Binocular Space Research Articles

The horopter: Old and new.

The horopter's history may partly be responsible for its ambiguous psychophysical definitions and obscured physiological significance. However, the horopter is a useful clinical tool integrating physiological optics and binocular vision. This article aims to help understand how it could come to such different attitudes toward the horopter. After the basic concepts underlying binocular space perception and stereopsis are presented, the horopter's old ideas that influence today's research show their inconsistencies with the conceptualized binocular vision. Two recent geometric theories of the horopter with progressively higher eye model fidelity that resolve the inconsistencies are reviewed. The first theory corrects the 200-year-old Vieth-Müller circle still used as a geometric horopter. The second theory advances Ogle's classical work by modeling empirical horopters as conic sections in the binocular system with the asymmetric eye model that accounts for the observed misalignment of optical components in human eyes. Its extension to iso-disparity conics is discussed.

Characterizing projective geometry of binocular visual space by Möbius transformation

Binocular vision involves the projection of objects in the 3-D visual space onto the two retinae and the comparison of spatial layout of objects in these retinal half-images. Here we characterize the unitary representation of the binocular space as a complex half-plane from the perspective of the cyclopean eye. We then investigate its automorphism group, namely the Möbius transformation group, and the associated invariants when the two eye positions are treated as fixed points of the automorphism. A three-point simple ratio from an object point to both eyes is constructed; as a complex number, its angle measures the difference in azimuth of the projected rays from the object point to each eye (i.e., horizontal disparity) while its modulus measures the ratio of the distances of the object point to the two eyes (i.e., relative vertical magnification or vertical size ratio). The four-point cross-ratio of two such simple ratios, as the only four-point invariant under Möbius transforms, reflects the fact that the relative disparity between any two object points remains unchanged when the eyes change fixation. Since the complex half-plane is biholomorphically equivalent to an open unit disk, both Poincaré model and Klein–Beltrami model give rise to a hyperbolic geometry, consistent with the empirically supported Luneburg’s (1947; 1950) model of binocular geometry (for the depth plane). Finally, the hyperbolic tangent function f(z)=tanh(z), with inverse f−1(z)=12log1+z1−z, is shown to act as the psychophysical function relating the physical representation of the binocular space to its cyclopean representation.

‘Perspectives bâtardes’: Stereoscopy, Cézanne, and the Metapictorial Logic of Spatial Construction

Stereoscopy initiated a dimensional shift that freed representation from the laws of geometric optics and longstanding conventions of linear perspective, creating aesthetic possibilities that were first explored by now obscure stereo-photographers. The late nineteenth-century ‘rupture in the pictorial order’ that Hubert Damisch attributes to Paul Cézanne has a genealogy that includes stereoscopy, the debates about binocular space perception that it occasioned, as well as innovative photographic work for the stereoscope. Through reading Hippolyte Taine, Cézanne was aware of the Berkelean idea that binocular space is not seen, but rather assembled from visual and tactile sensation sampled across time. The problem that this understanding posed for representation led many to insist that binocular space could only be invoked through synthetic apparatuses like the stereoscope; others argued for new conventions of planar representation, or ‘binocular painting’. The visual evidence from Cézanne’s work suggests that he paid attention to prescriptions for binocular painting and also to the new pictorial language developed by stereo photographers. Stereo photographers in the 1850s and 1860s, by abandoning perspectival conventions and proposing a constructivist conception of space, helped to transform Western visual culture in hitherto unappreciated ways.

GPS双目摄像机标定及空间坐标重建

GPS双目摄像机标定及空间坐标重建

The taller you are, the smaller the world appears: perceptual distortion of binocular space depends on the size of personal body space

The taller you are, the smaller the world appears: perceptual distortion of binocular space depends on the size of personal body space

Euclidean geometry of binocular space under natural viewing conditions

Euclidean geometry of binocular space under natural viewing conditions

Functional Equations in Binocular Space Perception

A general theory of the relationship of binocular visual space to physical space is formulated within a conjoint measurement framework. Psychophysically motivated invariance relations induce various functional equations. Another class of functional equations arises if we additionally assume the validity of a different psychophysical theory that is based on a formula suggested by A. A. Blank. We solve and interpret most of these equations and point out an unsolved problem. The obtained results lead to a measurement-theoretic foundation of the psychophysical assumptions underlying the Luneburg theory of binocular vision. They also contribute to clarifying the relationship between the presented general theory and Blank's approach.

