Tadamasa Sawada

Detection of skewed symmetry

This study examined the ability of human observers to discriminate between symmetric and asymmetric planar figures from perspective and orthographic images. The first experiment showed that the discrimination is reliable in the case of polygons, but not dotted patterns. The second experiment showed that the discrimination is facilitated when the projected symmetry axis or projected symmetry lines are known to the subject. A control experiment showed that the discrimination is more reliable with orthographic, than with perspective images. Based on these results, we formulated a computational model of symmetry detection. The model measures the asymmetry of the presented polygon based on its single orthographic or perspective image. Performance of the model is similar to that of the subjects.

A Bayesian model of binocular perception of 3D mirror symmetrical polyhedra.

In our previous studies, we showed that monocular perception of 3D shapes is based on a priori constraints, such as 3D symmetry and 3D compactness. The present study addresses the nature of perceptual mechanisms underlying binocular perception of 3D shapes. First, we demonstrate that binocular performance is systematically better than monocular performance, and it is close to perfect in the case of three out of four subjects. Veridical shape perception cannot be explained by conventional binocular models, in which shape was derived from depth intervals. In our new model, we use ordinal depth of points in a 3D shape provided by stereoacuity and combine it with monocular shape constraints by means of Bayesian inference. The stereoacuity threshold used by the model was estimated for each subject. This model can account for binocular shape performance of all four subjects. It can also explain the fact that when viewing distance increases, the binocular percept gradually reduces to the monocular one, which implies that monocular percept of a 3D shape is a special case of the binocular percept.

Symmetry Is the sine qua non of Shape

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.

Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation

Symmetry has been shown to be a very effective a priori constraint in solving a 3D shape recovery problem. Symmetry is useful in 3D recovery because it is a form of redundancy. There are, however, some fundamental limits to the effectiveness of symmetry. Specifically, given two arbitrary curves in a single 2D image, one can always find a 3D mirror-symmetric interpretation of these curves under quite general assumptions. The symmetric interpretation is unique under a perspective projection and there is a one parameter family of symmetric interpretations under an orthographic projection. We formally state and prove this observation for the case of one-to-one and many-to-many point correspondences. We conclude by discussing the role of degenerate views, higher-order features in determining the point correspondences, as well as the role of the planarity constraint. When the correspondence of features is known and/or curves can be assumed to be planar, 3D symmetry becomes non-accidental in the sense that a 2D image of a 3D asymmetric shape obtained from a random viewing direction will not allow for 3D symmetric interpretations.

Detecting mirror-symmetry of a volumetric shape from its single 2D image

We present a new computational model for verifying whether a 3D shape is mirror-symmetric based on its single 2D image. First, a psychophysical experiment which tested human performance in detection of 3D symmetry is described. These psychophysical results led to the formulation of a new algorithm for symmetry detection. The algorithm first recovers the 3D shape using a priori constraints (symmetry, planarity of contours and 3D compactness) and then evaluates the degree of symmetry of the 3D shape. Reliable discrimination by the algorithm between symmetric and asymmetric 3D shapes involves two measures: similarity of the two halves of a 3D shape and compactness of the 3D shape. Performance of this algorithm is highly correlated with that of the subjects. We conclude that this algorithm is a plausible model of the mechanisms used by the human visual system.

Detecting 3-D Mirror Symmetry in a 2-D Camera Image for 3-D Shape Recovery

In this paper, we take up the long-standing problem of how to recover 3-D shapes represented by a 2-D image, such as the image on the retina of the eye, or in a video camera. Our approach is biologically grounded in a theory of how the human visual system solves this problem, focusing on shapes that are mirror symmetrical in 3-D. A 3-D mirror-symmetrical shape can be recovered from a single 2-D orthographic or perspective image by applying several a priori constraints: 3-D mirror symmetry, 3-D compactness, and planarity of contours. From the computational point of view, the application of a 3-D symmetry constraint is challenging because it requires establishing 3-D symmetry correspondence among features of a 2-D image, which itself is asymmetrical for almost all viewing directions relative to the 3-D symmetrical shape. We describe new invariants of a 3-D to 2-D projection for the case of a pair of mirror-symmetrical planar contours, and we formally state and prove the necessary and sufficient conditions for detection of this type of symmetry in a single orthographic and perspective image.

