# Symmetry Is the *sine qua non* of Shape

## Abstract

Three-dimensional (3D) shape has been studied for centuries despite the absence of a commonly accepted definition of this property. The absence of a useful definition has been a major obstacle in making progress towards understanding the mechanisms that are responsible for the perception of shape. Today, in the absence of the needed new definition, there is no consensus about whether shapes are, or can be, perceived veridically. This chapter reviews the main definitions of shape in use before our new definition was formulated, calling attention to their shortcomings. It then describes our new definition, which is based on the assumption that 3D shape is based on 3D geometrical self-similarities (3D symmetries) of an object, rather than on similarities of an object with respect to other objects. We explain the new definition by discussing the invariants of three types of symmetry groups in 3D and then derive the invariants of the *perspective* projection from a 3D space to a 2D image. In our definition, the invariants of 3D symmetries serve as the basis for describing the 3D shapes, and the invariants of perspective projections are essential for recovering 3D shapes from one or more 2D images. We conclude by discussing several implications of this new definition which makes it clear: (i) that the veridicality of shape perception is no longer only an empirical concept—the new definition provides a principled theory of when and how the veridicality of shape can be achieved; (ii) how shape constancy applies to non-rigid objects; (iii) that there are *informative, but objective, shape priors that do not have to be learned from objects or updated on the basis of experience:* these priors are the object’s symmetries and (iv) that what had loomed as a controversy between view-invariant and view-dependent shape perception has been resolved.

## Keywords

Rigid Motion Translational Symmetry Projective Transformation Geodesic Line Shape Constancy## References

- 1.Ashby FG, Perrin NA (1988) Toward a unified theory of similarity and recognition. Psychol Rev 95(1):124–150 CrossRefGoogle Scholar
- 2.Biederman I (1987) Recognition-by-components: a theory of human image understanding. Psychol Rev 94(2):115–147 CrossRefGoogle Scholar
- 3.Biederman I, Gerhardstein PC (1993) Recognizing depth-rotated objects: evidence and conditions from three-dimensional viewpoint invariance. J Exp Psychol Hum Percept Perform 19(6):1162–1182 CrossRefGoogle Scholar
- 4.Binford TO (1971) Visual perception by computer. In: IEEE conference on systems and control, Miami Google Scholar
- 5.Chan MW, Stevenson AK, Li Y, Pizlo Z (2006) Binocular shape constancy from novel views: the role of a priori constraints. Percept Psychophys 68(7):1124–1139 CrossRefGoogle Scholar
- 6.Egan E, Todd J, Phillips F (2012) The role of symmetry in 3D shape discrimination across changes in viewpoint. J Vis 12(9):1048 CrossRefGoogle Scholar
- 7.Feldman J, Singh M (2005) Information along curves and closed contours. Psychol Rev 112(1):243–252 CrossRefGoogle Scholar
- 8.Feldman J, Singh M (2006) Bayesian estimation of the shape skeleton. Proc Natl Acad Sci 103(47):18014–18019 MathSciNetzbMATHCrossRefGoogle Scholar
- 9.Hartley R, Zisserman A (2003) Multiple view geometry in computer vision. Cambridge University Press, Cambridge Google Scholar
- 10.Knill DC, Richards W (1996) Perception as Bayesian inference. Cambridge University Press, New York zbMATHCrossRefGoogle Scholar
- 11.Li M, Vitanyi P (1997) An introduction to Kolmogorov complexity and its applications. Springer, New York zbMATHCrossRefGoogle Scholar
- 12.Li Y, Pizlo Z, Steinman RM (2009) A computational model that recovers the 3D shape of an object from a single 2D retinal representation. Vis Res 49(9):979–991 CrossRefGoogle Scholar
- 13.Li Y, Sawada T, Shi Y, Kwon T, Pizlo Z (2011) A Bayesian model of binocular perception of 3D mirror symmetrical polyhedra. J Vis 11(4):1–20 CrossRefGoogle Scholar
- 14.Li Y, Sawada T, Latecki LM, Steinman RM, Pizlo Z (2012) Visual recovery of the shapes and sizes of objects, as well as distances among them, in a natural 3D scene. J Math Psychol 56(4):217–231 MathSciNetCrossRefGoogle Scholar
- 15.Pizlo Z (2001) Perception viewed as an inverse problem. Vis Res 41(24):3145–3161 CrossRefGoogle Scholar
- 16.Pizlo Z (2008) 3D shape: its unique place in visual perception. MIT Press, Cambridge Google Scholar
- 17.Pizlo Z, Rosenfeld A, Weiss I (1997) The geometry of visual space: about the incompatibility between science and mathematics. Comput Vis Image Underst 65:425–433 CrossRefGoogle Scholar
- 18.Pizlo Z, Rosenfeld A, Weiss I (1997) Visual space: mathematics, engineering, and science. Comput Vis Image Underst 65:450–454 CrossRefGoogle Scholar
- 19.Pizlo Z, Sawada T, Li Y, Kropatsch W, Steinman RM (2010) New approach to the perception of 3D shape based on veridicality, complexity, symmetry and volume. Vis Res 50(1):1–11 CrossRefGoogle Scholar
- 20.Poggio T, Torre V, Koch C (1985) Computational vision and regularization theory. Nature 317(6035):314–319 CrossRefGoogle Scholar
- 21.Sawada T (2010) Visual detection of symmetry of 3D shapes. J Vis 10(6):4 (22pp) MathSciNetCrossRefGoogle Scholar
- 22.Shepard RN, Cooper LA (1982) Mental images and their transformations. MIT Press, Cambridge Google Scholar
- 23.Shi Y (2012) Recovering a 3D shape of a generalized cone from a single 2D image. Master’s thesis, Department of Psychological Sciences, Purdue University, Indiana Google Scholar
- 24.Thompson DW (1942) On growth and form. Cambridge University Press, Cambridge zbMATHGoogle Scholar