Skip to main content
SpringerLink
Log in

Variations in the Anisotropy and Affine Structure of Visual Space: A Geometry of Visibles with a Third Dimension

  • Published:
Topoi Aims and scope Submit manuscript

Abstract

A meta-analysis and an experiment show that the degree of compression of the in-depth dimension of visual space relative to the frontal dimension increases quickly as a function of the distance between the stimulus and the observer at first, but the rate of change slows beyond 7 m from the observer, reaching an apparent asymptote of about 50 %. In addition, the compression of visual space is greater for monocular and reduced cue conditions. The pattern of compression of the in-depth dimension as a function of distance is similar to the ratio of in-depth to frontal visual angles of stimuli, but is not as extreme as this ratio would suggest, implying that observers are incapable of fully ignoring size information provided by cues to depth. Size and distance judgments may be described by an Affine transformation of physical space; however, the compression parameter in this model changes as a function of distance from the observer and other experimental conditions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Baird JC, Biersdorf WR (1967) Quantitative functions for size and distance judgments. Percept Psychophys 2:161–166

    Article  Google Scholar 

  • Baird JC, Wagner M (1991) Transformation theory of size judgment. J Exp Psychol Hum Percept Perform 17:852–864

    Article  Google Scholar 

  • Bian Z, Andersen GJ (2011) Environmental surfaces and the compression of perceived visual space. J Vis 11:1–14

    Article  Google Scholar 

  • Bingham GP, Lind M (2008) Large continuous perspective transformations are necessary and sufficient for accurate perception of metric shape. Percept Psychophys 70:524–540

    Article  Google Scholar 

  • Bingham GP, Crowell JA, Todd JT (2004) Distortions of distance and shape are not produced by a single continuous transformation of reach space. Percept Psychophys 66:152–169

    Article  Google Scholar 

  • Blank AA (1953) The Luneberg theory of binocular visual space. J Opt Soc Am 43:717–727

    Article  Google Scholar 

  • Blank AA (1957) The geometry of vision. Br J Physiol Opt 14:1–30

    Google Scholar 

  • Blank AA (1958) Axiomatics of binocular vision. (The foundation of metric geometry in relation to space perception). J Opt Soc Am 48:911–925

    Article  Google Scholar 

  • Blank AA (1959) The Luneburg theory of binocular space perception. In: Koch S (ed) Psychology: a study of a science, vol 1. Mcgraw-Hill, New York, pp 395–426

    Google Scholar 

  • Brunswik E (1929) Zur Entwicklung der Albedowahrnehmung. Zeitschrift für Psychologie 109:40–115

    Google Scholar 

  • Brunswik E (1933) Die Zugänglichkeit von Gegenständen für die Wahrnehmung und deren quantitative Bestimmung. Archiv für die Gesamte Psychologie 88:357–418

    Google Scholar 

  • Brunswik E (1956) Perception and the representative design of psychological experiments, 2nd edn. University of California Press, Berkeley

    Google Scholar 

  • Cuijpers RH, Kappers AML, Koenderink JJ (2000) Investigation of visual space using an exocentric pointing task. Percept Psychophys 62:1556–1571

    Article  Google Scholar 

  • Cuijpers RH, Kappers AML, Koenderink JJ (2001) On the role of external reference frames on visual judgments of parallelity. Acta Psychol 108:283–302

    Article  Google Scholar 

  • Cutting JE (2003) Reconceiving perceptual space. In: Hecht H, Schwartz R, Atherton M (eds) Looking into pictures: an interdisciplinary approach to pictorial space. MIT Press, Cambridge, MA, pp 215–238

    Google Scholar 

  • Cutting JE, Vishton PM (1995) Perceiving layout and knowing distances: The integration, relative potency, and contextual use of different information about depth. In: Epstein W, Rogers S (eds) Perception of space and motion. Academic Press, San Diego, CA, pp 69–117

    Chapter  Google Scholar 

  • Doumen MJA, Kappers AML, Koenderink JJ (2005) Visual space under free viewing conditions. Percept Psychophys 67:1177–1189

    Article  Google Scholar 

  • Drösler J (1979) Foundations of multi-dimensional metric scaling in Cayley-Klein geometries. Br J Math Stat Psychol 19:185–211

    Article  Google Scholar 

  • Drösler J (1988) The psychophysical function of binocular space perception. J Math Psychol 32:285–297

