Aquilonius F

Ernst B

Gombrich E H

Herschel J F W

Küster F

Panofsky E

Pirenne M H

von Helmholtz H

White J

Crossref

Straightness and the Sphere of Vision

Tyler, C. W.

City Research Online

Tyler, C. W. (2009). Straightness and the sphere of vision. PERCEPTION, 38(10), doi: 10.1068/p3810ed City Research OnlineOriginal citation: Tyler, C. W. (2009). Straightness and the sphere of vision. PERCEPTION, 38(10), doi: 10.1068/p3810ed Permanent City Research Online URL: http://openaccess.city.ac.uk/11901/ Copyright & reuseCity University London has developed City Research Online so that its users may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact with the author(s) of this paper, please email the team at publications@city.ac.uk.Straightness and the sphere of visionThe nature of perceived visual space is an issue that has been as contentious withinits field as has the nature of physical space within the field of physics. In one of theearliest systemizations of visual space, Euclid (c. 300 BC) has been interpreted asconceptualizing visual projective space in terms of a sphere surrounding the viewer.He represents a tradition that `` considered our sphere of vision quite literally as a`sphere'öan assumption, incidentally, which more nearly agrees with physiological andpsychological reality than that which underlies Brunelleschi's rectilinear construction.''(Panofsky 1960).The same construct was subsequently affirmed by Leonardo daVinci in the late 1400sand by others (see Aguilonius 1613; Abraham Bosse 1665; John Herschel 1833, 1849;Hermann von Helmholtz 1910/1962; White 1957), in terms of the `natural perspective'of the free-viewing observer (as opposed to the `artificial perspective' of the projectiononto a flat picture plane).The contrast with artificial perspective is illustrated in figure 1 by Bosse's illustra-tion of the relation between the circular projection of the visual cone and the flat planeof the picture surface (inscribed with the phrase ``To prove that one can neither definenor paint as the eye sees''). As a consequence, there has been an incessant debate sinceat least the time of Leonardo about whether straight lines in the world are perceivedas straight or curved in a form of barrel distortion, and whether they should beEditorial essayPerception, 2009, volume 38, pages 1423 ^ 1427doi:10.1068/p3810edFigure 1. Diagram of the visual cone by Abraham Bosse (1665).depicted as straight in the picture plane of the artist's canvas. Both Panofsky (1924)and Gombrich (1972), for example, argued that straight lines in natural perspective areultimately curved, and Herschel (1849), Escher (1947ösee Ernst 1976), and White(1957) developed formal schemes for the definition of curved perspective, approximat-ing the projective geometry of the extreme wide-angle or `fish-eye' lens. Later authors(Pirenne 1970; Hansen 1973; Anstis 1998; Liu and Schor 1998; Oomes et al 2009;Rogers and Rogers 2009) have also discussed the issue of the apparent curvature ofphysically straight lines.A clear statement of the natural perspective concept may be found in Herschel's(1849) Outlines of Astronomy:`` In celestial perspective, every point to which the view is for the moment directed, is equallyentitled to be considered as the c`entre of the picture', every portion of the surface ofthe sphere being similarly related to the eye. Moreover, every straight line (supposed to beindefinitely prolonged) is projected into a semicircle of the sphere, that, namely, in whicha plane passing through the line and the eye cuts its surface. And every system of parallelstraight lines, in whatever direction, is projected into a system of semicircles of thesphere, meeting in two common apexes, or vanishing points, diametrically opposite to eachother'' (citation acquired from online version).The implication of such perspective schemes is that lines that are straight in theworld should not necessarily be perceived as straight by the viewer. The issue had beentaken up experimentally by Helmholtz (1910/1962), who observed that the lines in alarge-visual-angle grid seemed to curve inwards in the far periphery. This result is mosteasily checked by looking down at a tile floor in a large building (such as an airport).In the context of the eye moving around to explore the environment, a natural defini-tion of straightness is the path followed by the eye