On the Psychophysics of Binocular Space Perception

Binocular vision gives rise to a perceptual space that is endowed with a rich geometrical structure, which according to Luneburg is a non-Euclidean Riemannian geometry of constant Gaussian curvature. This hypothesis, together with certain psychophysical assumptions, provides a qualitative explanation of classical empirical phenomena. The Luneburg theory has been less successful in quantitatively predicting experimental data, a fact that has sometimes been taken as evidence against the presumed geometrical structure. Some of these shortcomings, however, may also be attributed to psychophysical assumptions that are independent of the geometry of visual space, and that were found to be violated systematically. We propose a generalization of Luneburg's psychophysical mapping that can account for these findings. The foundation of a psychologically significant recoordinatization of physical space is formulated within a conjoint measurement approach. It is shown that psychophysical invariances lead to restrictions on the possible form of the coordinate transformation, but its complete specification is left as an open problem. The proposed approach can at least partially explain some empirical observations that are considered to contradict Luneburg's conception of a psychophysical theory of binocular space perception.

Management of Aniseikonia after Intraocular Lens Implantation or Refractive Surgery

Aniseikonia is a difference in perceived image size in the two eyes resulting in asthenopic symptoms and distortions of binocular space perception. This is a complication of refractive and implant surgery in patients with binocular vision. While many individuals adapt readily to modest aniseikonic errors, quite a number do not. Few instruments exist to adequately test for aniseikonia. A means is described to determine if aniseikonia exists and a series of simple steps is suggested to minimize aniseikonia induced by refractive surgery or by surgery for removal of cataract and intraocular lens insertion. Also, using the proposed protocol, it is possible to determine if existing image size differences resulting from prior and/or induced aniseikonia are predominantly refractive or axial. A case report is presented. The techniques described are not costly, provide no risk, take little time, and resolve or meaningfully reduce aniseikonia in many patients having non-extreme errors of this type.

The Stereoscope and Photographic Depiction in the 19th Century

tion of the role of the interocular discrepancy for binocular space perception. Prior to his researches, several individuals had observed an essential component of Wheatstone's innovation: in binocular vision, the two eyes receive slightly different images.2 Kepler and Descartes had surmised the muscular sensations arising from the convergence of the eyes in binocular vision might play a role in measuring the proximity of objects. But Wheatstone was the first to propose the mind fathoms visual space by combining the information from a pair of two-dimensional, monocular pictures. It being thus established, Wheatstone wrote, that the mind perceives an object of three dimensions by means of the two dissimilar pictures projected by it on the two retinae, the following question occurs: What would be the visual effect of simultaneously presenting