    Article  Google Scholar 

  • Drösler J (1995) The invariances of Weber’s and other laws as determinants of psychophysical structures. In: Luce RD, D’Zmura M, Hoffman D, Iverson GJ, Romney AK (eds) Geometric representations of perceptual phenomena. Lawrence Erlbaum Associates, Mahwah, NJ, pp 69–93

    Google Scholar 

  • Flash T, Handzel AA (2007) Affine differential geometry analysis of human arm movements. Biol Cybern 96:577–601

    Article  Google Scholar 

  • Flückiger M (1991) La perception d’objets lointains. In: Flückiger M, Klaue K (eds) La perception de L’Environnement. Delachaux and Niestlé, Lausanne, pp 211–238

    Google Scholar 

  • Foley JM, Ribeiro-Filho NP, Da Silva JA (2004) Visual perception of extent and the geometry of visual space. Vis Res 44:147–156

    Article  Google Scholar 

  • Fry GA (1950) Visual perception of space. Am J Optom 27:531–553

    Article  Google Scholar 

  • Gibson JJ (1950) The perception of the visual world. Houghton Mifflin Company, Boston

    Google Scholar 

  • Gibson JJ (1959) Perception as a function of stimulation. In: Koch S (ed) Psychology: a study of a science, vol 1. McGraw-Hill, New York, pp 456–501

    Google Scholar 

  • Gibson JJ (1966) The senses considered as perceptual systems. Houghton Mifflin Company, Boston

    Google Scholar 

  • Gilinsky AS (1951) Perceived size and distance in visual space. Psychol Rev 58:460–482

    Article  Google Scholar 

  • Gogel WC (1990) A theory of phenomenal geometry and its applications. Percept Psychophys 48:104–123

    Article  Google Scholar 

  • Gogel WC (1993) The analysis of perceived space. In: Masin SC (ed) Foundations of perceptual theory. Elsevier Science Publishers, New York, pp 113–182

    Chapter  Google Scholar 

  • Gogel WC (1998) An analysis of perceptions from changes in optical size. Percept Psychophys 60:805–820

    Article  Google Scholar 

  • Granrud CE (2009) Development of size constancy in children: a test of the metacognitive theory. Atten Percept Psychophys 71:644–654

    Article  Google Scholar 

  • Granrud CE (2012) Judging the size of a distant object: Strategy use by children and adults. In: Hatfield G, Allred S (eds) Visual experience: Sensation, cognition, and constancy, Chap. 1. Oxford University Press, Oxford, pp 13–34

  • Haber RN, Haber LR, Levin CA, Hollyfield R (1993) Properties of spatial representations: data from sighted and blind subjects. Percept Psychophys 54:1–13

    Article  Google Scholar 

  • Hatfield G (2003) Representation and constraints: the inverse problem and the structure of visual space. Acta Psychol 114:355–378

    Article  Google Scholar 

  • Hatfield G (2009) Perception & cognition: essays in the philosophy of psychology. Oxford University Press, Oxford

    Google Scholar 

  • Hatfield G (2012) Phenomenal and cognitive factors in spatial perception. In: Hatfield G, Allred S (eds) Visual experience: sensation, cognition, and constancy, Chap. 2. Oxford University Press, Oxford, pp 35–62

  • Hecht H, van Doorn A, Koenderink JJ (1999) Compression of visual space in natural scenes and in their photographic counterparts. Percept Psychophys 61:1269–1286

    Article  Google Scholar 

  • Heller J (1997) On the psychophysics of binocular space perception. J Math Psychol 41:29–43

    Article  Google Scholar 

  • Hiro O (1997) Distance perception in driving. Tohoku Psychologica Folia 55:92–100

    Google Scholar 

  • Hoffman WC (1966) The Lie algebra of visual perception. J Math Psychol 3:65–98

    Article  Google Scholar 

  • Hoffman WC, Dodwell PC (1985) Geometric psychology generates the visual Gestalt. Can J Psychol 39:491–528

    Article  Google Scholar 

  • Indow T (1967) Two interpretations of binocular visual space: hyperbolic and Euclidean. Annu Jpn Assoc Philos Sci 3:51–64

    Google Scholar 

  • Indow T (1974) On geometry of frameless binocular perceptual space. Psychologia 17:50–63

