation ($p 1 or $p 2 ) of one tone pair ( p 1 , p 2 ) is equal to the same number of semitones as the corresponding transformation ($p$1 or $p$2 ) of a second tone pair ( p$1 , p$2 ). This implies that the observer's affine connection is flat. Therefore, it is mathematically compatible with a variety of Riemannian metrics, including the Euclidean definition of length: |$p| =- $p 21 +$p 22 . However, the observer may not possess the neural circuitry that is necessary to sense lengths defined in this way; it may not even be possible to train the observer to compute this quantity accurately. For example, the observer may not sense the fact that the transformation ($p 1 , $p 2 )=(3 semitones, 4 semitones) has the same ``size'' as the transformation ($p 1 , $p 2 )=(0 semitones, 5 semitones). In short, the existence of a metric is a significant experimental question, above and beyond the existence of an affine connection. The method presented in this paper does not assume that the observer has such a sense of ``distance,'' unlike certain variants of multidimensional scaling (Shepard, 1964; Beals, Krantz 6 Tversky, 1968; Baird 6 Noma, 1978; Lindman 6 Caelli, 1978; Carroll 6 Arabie, 1980; Townsend 6 Thomas, 1993). The approach in this paper differs significantly from that of Dzhafarov and Colonius (1999) who also proposed a multidimensional generalization of Fechnerian psychophysics. First of all, they derived their scales from the psychometric function, which is a measure of discriminability. In contrast, there is no fundamental connection between stimulus discrimination (e.g., JNDs) and our method of describing stimuli. Second, their method involves the construction of a metric on the stimulus manifold. As mentioned above, we do not assume that the observer imposes a metric on the manifold. The curvature and torsion of the affine connection must also be determined experimentally. However, there are theoretical grounds for expecting common perceptual experiences to be described by nearly flat manifolds with nearly vanishing torsion. Flat manifolds have the feature that the perception of a stimulus transformation does not depend on the path used to create that stimulus (Fig. 7). Thus, the observer perceives ``local'' transformations in a consistent and reproducible way; i.e. he does not ``lose his bearings.'' A flat affine connection would arise naturally if the observer's local sense of direction in stimulus space (i.e., the h a (x) in Eq. (8)) was derived from information intrinsic to the stimuli at hand. Such an observer would not have to use information from previous segments of hisher traversal through stimulus space to determine local equivalents of the reference transforms. As a hypothetical example, consider the two-dimensional continuum of stimuli consisting of rectangles with sides of varying dimensions. Suppose that the observer was ``wired'' or trained to associate each stimulus (i.e., each rectangle) with local vectors h a (x), which described fixed fractional increases in the length of one side (a=1) or the other (a=2). If these transformations were judged to be perceptually equivalent across the manifold, the manifold would be flat, and it would have the affine connection given by Eq. (8). If the affine connection happens to be torsionless in addition to being flat, every sequence of transformations connecting two stimuli is associated with the same net coordinate-independent description s a ;