The Stereoscope and Photographic Depiction in the 19th Century

The Stereoscope and Photographic Depiction in the 19th Century ROBERT J. SILVERMAN In 1838, Sir Charles Wheatstone, a British physicist who was one of the inventors of the telegraph, published his “Contributions to the Physiology of Vision.”1 This paper announced Wheatstone’s explana­ tion of the role of the interocular discrepancy for binocular space perception. Prior to his researches, several individuals had observed an essential component of Wheatstone’s innovation: in binocular vision, the two eyes receive slightly different images.2 Kepler and Descartes had surmised that the muscular sensations arising from the convergence of the eyes in binocular vision might play a role in measuring the proximity of objects? But Wheatstone was the first to propose that the mind fathoms visual space by combining the information from a pair of two-dimensional, monocular pictures. “It being thus established,” Wheatstone wrote, “that the mind perceives an object of three dimensions by means of the two dissimilar pictures projected by it on the two retinae, the following question occurs: What would be the visual effect of simultaneously presenting Dr. Silverman received his Ph.D. from the University of Washington in 1992, with the dissertation “Instrumentation, Representation, and Perception in Modern Science: Imitating Human Function in the Nineteenth Century.” Research for this article was supported by a grant from the National Endowment for the Humanities and by assistance from the University of Washington. The author wishes to thank Thomas L. Hankins, Keith R. Benson, and Bruce W. Hevly for their guidance and insight. 'Charles Wheatstone, “Contributions to the Physiology of Vision—Part the First. On some Remarkable, and hitherto Unobserved, Phenomena of Binocular Vision,” Philo­ sophical Transactions of the Royal Society 128 (1838): 371-94. Also in Nicholas J. Wade, ed., Brewster and Wheatstone on Vision (London, 1983), sec. 2.4. Since many of the important papers on this subject by Brewster and Wheatstone have been conveniently collected in Wade’s book, page references for articles contained in this volume will be given, wherever it is possible to do so, rather than the original sources. On Wheatstone, see Brian Bowers, Sir Charles Wheatstone, F.R.S., 1802-1875 (London, 1975). -Antecedents who are often cited include Euclid, Galen, Leonardo da Vinci, Giambattista della Porta, Aguilonius, Joseph Harris, and William Wells. "A. C. Crombie, “The Mechanistic Hypothesis and the Scientific Study ofVision: Some Optical Ideas as a Background to the Invention of the Microscope,” in Historical Aspects of Microscopy, ed. S. Bradbury and G. L’E. Turner (Cambridge, 1967), pp. 3-112.© 1993 by the Society for the History of Technology. All rights reserved. 0040-165X/93/3404-0006S01.00 729 730 Robert J. Silverman to each eye, instead of the object itself, its projection on a plane surface as it appears to that eye?”4 Wheatstone’s paper introduced an instrument that facilitated this test. The apparatus employed two mirrors mounted in a right angle in order to present the reflection of one perspectival drawing to each eye, thus creating a single percep­ tion of marked relief (figs. 1 and 2). Wheatstone called his device “a Stereoscope to indicate its property of representing solid figures.”5 Since its invention, the stereoscope has served as a tool for the study of vision. Wheatstone’s investigation of the mental aspect of depth perception offered a fundamental contribution to experimental psy­ chology, a field that became prominent in American and European universities in the late 19th century.6 Yet, despite its crucial role in the laboratory, the stereoscope is most immediately recognized as the consummate Victorian amusement. The stereoscope belonged to the class of “philosophical toys,” such as the kaleidoscope and the zoetrope , which provided entertainment but also illustrated scientific principles.7 The stereoscope occupied a curious cultural position during the second half of the 19th century. As Robert Hunt, a British photo­ graphic chemist, noted in 1856: “The stereoscope is now seen in every drawing room; philosophers talk learnedly upon it, ladies are de­ lighted with its magic representations, and children play with it.”8 The instrument was discussed in newspapers and magazines, in art jour­ nals, and in scientific treatises. Its widespread prominence affords an opportunity for historians to examine some features...

The psychophysical function of binocular space perception

A psychophysical study of binocular space perception is conducted because our understanding about this psychological phenomenon is inadequate for several reasons. The traditional psychophysical function induces a dependency of apparent distance on apparent direction. However, it fails to capture the uniqueness of the visual origin, as opposed to the location of two eyes in physical space, and is highly irregular at the boundaries of visual space. All of these deficiencies may be attributed to the fact that the traditional psychophysical function was developed with the sole objective of fitting the data. A general postulate concerning the generic psychophysical function is presented first in order to introduce a psychologically meaningful invariance property. Invariance is defined as perceptability of rigid motion. The postulate amounts to the application of R. D. Luce's ((1959). On the possible psychophysical laws. Psychological Review, 66, 81–95) principle for “possible psychophysical laws,” weakened for non-Euclidean perceptual spaces. Second, the paper presents different possible psychophyscaal laws for binocular space perception. This presentation leads to ζ = √ z 2+ c 2, with z as the (complex-valued) stimulus localization, c as the (real-valued) interpupiliary distance of the subject, and ζ as the (complex-valued) localization in the visual plane. This function captures most of the empirical horopters in a qualitatively valid manner. Moreover, the function is shown to be a conformal map and thus satisfies the general principle of independence of apparent distance and apparent direction. Finally, the relationship among the laboratory plane, the visual plane, and Euclidean maps of the visual plane is discussed.

The role of visual experience in the formation of binocular projections in frogs.