as it moves between two points.Such motions are determined by the space-orientation of the axis around which theeye rotates, which is governed by Listing's law.The question raised by Helmholtz was: what line geometry provides the ocular motionpath for lines in the visual periphery, ie lines that do not pass through the fovea?Geometrically, the answer is direction circles (Klu« ster 1876; Helmholtz 1910/1962;Simonsz and Den Tonkelaar 1990), which have the property of being circles inscribedon the visual sphere that all converge to pass through the `occipital point', or the pointdirectly behind the viewer opposite the fixation point (figure 2). Direction circles arethus not great circles unless they pass through the fovea (along any meridian). Relativeto these great circles, the other direction lines may be considered to be curved, butthey will all remain self-congruent as the eye is rotated (as long as the rotation adheresto the minimum-energy principle of Listing's law). The direction circles thus corre-spond to a visual version of the principle of conservation of energy, first enunciated byHelmholtz (1847). They correspond to the spatial invariants, or null distortion field, for eyetransformation under minimum-energy conditions.As such, the direction circles form a kind of invariance `horopter' for a givendirection of eye-movement transformation, which in physics terminology would betermed the invariance domains of the minimum-energy rotational group. There is thusconsiderable interest in determining whether the visual perception of straightnessconforms to this combination of minimum-energy principles. The reason that thespherical issue enters the question of natural perspective is that the eye cannot trans-late across a surfaceöthe eye can only rotate about its center. A person can translatethrough the world (although only with some difficulty, unless he/she is on a movingvehicle). However, natural perspective is more typically associated with the stationaryobserver with a moving eye, which is capable only of rotation. It is this distinction thatbrings the direction lines into play as the candidate invariants for natural perspective.1424 Editorial essayIndeed, Panofsky (1924) mentions that the curved retina is a key reason for expectingstraight lines to be curved in natural perspective. However, the preceding developmenthas said nothing about the shape of the retina, and is framed purely in terms of theoperational invariants of the optic array. On this view, it is not the curvature ofthe retina that is relevant, but the sphericity of the eye that determines the form of theinvariants by constraining its motion to rotation around a point and adherence toListing's law controlling the rotation axis for trajectories not passing through the primaryposition. In one sense, the two properties both derive from the curvature of the eyeball,but differ in the way its curvature influences the geometry of perspective.This conflict between domains of operation may account for some of the discrep-ancies among the results of recent empirical studies of this issue. As reviewed by Rogersand Rogers (2009), for example, a wide range of results on the geometry of the perceivedstraightness of lines under 2-D and 3-D viewing, depend on the specifics of the viewingconditions. The above considerations suggest four factors that might interact to affecteach other:(i) the degree to which the tasks allow eye movements (enhancing the direction-lineinvariant) or require fixation;(ii) the relative strength of binocular versus monocular cues to the flatness or curvatureof the surrounding field;(iii) the relative strength of the monocular and binocular cues to 3-D structure defining theedge to be judged;(iv) the degree of realism and appeal to Bayesian priors for known straight edges ofobjects (in the translational sense).Interestingly enough, both the study by Helmholtz and the rigorous replication byRogers and Rogers (2009) find a highly accurate adherence to the minimum-energyprediction for the perceived visual straightness of peripheral lines under conditions ofdirectioncircleoccipitalpointyxzFigure 2. Ocular rotation circles, known as d`irection circles' and their projection to a frontoparallelplane. From Rogers and Rogers (2009).Editorial essay 1425precise fixation, although this result is immediately violated as soon as the eye movesto