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i.e., the observer perceives the same net relationship between two stimuli no matter how one of them was transformed into the other one (Fig. 6). Equivalently, upon revisiting a stimulus, the observer is able, to recognize it because heshe correctly perceives no net transformation to have occurred. This situation, which implies that the observer imposes psychophysical scale functions on the stimulus manifold, certainly matches the common perceptual experience of most observers, and in this case, the methodology of this paper can be greatly simplified (Levin, 1999b). Of course, there are many situations in which an observer has two ostensibly identical experiences and perceives differences because of intervening changes in hisher physical condition. For example, a weight may be perceived to be heavier after an observer becomes fatigued during a work-out. However, such an observer has not really experienced the same stimulus twice; i.e., heshe has not made a round trip in stimulus space. Rather, heshe has experienced two stimuli corresponding to two different points in a higher dimensional stimulus space, the additional dimensions being used to parameterize the changing attributes of hisher physical condition (e.g., muscle fatigue). Curved manifolds describe more confusing and unfamiliar perceptual experiences. The transformation of one stimulus that is perceptually equivalent to a transformation of another stimulus may depend on the path traversed between the two stimuli (Fig. 7). For example, this might happen if the observer derived the transformation at any point from the equivalent transformation at previously visited points, instead of deriving it from cues that are intrinsic to the current stimulus (see the previous paragraph). In this situation, the observer's interpretation of a transformation may depend on that observer's recent perceptual history; i.e., on how that stimulus state was reached in the first place. If the observer was not aware of the curvature of the manifold, heshe could become ``disoriented'' and confused if a stimulus was revisited and the new perception of a transformation conflicted with a perception of the same transformation recalled from memory. Furthermore, curvature andor torsion of the affinity imply that different trajectories between two stimuli may be associated with different values of the net transformation perceived by the observer (Fig. 8). In this situation, there are no well-defined psychophysical scale functions which describe the observer's perceptions of stimuli. Unless the observer explicitly accounts for this effect, the relationship perceived between two states could depend on how one of them was transformed into the other. This pathdependence of the recognition process is reminiscent of some cases of abnormal perception described in the neurological literature on agnosia 7 (Farah, 1990). On the 7 As a speculative example, it is conceivable that Fig. 8 describes the patient in ``The Man Who Mistook His Wife for a Hat'' (Sacks, 1985). He looked at his wife (stimulus A), looked away at another stimulus (B), and then looked back at her, completing a round-trip in stimulus space (x-space). However, he did not recognize her because his internal representation of her had changed (A Ä A in s-space). As another speculative example, imagine a hypothetical individual who uses a curved or asymmetric affine connection to describe the perceived relationships among a collection of thoughts (considered to be internally generated stimuli). If this individual took the curvature or torsion into account, heshe could ``navigate'' among such internal states in a consistent manner. However, if the individual did not account for these factors, he might experience a kind of mental pathology in which he could not recognize his own thoughts upon revisiting them. These comments are simply meant to illustrate the theoretical potential of differential geometry to describe complex and bizarre perceptual phenomena. Of course, in any particular case, detailed experiments must be performed to test this type of speculation.

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other hand, it is conceivable that a curved affine connection could be an asset, instead of an indication of neuropathology. If the observer was aware of the curvature and accounted for it, hisher description of the stimulus space could be simpler than that of an observer who insisted on imposing a flat affine connection on the manifold. For instance, the relationships between, points on a spherical surface are simply described by the usual curved affine connection induced on the sphere by the Euclidean metric of the embedding 3D space; a more complex description would result if one insisted on imposing a flat connection on the sphere. These are exotic examples, but they serve to make the following point: the methods in this paper are quite general, and, at least in principle, they have the potential to describe some bizarre or pathological perceptual experiences. It has long been known that an observer's perception of a stimulus can be influenced by the nature of other stimuli to which heshe has recently been exposed (Baird, 1997). However, this phenomenon need not be interpreted as path dependence of perceptual judgment due to curvature or torsion of the manifold. To see this, note that most of these experiments involve sequences of ID stimuli; i.e., stimuli with different values of a single physical variable (e.g., a sound with variable intensity). If a sequence of such stimuli is treated as a path through a 1D stimulus continuum, then temporal sequence effects are definitely not manifestations of curvature or torsion, because all one-dimensional manifolds have vanishing curvature and torsion. On the other hand, suppose that the entire sequence of 1D stimuli is treated as a single stimulus on a higher dimensional manifold, e.g., pairs of tones with different intensities are treated as different points on a 2D manifold. Then, two sequences with different early 1D stimuli and identical terminal 1D stimuli correspond to two different points in the higher dimensional manifold. The fact that the perception of the terminal 1D stimulus depends on the identity of an earlier one just means that stimulus sequences corresponding to different points in the higher dimensional manifold are perceived differently. This can certainly happen even if the higher dimensional stimulus manifold has no curvature or torsion. A few comments should be made about the meaning of the s-space description of an evolving stimulus. An s-space trajectory encodes an observer's description of how stimulus B can be created by beginning with reference stimulus A and subjecting it to a series of reference transformations (or their perceptual equivalents). This is analogous to describing a location B on an urban landscape by specifying a path that reaches it from starting point A (e.g., ``one block in direction 1, followed by 3 blocks in direction 2, followed by 3 blocks in direction 1''). If the manifold has a flat and torsionless affine connection, it is not necessary to describe the entire path, because each stimulus can be assigned path-independent coordinates s a . In that case, the stimulus B can be specified by simply giving its coordinates s a$ as well as the identities of the reference stimulus A and reference transformations. This is analogous to imposing a grid with a certain origin and directionality on a flat landscape and then representing locations by their coordinates on this grid (e.g., in the above example, ``B is located at coordinates [4, 3] relative to A''). This is a ``natural'' internal coordinate system, because it is independent of the x coordinate system and is completely determined by the attributes of the observer's ``point of view'' (i.e., hisher affine connection and choice