Many parts of the visual system contain topographic maps of the visual field. In such structures, the binocular portion of the visual field is generally represented by overlapping, matching projections relayed from the two eyes. One of the developmental factors which helps to bring the maps from the two eyes into register is visual input. The role of visual input is especially dramatic in the frog, Xenopus laevis. In tadpoles of this species, the eyes initially face laterally and have essentially no binocular overlap. At metamorphosis, the eyes begin to move rostrodorsally; eventually, their visual fields have a 170 degree region of binocular overlap. Despite this major change in binocular overlap, the maps from the ipsilateral and contralateral eyes to the optic tectum normally remain in register throughout development. This coordination of the two projections is disrupted by visual deprivation. In dark-reared Xenopus, the contralateral projection is nearly normal but the ipsilateral map is highly disorganized. The impact of visual input on the ipsilateral map also is shown by the effect of early rotation of one eye. Examination of the tectal lobe contralateral to the rotated eye reveals that both the contralateral and the ipsilateral maps to that tectum are rotated, even though the ipsilateral map originates from the normal eye. Thus, the ipsilateral map has changed orientation to remain in register with the contralateral map. Similarly, the two maps on the other tectal lobe are in register; in this case, both projections are normally oriented even though the ipsilateral map is from the rotated eye. The discovery that the ipsilateral eye's map reaches the tectum indirectly, via a relay in the nucleus isthmi, has made it possible to study the anatomical changes underlying visually dependent plasticity. Retrograde and anterograde tracing with horseradish peroxidase have shown that eye rotation causes isthmotectal axons to follow abnormal trajectories. An axon's route first goes toward the tectal site where it normally would arborize but then changes direction to reach a new tectal site. Such rearrangements bring the isthmotectal axons into proximity with retinotectal axons which have the same receptive fields. Anterograde horseradish peroxidase filling has also been used to study the trajectories and arborizations of developing isthmotectal axons. The results show that the axons enter the tectum before the onset of eye migration but do not begin to branch profusely until eye movement begins to create a zone of binocular space.(ABSTRACT TRUNCATED AT 400 WORDS)

Visual field size, binocular domain and the ommatidial array of the compound eyes in worker honey bees

1. The limits of the visual fields and the binocular region of the compound eyes of the worker honey bee were determined by observing the ‘pseudopupil’. The visual field data (Table 1) are valid for a defined, reproducible head-position (Fig. 2). This also means that a stereotaxic system for the bee's compound eyes is now available. The relevant angles are defined in Figs. 1, 3, 4, 5. 2. The extent of the binocular domain in the horizontal plane which lies 490 μm above the bases of the antennae (Figs. 2A, 5) is 29 °. The dorsal binocular field in the vertical plane which is perpendicular to the longitudinal body-axis extends over 42 °; the corresponding value for the ventral binocular field is 31 ° (Figs. 5, 6). 3. The lines of view of the medial peripheral ommatidia of the compound eyes intersect 5.6 mm in front of the head in the horizontal plane. In the vertical plane, we obtained a corresponding value of 2.8 mm dorsally and 4.7 mm ventrally. Each of these points of intersection marks the start of the binocular space in the respective plane (Figs. 5, 6). 4. The visual field is almost a complete sphere (Figs. 6, 7). The extent of the caudally-situated dead space is not due to the shape of the thorax and abdomen; rather, the lines of view of the peripheral lateral ommatidia skirt the outlines of the thorax. 5. The ‘monocular-binocular-boundary’ in the facetal array of the compound eye was established. This boundary separates the ommatidia whose lines of sight are directed towards the binocular domain from those ommatidia which look solely into the visual field of the respective eye (= monocular visual field). We further determined the lines of sight and the x-, y- and z-coordinate values of certain ommatidia which lie on this monocular-binocular-boundary (Table 1). 31% of all ommatidia of the bee belong to the binocular region (Fig. 8). 6. Medial peripheral ommatidia of one eye and corresponding ommatidia on the monocular-binocular-boundary of the other eye, i.e., ommatidia which look at the same point in space, usually are located ondifferent z-rows (Fig. 9, Table 1). 7. The ommatidial array of the compound eye was examined. A regular hexagonal lattice occurs at the centre of the eye; irregularities are present in the dorsal and ventral regions. An x-y-z coordinate system can be used to quantitatively describe the position of particular facets in the ommatidial array (Fig. 8). 8. Both the number of x-, y- and z-rows as well as the total number of ommatidia vary from animal to animal and — to a lesser extent — between the right and left eyes of any one bee. The average number of ommatidia per eye is 5,432. 9. An estimate was made of the visual fields of the 3 ocelli of the bee (Fig. 10).

