The Role of Orthogonality and Compactness in Recovering 3D Symmetrical Shapes

We explore the role of geometrical projections from 3D to 2D as well as the role of compactness, symmetry, and orthogonality constraints in monocular 3D shape recovery. This recovery is an ill-posed inverse problem, but our prior work has shown that subjects’ performance is quite reliable. On each trial, a static image is shown on the left side of a computer screen, and on the right side of the screen a user-adjustable 3D shape is shown rotating. The subject adjusts either one or two parameters of the 3D shape on the right until it is as close as possible to their 3D percept produced by the static 2D image on the left. The images are either orthographic or perspective projections of the 3D shapes. To test the role of orthogonality constraint, we generated three classes of objects: (i) synthetic polyhedral objects composed of boxes which have no 90deg angles; (ii) similar polyhedral objects but with a number of 90deg angles; and (iii) symmetrical or approximately symmetrical, real-world objects selected from the ModelNet-40 dataset. Many of these real-world objects have a high degree of orthogonality. The computational model uses regularization to perform 3D shape recovery. The cost function has terms that penalize departure from orthogonality and from 3D compactness of the 3D shape’s convex hull. Mirror symmetry is an implicit constraint. Perspective images lead to more accurate recovery than orthographic images. This is related to the fact that perspective images determine a unique 3D interpretation of symmetrical shapes. Monocular recovery from perspective images is as good as binocular recovery reported in prior studies. Recovery of natural objects is more accurate than recovery of synthetic polyhedral objects. This result suggests that familiarity plays a role. We will discuss the role of familiarity with individual objects and familiarity with categories of objects.