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of reference stimulus and reference transformations). It also constitutes a common coordinate system among observers who share these attributes but may use different x coordinate systems. For example, consider two observers whose primary sensory cortices record different linear (or nonlinear) combinations of the same physical attributes of the stimulus; e.g., their sensory inputs correspond to x coordinate systems that are rotated andor rescaled with respect to one another. Their s-space descriptions of an evolving stimulus will still be the same, as long as they use identical affine connections and identical reference stimulitransformations. As stated in IIA, this is because the s a are scalar quantities. Up to this point, we have focused on the perception of evolving stimuli. However, in principle, it is possible for an observer to internally generate s-space representations of static stimuli. If an observer is capable of mentally transforming a memorized template to match a static stimulus, the stimulus could be represented by the s a coordinates of the endpoint of the trajectory describing the transformation process. In other words, the stimulus could be represented by encoding the evolution of the template-matching process with certain reference vectors (and their perceptual equivalents). This would be possible if the observer had a simple way of generating the reference vectors at each point on the manifold; e.g. if heshe could derive these vectors from cues that are intrinsic to the stimulus itself 8 (see the above-described hypothetical example of rectangle stimuli). The foregoing considerations suggest that s-space trajectories are the natural way to represent stimuli in a mind that is aware of relationships among stimuli but is not conscious of their absolute physical characteristics (e.g., x-coordinates). Needless to say, it must be determined experimentally if the human brain actually uses s-space in this way. The methodology of this paper can be used to compare the perceptions of two observers of the same evolving physical phenomenon. As shown in Fig. 5a, if the observers agree to describe the stimulus in terms of the same reference stimulus and reference transformations, they will generate equivalent descriptions (s-space trajectories) if and only if their affine connections are equal. In this sense, observers with different affine connections have intrinsically different views of the world. As an example, consider two observers, O and O$, of the tone pairs described earlier in this section. They will have different affine connections if O accurately recognizes semitones throughout the musical scale, but O$ has a warped sense of semitones at higher points on the scale (e.g., at high points on the scale, the ``semitones'' of O$ are actually greater than the conventional semitone pitch increment). The observer O might sense that O$ is singing out of tune at high pitches (i.e., that he is not using an equally tempered scale). Two observers will differ in a more superficial manner if they have the same affine connections, but simply encode their observations with different reference stimuli and reference transformations. A simple example of 8 There is an added benefit to be gained if the h a (x) are derived from the intrinsic features of a stimulus, i.e., if they are ``embedded'' in the stimulus in the sense that they bear a constant relation to it. In that case, the coordinates s that are derived by encoding an object during a template-matching process will be independent of its orientation with respect to the observer; e.g., matching the object to its template will generate the same s coordinates as matching the rotated object to a suitably rotated template. The internal (s) representation of such an object would be ``constant'' or intrinsic to the object in this sense; only the reference stimulus (the template) would differ (by its degree of rotation).