    Google Scholar 

  • Indow T (1990) On geometrical analysis of global structure of visual space. In: Geissler HG (ed) Psychological explorations of mental structures. Hogrefe & Huber, Toronto, pp 172–180

    Google Scholar 

  • Indow T (1995) Psychophysical scaling: scientific and practical applications. In: Luce RD, D’Zmura M, Hoffman D, Iverson GJ, Romney AK (eds) Geometric representations of perceptual phenomena. Lawrence Erlbaum Associates, Mahwah, NJ, pp 1–34

    Google Scholar 

  • Kant I (1781/1929) Critique of pure reason (trans: Kemp Smith N). Macmillan, London

  • Koenderink JJ, van Doorn AJ, Lappin JS (2000) Direct measurement of the curvature of visual space. Perception 29:69–79

    Article  Google Scholar 

  • Koffka K (1935) Principles of gestalt psychology. Harcourt Brace, New York

    Google Scholar 

  • Köhler W (1926) An aspect of gestalt psychology. In: Murchison C (ed) Psychologies of 1925. Clark University, Worcester, MA

    Google Scholar 

  • Kong Q, Zhang T, Ding B, Ge H (1995) Psychological measurements of railroad drivers. Chin Ment Health J 9:213–214

    Google Scholar 

  • Levin CA, Haber RN (1993) Visual angle as a determinant of perceived interobject distance. Percept Psychophys 54:250–259

    Article  Google Scholar 

  • Li Z, Durgin FH (2010) Perceived slant of binocularly viewed large scale surfaces: a common model form explicit and implicit measures. J Vis 10:1–16

    Google Scholar 

  • Li Z, Durgin FH (2012) A comparison of two theories of perceived distance on the ground plane: the angular expansion hypothesis and the intrinsic bias hypothesis. i-Perception 3:368

  • Li Z, Durgin FH (2013) Depth compression based on mis-scaling of binocular disparity may contribute to angular expansion in perceived optic slant. J Vis 13:1–18

    Google Scholar 

  • Li Z, Sun E, Cassandra JS, Spiegel A, Klein B, Durgin FH (2013) On the anisotropy of perceived ground extents and the interpretation of walked distance as a measure of perception. J Exp Psychol Hum Percept Perform 39:477–493

    Article  Google Scholar 

  • Loomis JM, Philbeck JW (1999) Is the anisotropy of perceived 3-D shape invariant across scale? Percept Psychophys 61:397–402

    Article  Google Scholar 

  • Loomis JM, Da Silva JA, Fujita N, Fukusima SS (1992) Visual space perception and visually directed action. J Exp Psychol Hum Percept Perform 18:906–921

    Article  Google Scholar 

  • Loomis JM, Philbeck JW, Zahorik P (2002) Dissociation between location and shape in visual space. J Exp Psychol Hum Percept Perform 28:1202–1212

    Article  Google Scholar 

  • Luneburg RK (1947) Mathematical analysis of binocular vision. Oxford University Press, New York

    Google Scholar 

  • Luneburg RK (1948) Metric methods in binocular visual perception. Courant anniversary volume. Interscience, New York, pp 215–240

  • Luneburg RK (1950) The metric of binocular visual space. J Opt Soc Am 40:637–642

    Article  Google Scholar 

  • Luria SM, Kiney JS, Weissman S (1967) Distance estimates with “filled” and “unfilled” space. Percept Mot Skills 24:1007–1010

    Article  Google Scholar 

  • Matsushima EH, de Oliveira AP, Ribeiro-Filho NP, Da Silva JA (2005) Visual angle as determinant factor for relative distance perception. Psicológica 26:97–104

    Google Scholar 

  • Moore CC (1907) Estimate of distance. Law Notes, p. 5-7

  • Norman JF, Lappin JS, Norman HF (2000) The perception of length on curved and flat surfaces. Percept Psychophys 62:1133–1145

    Article  Google Scholar 

  • Predebon J (1990) Relative distance judgments of familiar and unfamiliar objects viewed under representatively natural conditions. Percept Psychophys 47:342–348

    Article  Google Scholar 

  • Predebon J (1992) The influence of object familiarity on magnitude estimates of apparent size. Perception 21:77–90

    Article  Google Scholar 

  • Proffitt DR (2006) Embodied perception and the economy of action. Perspect Psychol Sci 1:110–122