fixate a peripheral line. This apparently paradoxical violation can be seen to makesense as soon as it is realized that the minimum-energy principle applies only tolines parallel to the direction of the eye movement. The eye movement required to fixatethe peripheral line cannot be parallel to that line, and therefore does not bring theminimum-energy principle into play with respect to that line. Similarly, the checkerboardof figure 2 does not form a minimum-energy field because it has roughly orthogonalsets of features. Each set alone would form a minimum-energy field, but the two setscombined together violate the invariance property of the field for eye movements ofany kind.In conclusion, I have argued that straightness is most satisfactorily defined interms of the minimum energy principle of eye rotation, both in a formal sense andas an operational definition available to an organism. For the physicist, translation isperformed by hand or machine. Since the eye is not amenable to the operationof translation, its closest approximation is rotation about its center in the direction ofthe local line orientation. For such motions, the lines of eye movements away from theprimary position form the direction circles identified by Helmholtz, which his and laterstudies appear to validate as the operative principle determining perceived straightnessin human observers. These studies therefore appear to put the concept of perceptualstraightness on a firm theoretical footing based on the kinetics of eye movements, andone that was anticipated by a wide variety of thinkers on the issue of curved perspectiveover the past millennia.Christopher W TylerSmith-Kettlewell Eye Research Institute, 2318 Fillmore Street, San Francisco, CA 94115, USA;e-mail: cwt@ski.orgReferencesAnstis S M, 1998 ``Picturing peripheral acuity'' Perception 27 817 ^ 825Aquilonius F, 1613 Opticorum Libri Sex. Philosophis iuxta ac Mathematicis utiles (Antwerp:Moreti)Bosse A, 1665 Traite¨ des pratiques ge¨ome¨trales et perspectives: enseigne¨es dans l'Acade¨mie royale de lapeinture et sculpture (Paris)Ernst B, 1976 The Magic Mirror of M. C. Escher (New York: Random House)Gombrich E H, 1972 `` The `what' and the `how': perspective representation and the phenomenalworld'', in Logic and Art: Essays in Honor of Nelson Goodman Eds R Rudner, I Scheffler(Indianapolis, IN: Bobbs-Merrill) pp 129 ^ 149Hansen R, 1973 `` This curving world'' Journal of Aesthetics and Art Criticism 32 147 ^ 161Helmholtz H von, 1847Ûber die Erhaltung der Kraft. Eine physikalische Abhandlung (Berlin: G Reimer)Helmholtz H von, 1910/1962 Treatise on Physiological Optics (New York: Dover, 1962. Englishtranslation by J P C Southall from the 3rd German edition of Handbuch der physiologischenOptik Eds A Gullstrand, J von Kries (Hamburg: Voss, 1910; first edition published in 1867,Leipzig: Voss)Herschel J F W, 1833 A Treatise on Astronomy [4th edition, Ed. S C Walker (Philadelphia, PA:Lea & Blanchard, 1869)]Herschel J F W, 1849 Outlines of Astronomy (London: Spottiswoode)Ku« ster F, 1876 `` Die Direktionskreise des Blickfeldes''Archiv fu« r Ophthalmologie 22 149 ^ 210Liu L, Schor C M, 1998 `` Functional division of the retina and binocular correspondence'' Journalof the Optical Society of America A 15 1740 ^ 1755Oomes A H J, Koenderink J J, van Doorn A J, de Ridder H, 2009 `` What are the uncurved linesin our visual field? A fresh look at Helmholtz's checkerboard'' Perception 38 1284 ^ 1294Panofsky E, 1924 `` Die Perspektive als symbolische Form'', in Vortra« ge der Bibliothek Warburg,1924 ^ 25 (Leipzig/Berlin, 1927) pp 258 ^ 330Panofsky E, 1960 Renaissance and Renascences in Western Art (Stockholm: Almquist & Wiksell)Pirenne M H, 1970 Optics, Painting and Photography (London: Cambridge University Press)Rogers B J, Rogers C, `` Visual globes, celestial spheres, and the perception of straight and parallellines'' Perception 38 1295 ^ 13121426 Editorial essaySimonsz H J, Den Tonkelaar I, 1990 `` 19th Century mechanical models of eye movements,Donders' law, Listing's law and Helmholtz' direction circles'' Documenta Ophthalmologica 7495 ^ 112White J, 1957 The Birth and Rebirth of Pictorial Space (London: Faber and Faber)Editorial essay 1427

Straightness and the sphere of vision

Straightness and the sphere of vision

Abstract

Similar works

Full text

Available Versions

Crossref

City Research Online