Symmetry detection in 3D scenes

Retinal image of a symmetric object is itself symmetric only for a small set of viewing directions. Interestingly, human subjects have little difficulty in determining whether a given retinal image was produced by a symmetric object, regardless of the viewing direction. We tested perception of planar (2D) symmetric figures (dotted patterns and polygons) when the figures were slanted in depth. We found that symmetry could be detected reliably with polygons, but not with dotted patterns. Next, we tested the role image features representing the symmetry of the pattern itself (orientation of projected symmetry axis and symmetry lines) vs. those representing the 3D viewing direction (orientation of the axis of rotation). We found that symmetry detection is improved when the projected symmetry axis or lines are known to the subject, but not when the axis of rotation is known. Finally, we showed that performance with orthographic images is higher than that with perspective images. A computational model, which measures the asymmetry of the presented polygon based on its single orthographic or perspective image, is presented. Performance of the model is similar to the performance of human subjects.

Smooth-shape assumption for perceiving shapes from shading.

Humans can perceive three-dimensional shapes from shading, but reconstructing the original shape of an object from shading alone (luminance distribution) is mathematically impossible. Researchers have used different assumptions and reported that the human visual systems can resolve this difficulty. Here, we propose an assumption for perceiving shape from shading: that the object shape is assumed to be smooth rather than angular. In experiment 1, we investigated the effect of shape smoothness by manipulating the shading profile of the test region. In experiment 2, we further investigated the effect of shape smoothness by manipulating shapes of the regions bordering on the test region using binocular disparity. Each stimulus in our experiments is interpretable from shading as having either smooth or angular edges. Observers responded to the perceived shape while viewing the stimuli, and most tended to perceive smooth rather than angular edges. These results support the idea that the smooth-shape assumption is effective for perceiving shape from shading.

Perception of 3D Symmetrical and Nearly Symmetrical Shapes

A 2D orthographic image of a 3D mirror-symmetrical shape determines a one-parameter family of 3D symmetrical shapes. When the 2D orthographic projections of the 3D symmetry lines of the 3D shape are set parallel to the x-axis on the image plane, any pair of 3D shapes in this one parameter family are related to one another by the following 3D affine transformation: [ X Y Z2 ] =  1 0 0 0 1 0 cos (2α1)−cos (2α2) sin (2α2) 0 sin (2α1) sin (2α2) [ XY Z1 ]

The divisive normalization model of V1 neurons: a comprehensive comparison of physiological data and model predictions

The physiological responses of simple and complex cells in the primary visual cortex (V1) have been studied extensively and modeled at different levels. At the functional level, the divisive normalization model (DNM; Heeger DJ. Vis Neurosci 9: 181-197, 1992) has accounted for a wide range of single-cell recordings in terms of a combination of linear filtering, nonlinear rectification, and divisive normalization. We propose standardizing the formulation of the DNM and implementing it in software that takes static grayscale images as inputs and produces firing rate responses as outputs. We also review a comprehensive suite of 30 empirical phenomena and report a series of simulation experiments that qualitatively replicate dozens of key experiments with a standard parameter set consistent with physiological measurements. This systematic approach identifies novel falsifiable predictions of the DNM. We show how the model simultaneously satisfies the conflicting desiderata of flexibility and falsifiability. Our key idea is that, while adjustable parameters are needed to accommodate the diversity across neurons, they must be fixed for a given individual neuron. This requirement introduces falsifiable constraints when this single neuron is probed with multiple stimuli. We also present mathematical analyses and simulation experiments that explicate some of these constraints.