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observers using different reference stimuli are the proverbial persons who see the same glass as half-full and half-empty, respectively. As another example, consider observers O and O$ who correctly recognize semitones (i.e, they have the same affine connection). However, suppose that they differ on the definition of middle C, because O has an accurate sense of absolute pitch and O$ does not. This corresponds to a difference in choice of reference stimulus (i.e., ``frame shift''). In this case, O would recognize that O$ is singing in the wrong key and could show O$ the ``right'' key by intoning a middle C (or playing it on a pitch pipe). Finally, suppose that O and O$ differed by a uniform scale factor in their definition of a semitone. For example, suppose O correctly recognized a semitone, but O$ thought that a smaller pitch increment was a semitone. In this case, the two observers would have the same affine connection, but they would differ by their choice of reference transformation. The more musical observer O could demonstrate the ``correct'' reference transformation (a semitone) by singing a scale or playing it on an instrument. In short, even if two observers have the same affine connection, they may still ``look at the world from different perspectives'' if they differ in their choice of the reference stimulus and reference transformations. By the same token, a constant choice of reference information is necessary if a single observer is to experience the same physical phenomenon in the same way on two different occasions. This kind of stability of perception could be achieved if the observer had a fixed way of deriving the reference information from the stimulus itself. In general, the affine connections of two observers may differ because of variations at any point in the perceptual process, i.e., because of any sort of acquired or congenital differences in the structures of their sensory organs or brains. Since the methods in this article do not model the perceptual mechanisms of the observer, they cannot identify the reason why two observers do not ``see'' stimuli in the same way. However, as illustrated in Figs. 5a and 5b, these methods could be used to ``translate'' between these two ways of encoding stimuli. Specifically, any evolving stimulus perceived by one observer could be mapped onto another stimulus that a second observer would perceive in the same way; i.e. both observers would perceive their respective stimuli to be the result of the same sequence of transformations, starting with the same initial stimulus state. This type of translation process is illustrated in the left panels of Figs. 11c and 11e. The two observers of the dot on the Ponzo background use the same language (``two moves rightward, followed by two moves upward'') to describe slightly different physical trajectories (origin Ä A Ä B and origin Ä E Ä F). In principle, this stimulus translation procedure provides a method of enhancing communication by enabling observers with different neurosensory systems to share the same perceptual experience. There are a number of potential applications of the techniques reported in this article (Levin, 1999a). In principle, it is possible to build a machine that emulates the perceptual performance of a given observer, at least for simple stimulus manifolds. Such a device would store the equivalence relations of the observer to be emulated. This could be done by measuring the observer's affine connection and explicitly encoding it as a parametric representation (e.g., a Taylor series) in the machine's control program or hardware. Alternatively, the device could be based on a nonparametric ``connection engine'' consisting of a neural net that performed

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parallel transport, after it was ``trained'' to reproduce the equivalence relations of the observer. Because of the coordinate independence of the approach, it is insensitive to the method used to implement parallel transport. For example, even if two neural nets had different architectures, they would generate identical s-space representations of an evolving stimulus, as long as they were trained on the same data on perceptually equivalent transformations and as long as they used the same reference information. One could also imagine other possible applications of these methods. For example, measurements of the affine connection might be used to detect and characterize individual differences in perception for the purpose of neurological, psychological, and psychiatric diagnosis. At least in principle, if such differences are found, the previously mentioned method of perceptual translation could be used to ``predistort'' stimuli so that an individual with ``suboptimal'' perception could see the ``world'' with the perception of an ``expert'' observer. This would allow two individuals to communicate accurately, even in the presence of perceptual differences. The main obstacle to the practical application of the methodology of this paper is the necessity of acquiring large amounts of data from each subject. For example, suppose that it is desired to measure the first two terms in the Taylor series of an affine connection on an n-dimensional stimulus manifold. The number of parameters to be determined is n 3(1+n). The measurement of each pair of perceptually equivalent stimulus transformations provides n experimental constraints on the connection (e.g., the n linear relationships in Eq. (1)). Therefore, at least n 2(1+n) measurements must be made in order to fully determine the affine connection. This means that at least 12, 36, 80, and 150 measurements are required for stimulus manifolds that have 2, 3, 4, and 5 dimensions, respectively. In order to reduce noise, it might be necessary to overdetermine the affine connection by acquiring five or ten times as much data. 9 Finally, the possibility of perceptual spaces with unusual geometries and topologies should at least be mentioned. For instance, suppose that the stimulus manifold was closed (e.g., topologically equivalent to a sphere, cylinder, or torus). Such a space might have reentrant geodesics. This means that repeated applications of perceptually equivalent transformations would return the stimulus to its original physical state. Because the observer would perceive a nonvanishing change in s-space, heshe might fail to recognize the original state when it was revisited. 10 This is analogous to the ``circular tone'' phenomenon in music (Shepard, 1978, 1982). It

9 Although it is tedious to gather the data necessary to characterize the perception of an individual, it may be easier to deduce the parallel transport rules that characterize the perception of an ``average'' human. This can be done if one assumes that the average human mind is built (or trained) to perceive most ``naturally occurring'' stimulus transformations as geodesics. One can argue that this tendency is reflected in the laws of physics, which represent human perceptions of the motions of many physical systems as geodesics of an action principle. If one is willing to entertain this speculation, then the full range of ``natural'' motions across a stimulus manifold can be used to derive a parallel transport mechanism (1 klm )) that will represent them as geodesic (or nearly geodesic) motions. The perceptual performance of the resulting system (i.e., its s-space representation of stimuli) should resemble that of the average human observer. Experiments of this kind are underway (D. Levin, unpublished). 10 As mentioned previously, this type of perceptual ``dislocation'' could also occur if the observer did not account for the curvature andor torsion of the affine connection.