    Article  Google Scholar 

  • Reid T (1764/1813) Inquiry into the human mind (D. Stewart, Ed., Vol. 1). Samuel Etheridge, Charlestown

  • Reinhardt R, Anthony H (1996) Remote operation: a selective review of research into visual depth perception. J Gen Psychol 123:237–248

    Article  Google Scholar 

  • Roscoe SN (1985) Bigness is in the eye of the beholder. Hum Factors 27:615–636

    Google Scholar 

  • Schoumans N, Koenderink JJ, Kappers AML (2002) Scale invariance in near space: pointing under influence of context. Acta Psychol 110:63–81

    Article  Google Scholar 

  • Shaffer DM, Krauchunas SM, Eddy M, McBeath MK (2004) How dogs navigate to catch Frisbees. Psychol Sci 15:437–441

    Article  Google Scholar 

  • Shaffer DM, Maynor AB, Roy WL (2008) The visual perception of lines on the road. Percept Psychophys 70:1571–1580

    Article  Google Scholar 

  • Sipes DE (1997) Hyperstereopsis as an attenuator for perceptual depth compression (Doctoral dissertation, The Johns Hopkins University, 1997). Dissertation Abstracts International, 58 (4-B), 2161

  • Smeets JBJ, Sousa R, Brenner E (2009) Illusions can warp visual space. Perception 38:1467–1480

    Article  Google Scholar 

  • Teghtsoonian M (1974) The doubtful phenomenon of over-constancy. In: Moskowitz HR, Scharf B, Stevens JC (eds) Sensation and measurement. Reidel, Dordrecht, pp 411–420

    Chapter  Google Scholar 

  • Thorndyke PW (1981) Distance estimation from cognitive maps. Cogn Psychol 13:526–549

    Article  Google Scholar 

  • Thouless RH (1931) Phenomenal regression to the “real” object: I. Br J Psychol 21:339–359

    Google Scholar 

  • Tittle JS, Todd JT, Perotti VJ, Norman JF (1995) Systematic distortion of perceived three-dimensional structure from motion and binocular stereopsis. J Exp Psychol Hum Percept Perform 21:663–678

    Article  Google Scholar 

  • Todd JT, Oomes AHJ, Koenderink JJ, Kappers AML (2001) On the affine structure of perceptual space. Psychol Sci 12:191–196

    Article  Google Scholar 

  • Toye RC (1986) The effect of viewing position on the perceived layout of space. Percept Psychophys 40:85–92

    Article  Google Scholar 

  • Wagner M (1982) The metric of visual space. Unpublished doctoral dissertation, Dartmouth College

  • Wagner M (1985) The metric of visual space. Percept Psychophys 38:483–495

    Article  Google Scholar 

  • Wagner M (2006) The geometries of visual space. Lawrence Erlbaum Associates, Mahwah, NJ

    Google Scholar 

  • Wagner M (2008) Comparing the psychophysical and geometric characteristics of spatial perception and cognitive maps. Cogn Stud 15:1–16

    Google Scholar 

  • Wagner M (2012) Sensory and cognitive explanations for a century of size constancy research. In: Hatfield G, Allred S (eds) Visual experience: sensation, cognition, and constancy, Chap. 3. Oxford University Press, Oxford, pp 63–86

  • Wagner M, Feldman E (1989) The metric properties of three dimensional visual space. In: Canevet G, Scharf B, Bonnel AM, Possamai CA (eds) International society for psychophysics annual, vol 5. International Society for Psychophysics, Cassis, pp 96–101

    Google Scholar 

  • Wagner M, Hatfield G, Cassese K, Makwinski AN (2013) Testing three models of the geometry of visual space. Paper presented at the meetings of the Association for Psychological Science, Washington, DC

  • Wu J, He ZJ, Ooi TL (2008) Perceived relative distance on the ground affected by the selection of depth information. Percept Psychophys 70:707–713

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wagner College, One Campus Road, Staten Island, NY, 08826, USA

    Mark Wagner & Anthony J. Gambino

Authors
  1. Mark Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anthony J. Gambino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mark Wagner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wagner, M., J. Gambino, A. Variations in the Anisotropy and Affine Structure of Visual Space: A Geometry of Visibles with a Third Dimension. Topoi 35, 583–598 (2016). https://doi.org/10.1007/s11245-015-9303-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11245-015-9303-x

Keywords

Search

Navigation