Geometrical properties of a generalized cone and its 2D image

Shapes of many natural and man-made objects in real scenes can be characterized by symmetry or they can be decomposed into parts that can be characterized by symmetry. The sensitivity of the human visual system to symmetry is critical for detecting and interacting with these objects.

(Mach, 1906,1959, see also Julesz, 1971, Kohler et al., 2016, Makin et al., 2013, Makin et al., 2012) pointed out that the visual system is sensitive to the following three types of 2D symmetry: mirror (or bilateral or reflectional), rotational (or cyclic or radial), and translational (or repetition). The same 3 types of symmetry can be found in 3D objects. However, the 2D retinal or camera images of 3D symmetrical objects are not themselves symmetrical except when the viewing direction is degenerate (Sawada et al., 2011, Sawada et al., 2014, Sawada and Zaidi, 2018, see also Sawada, 2020). It follows, that the sensitivity of the visual system to different types of 2D symmetry cannot be generalized to sensitivity to the various types of 3D symmetry.

Kanade’s (1981) paper was the first attempt to analyze 2D images of patterns that are symmetrical in a 3D scene, but not in the camera image. The camera image is not symmetrical, but symmetry is not completely lost in the projection from a 3D scene to the 2D image. It is now conventional to call such images skew-symmetrical. Research conducted during the last 30 years has shown that the human visual system has mechanisms to handle skew-symmetrical images in ways that are computationally optimal, or nearly so.

It has been shown that the visual system is sensitive to 3D mirror-symmetry (Sawada, 2010, Sawada and Farshchi, 2022) and that the 3D mirror-symmetry of an object is critical for perceiving the shape of the object and for recognizing the object from a 2D image (Jayadevan et al., 2018, Li and Pizlo, 2011, Li et al., 2009, Li et al., 2011, see Pizlo, 2019, Pizlo and de Barros, 2021, Pizlo et al., 2014 for a further discussion about a role of symmetry in the visual perception). The 3D mirror-symmetry of an object introduces invariant features in its 2D image that can be used by the visual system to detect the 3D symmetry in a 2D skew-symmetrical image (Li et al., 2013, Sawada et al., 2015). 3D symmetry also works as an a priori constraint that is used to recover the 3D shape of the object from the 2D image (Gordon, 1989, Hong et al., 2004, Li et al., 2009, Rothwell, 1995, Vetter and Poggio, 1994). Note that there is also a similar geometrical property that works with 3D rotational-symmetry (Sawada et al., 2014, Sawada and Zaidi, 2018, see also Sawada, 2020) but the sensitivity of the visual system to 3D rotational-symmetry is, at best, very low (Sawada and Farshchi, 2022, Sawada and Zaidi, 2018). It is possible that the human ability to detect 3D rotational symmetry is related to the fact that such symmetry can often be described (at least approximately) as 3D mirror-symmetry. This is true with objects such as flowers. The question as to whether the human visual system has a separate mechanism for detecting and using 3D rotational-symmetry as an invariant and a priori constraint, is still open. The present paper explores the mathematical and computational characteristics of the 3rd type of 3D symmetry, the translational-symmetry. More specifically, it is shown how a 3D mirror-symmetry can be used to recover 3D translationally-symmetrical objects. This connection between mirror-symmetry and translational-symmetry is important because we already know that the human visual system uses a mirror-symmetry constraint, and it does so very well. If translationally-symmetrical objects can be recovered by using a mirror-symmetry constraint, it is likely that translationally-symmetrical shapes are also perceived by human observers veridically. Our preliminary psychophysical results suggest that this is indeed the case (Pizlo et al., 2014, Shi, 2012).

An object is with translational-symmetry when the object can be decomposed into components that are identical to one another and when the components align along a straight axis with equal steps. This type of translational-symmetry is discrete. Note that there should be an infinite number of components along an infinitely-long axis for the object to be exactly translationally-symmetrical. However, it can be often said that the object is with translational-symmetry even if the number of components is finite and the axis has a finite length. A more interesting version of translational-symmetry is continuous symmetry. Such symmetry exists when the object is a right-cylinder, which is produced by sweeping a planar closed curve along a straight axis that is normal to the plane of the curve. Note that in principle the cylinder should be infinitely long to be exactly translationally-symmetrical. However, it has been conventional to say that an object is translationally-symmetrical even if the cylinder is finitely long. The planar curve that was swept in the process of producing the cylinder shows at both ends of the cylinder as its end-sections. This cylinder is also 3D mirror-symmetrical about a plane that bisects the cylinder, is normal to the axis, and is parallel to the end-sections.

The generalized cone proposed by Binford (1971) can be regarded as a generalization of the continuous type of translational-symmetry. This generalized cone is produced by sweeping a planar cross-section along an axis so that the cross-section is always normal to a tangent of the axis at an intersection of the axis with the cross-section. The axis can be smoothly curved. The size of the cross-section of the generalized cone can change smoothly as a function of the position of the cross section relative to the axis. The orientation of the cross-section around the tangent of the axis at the intersection can also change smoothly. The resulting cone is composed of the two planar end-sections, the meridians connecting pairs of corresponding vertices of the cross-sections, and the surfaces connecting the end-sections and the meridians. The relationship between the cross-sections including the end-sections of the cone can be regarded as generalized-translational-symmetry.

Many complex and articulated 3D objects can be represented as composed of parts, where each part is a generalized cone (Biederman, 1987, Binford, 1971, Marr, 1977, Pentland, 1986). It has been shown that observers can reliably recognize objects composed of generalized cones from contour drawings of the objects (Biederman, 1987, Biederman, 2001, Biederman and Gerhardstein, 1993, Stankiewicz, 2002). It was also shown that a 3D shape of a generalized cone can be veridically perceived from an image of the cone when the two end-sections of the cone are visible (Pizlo et al., 2014, Shi, 2012).

A contour drawing of a generalized cone is composed of contours representing its end-sections and meridians.1 If one wants to discuss the perception of the shape of a 3D generalized cone from a 2D contour-drawing of the cone, it is important to understand the geometrical properties of these contours in the drawing. In this study, we analyzed the geometrical properties of the contours that represented the end-sections of the generalized cone with the meridians around the end-sections in a drawing of the cone. Our analysis showed that there are model-based invariants of a generalized cone under a projection from the 3D scene to a 2D drawing of it. The end-sections and the axis of the generalized cone can be recovered from the 2D drawing when the axis is planar. We found that the 3D translational symmetry of generalized cones can be analyzed using tools designed for 3D mirror-symmetry. Based on the results of our analysis, we discussed the role the end-sections play in the perception of a generalized cone from a contour-drawing.

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