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is also amusing to imagine more exotic perceptual spaces having ``worm holes,'' ``black holes,'' and other ``mind-bending'' characteristics. APPENDIX

The derivation of Eq. (3) is presented in this appendix. Let x(t) be any trajectory originating at the origin of the x coordinate system; i.e., x(0)=(0, 0). Each increment $x along the trajectory can be decomposed into components $s a along h a (t), the transformations at that point that are perceptually equivalent to reference transformations h a at the origin (Fig. 4), $s a =h 1a (t) $x 1 +h 2a (t) $x 2 ,

(A1)

where h ka (t) is defined to be the matrix inverse of h ka(t): h k 1(t) h l1(t)+h k 2(t) h l2(t)=$ lk .

(A2)

When the trajectory is traversed from the origin to x(t), the perceived sequence of transformations is s a (t)=

|

t

ds a = 0

|

t 0

_

h 1a (u)

dx 1 dx 2 du. +h 2a (u) du du

&

(A3)

It follows from Eq. (A2) that h ka (t) transforms as a covariant vector with respect to k and that its values along the trajectory change according to the covariant version of Eq. (1): $h la =

1 klm(x) h ka $x m .

:

(A4)

k, m=1, 2

This is equivalent to the integral equation h la (t)=h la (0)+

: k, m=1, 2

|

t O

1 klm(x) h ka (u)

dx m du. du

(A5)

When Eq. (A5) is substituted into Eq. (A3), the second term of the resulting expression can be integrated by parts to give s a (t)= : h la (0) x l (t)& l=1, 2

: k, l, m=1, 2

|

t 0

[x l (u)&x l (t)] 1 klm(x) h ka (u)

dx m du. du

(A6)

When Eq. (A5) is expanded as a power series in x(t), the first two terms are h la (t)=h la (0)+

: k, m=1, 2

1 klm(0) h ka (0) x m(t)+ } } } .

(A7)

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The first three terms in the power series expansion of s(t) can be derived by substituting Eq. (A7) into Eq. (A6), along with the first order Taylor series for the affinity in the region of the origin. The resulting expression is the same as Eq. (3), once the origin of the coordinate system is translated away from the origin of the trajectory. Consider the special case of a trajectory x(t) which forms a loop returning to the origin and enclosing an area A (e.g., Figs. 6 and 8). Then, Stoke's theorem can be used to simplify the integrals in Eq. (3) in order to derive the following expression for the net perceived transformation around the circuit,

\

2s a = & A+ &

:

1 llm

l, m=1, 2

: k, m=1, 2

\

h ka

|

A

|

xm d2 x A

x m d 2x

+\

+

: V kh ka k=1, 2

B km 12 +

2V k , 2x m

+

(A8)

where 2V k V k = + : (1 k V l &1 llm V k ) 2x m x m l=1, 2 lm

(A9)

is the covariant derivative of V and all of the quantities in Eq. (A8) are evaluated at the origin. If there is no net perceived transformation for every circuit-like trajectory (Fig. 6), Eq. (A8) implies that the curvature and torsion are zero, as stated in Eq. (5). If Eq. (5) is not true, Eq. (A8) shows that the net perceived transformation in the a direction may still be small if h ka is nearly perpendicular to V k and the first moment of the trajectory is along an eigenvector of B km 12 +(2V k2x m ) with a small eigenvalue. Equation 4 can be derived by using methods analogous to those presented above. ACKNOWLEDGMENTS The author is grateful to Robert Mintzer for his expert writing of the computer program that collected experimental measurements, described in Section IIIB. The author is also indebted to James Townsend for informing him about the existence and relevance of Fechner's work. Michael A. Levin is gratefully acknowledged for stimulating comments about the comparison of the perception of different observers.

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