Foundations of multi‐dimensional metric scaling in Cayley‐Klein geometries

Two‐dimensional metric scaling is based on the relations of incidence and orthogonality and on the motion of reflection in a straight line. Together with a set of optional postulates concerning different types of orthogonality these primitives are sufficient for operationalizing the Cayley‐Klein geometries for use in the psychological laboratory. They include the three Riemannian geometries (elliptic, Euclidean and hyperbolic) as special cases. The representation refers to a vector space over an unspecified field. The advantage of this lies in the identifiability of the geometry present, before technical axioms have been invoked in order to secure a field of real numbers.The approach appears to be commendable because it yields new solutions to classical problems from monocular and binocular space perception as well as from colour vision. Extensions to derived measurement outside of psychophysics and to three and more dimensions of multidimensional scaling are reported.

The role of convergence in visual space perception

Earlier experiments suggest that the perception of relative and absolute distance in binocular space is affected by the convergence angle to the stimulus. The question is how? A hypothesis was proposed in which the obtained effects are accounted for in terms of convergence differences. The hypothesis states that binocular stimuli are related to the rest convergence of the eyes which is assumed to be stable. Two experiments were conducted in which distance estimations were made to single binocular dots, viewed through a polarization stereoscope. The experimental results support the proposed hypothesis.

The Geometry of Visibles

The thesis I wish to defend is this: the geometry which precisely and naturally fits the actual configurations of the visual field is a non-Euclidean, two-dimensional, elliptical geometry. In substance, this thesis was advanced by Thomas Reid in 1764, although, since this preceded non-Euclidean geometries by 65 to 90 years, no mention was made of them.2 The thesis conflicts now, as in Reid's time, with the view held by most psychologists and philosophers that the actual two-dimensional configurations of the visual field are Euclidean. It also appears to conflict with the recent theory of Luneburg [8] that the geometry of binocular space perception is, though non-Euclidean, hyperbolic rather than elliptical; but we shall see that this latter difference is merely apparent-Luneburg's theory deals with a different problem. To grasp the import of our thesis it is important to distinguish at the outset:

Sensory direction inhomogeneous binocular visual space

Shifts in subjective sensory direction (local signature) occur in all sense modalities, and the dynamics of these shifts are quite similar. This is also true for visual direction in homogeneous binocular space. Binocular vision may, therefore, be interpreted in accordance with principles of intersensory processing.

Theoretical analysis of binocular space perception

A formalism is developed for dealing with the mapping of physical configurations onto the retinae and into the visual cortex. In this context we discuss the Luneburg-Blank theory of binocular vision, its range of validity and some of its basic difficulties. It is shown that Vieth-Müller circles and Hillebrand hyperbolae can be characterized by the invariance under shifting fixation and bihemispherical symmetry of their neural counterparts. This property justifies the use of a coordinate system based on these two families of curves and a corresponding polar coordinate system in visual space. We derive the complete set of curves in the horizontal plane which project onto the same neural units under change of fixation. Their significance lies in shewing that there exists only a restricted set of configurations which can have invariant neural representations and that the Vieth-Muller circles and Hillebrand hyperbolae are a subclass of this set. The implications for pattern recognition are treated in a separate paper.

Binocular space and its representation by cellular assemblies in the brain

It is shown that Luneburg's essentially psychophysical theory suffers from the limitations of a metric, whose coefficients are functions of the coordinates only. This ignores the contributions to space perception of the total stimulus configuration. Blank's reference of the co-ordinates to the horizon does not suffice to remove this difficulty. To say that the iseikonic transformation takes one stimulus configuration into a perceptually equivalent one is tantamount to the proposition that the only cues for binocular space perception are the relative stimulus points on the two retinae. It might then have been more likely that there should exist fixed “wired circuits” in the brain, which act as pattern recognizers. This, apparently, does not agree with the experimental evidence. Instead we propose that only a co-ordinate system could be “wired” into the brain and that the rest of pattern recognition is based on computation which involves other than merely the usual stereoscopic cues.