Abstract
The encounter of perception and action happens at the intermediate representations of space-time. In many of the computational models employed in the past, it has been assumed that a metric representation of physical space can be derived by visual means. Psychophysical experiments, as well as computational considerations, can convince us that the perception of space and shape has a much more complicated nature, and that only a distorted version of actual, physical space can be computed. This paper develops a computational geometric model that explains why such distortion might take place. The basic idea is that, both in stereo and motion, we perceive the world from multiple views. Given the rigid transformation between the views and the properties of the image correspondence, the depth of the scene can be obtained. Even a slight error in the rigid transformation parameters causes distortion of the computed depth of the scene. The unified framework introduced here describes this distortion in computational terms. We characterize the space of distortions by its level sets, that is, we characterize the systematic distortion via a family of iso-distortion surfaces which describes the locus over which depths are distorted by some multiplicative factor. Clearly, functions of the distorted space exhibiting some sort of invariance, produce desirable representations for biological and artificial systems [13]. Given that humans' estimation of egomotion or estimation of the extrinsic parameters of the stereo apparatus is likely to be imprecise, the framework is used to explain a number of psychophysical experiments on the perception of depth from motion or stereo.
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J. Y. Aloimonos and D. Shulman. Learning early-vision computations. Journal of the Optical Society of America A, 6:908–919, 1989.
A. Ames, K. N. Ogle, and G. H. Glidden. Corresponding retinal points, the horopter and the size and shape of ocular images. Journal of the Optical Society of America A, 22:538–631, 1932.
G. Baratoff. Qualitative Space Representations Extracted from Stereo. PhD thesis, Department of Computer Science, University of Maryland, 1997.
L. Cheong, C. Fermüller, and Y. Aloimonos. Interaction between 3D shape and motion: Theory and applications. Technical Report CAR-TR-773, Center for Automation Research, University of Maryland, June 1996.
K. Daniilidis. On the Error Sensitivity in the Recovery of Object Descriptions. PhD thesis, Department of Informatics, University of Karlsruhe, Germany, 1992. In German.
K. Daniilidis and H. H. Nagel. Analytical results on error sensitivity of motion estimation from two views. Image and Vision Computing, 8:297–303, 1990.
K. Daniilidis and M. E. Spetsakis. Understanding noise sensitivity in structure from motion. In Y. Aloimonos, editor, Visual Navigation: From Biological Systems to Unmanned Ground Vehicles, chapter 4. Lawrence Erlbaum Associates, Hillsdale, NJ, 1997.
O. D. Faugeras. Three-Dimensional Computer Vision. MIT Press, Cambridge, MA, 1992.
C. Fermüller and Y. Aloimonos. Direct perception of three-dimensional motion from patterns of visual motion. Science, 270:1973–1976, 1995.
C. Fermüller and Y. Aloimonos. Algorithm independent stabiity analysis of structure from motion. Technical Report CAR-TR-840, Center for Automation Research, University of Maryland, 1996.
C. Fermüller and Y. Aloimonos. Ordinal representations of visual space. In Proc. ARPA Image Understanding Workshop, pages 897–903, February 1996.
C. Fermüller and Y. Aloimonos. Towards a theory of direct perception. In Proc. ARPA Image Understanding Workshop, pages 1287–1295, February 1996.
C. Fermüller and Y. Aloimonos. Ambiguity in structure from motion: Sphere versus plane. International Journal of Computer Vision, 1997. In press.
J. M. Foley. Effects of voluntary eye movement and convergence on the binocular appreciation of depth. Perception and Psychophysics, 11:423–427, 1967.
J. M. Foley. Binocular distance perception. Psychological Review, 87:411–434, 1980.
J. J. Gibson. The Perception of the Visual World. Houghton Mifflin, Boston, 1950.
W. C. Gogel. The common occurrence of errors of perceived distance. Perception & Psychophysics, 25(1):2–11, 1979.
W. C. Hoffman. The Lie algebra of visual perception. Journal of Mathematical Psychology, 3:65–98, 1966.
B. K. P. Horn. Robot Vision. McGraw Hill, New York, 1986.
E. B. Johnston. Systematic distortions of shape from stereopsis. Vision Research, 31:1351–1360, 1991.
R. Julesz. Foundations of Cyclopean Perception. University of Chicago Press, Chicago, IL, 1971.
J. J. Koenderink and A. J. van Doorn. Two-plus-one-dimensional differential geometry. Pattern Recognition Letters, 15:439–443, 1994.
J. J. Koenderink and A. J. van Doorn. Relief: Pictorial and otherwise. Image and Vision Computing, 13:321–334, 1995.
S. Kosslyn. Image and Brain. MIT Press, Cambridge, MA, 1993.
J. M. Loomis, J. A. D. Silva, N. Fujita, and S. S. Fukusima. Visual space perception and visually directed action. Journal of Experimental Psychology, 18(4):906–921, 1992.
R. K. Luneburg. Mathematical Analysis of Binocular Vision. Princeton University Press, Princeton, NJ, 1947.
H. A. Mallot, S. Gillner, and P. A. Arndt. Is correspondence search in human stereo vision a coarse-to-fine process? Biological Cybernetics, 74:95–106, 1996.
D. Marr. Vision. W.H. Freeman, San Francisco, CA, 1982.
S. J. Maybank. Theory of Reconstruction from Image Motion. Springer, Berlin, 1993.
J. E. W. Mayhew and H. C. Longuet-Higgins. A computational model of binocular depth perception. Nature, 297:376–378, 1982.
K. N. Ogle. Researches in Binocular Vision. Hafner, New York, 1964.
J. G. Semple and L. Roth. Inroduction to Algebraic Geometry. Oxford University Press, Oxford, United Kingdom, 1949.
J. S. Tittle, J. T. Todd, V. J. Perotti, and J. F. Norman. Systematic distortion of perceived three-dimensional structure from motion and binocular stereopsis. Journal of Experimental Psychology: Human Perception and Performance, 21:663–678, 1995.
J. T. Todd and P. Bressan. The perception of three-dimensional affine structure from minimal apparent motion sequences. Perception and Psychophysics, 48:419–430, 1990.
R. Y. Tsai and T. S. Huang. Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:13–27, 1984.
H. L. F. von Helmholtz. Treatise on Physiological Optics, volume 3. Dover, 1962. J. P. C. Southhall, trans. Originally published in 1910.
M. Wagner. The metric of visual space. Perception and Psychophysics, 38:483–495, 1985.
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Fermüller, C., Cheong, L., Aloimonos, Y. (1997). The geometry of visual space distortion. In: Sommer, G., Koenderink, J.J. (eds) Algebraic Frames for the Perception-Action Cycle. AFPAC 1997. Lecture Notes in Computer Science, vol 1315. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017872
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DOI: https://doi.org/10.1007/BFb0017872
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