Shape from Angle
In this chapter, we study the problem of 3D recovery of polyhedra on the assumption that the corners are rectangular (rectangularity hypothesis). First, we study how the 3D orientations of edges defining a rectangular corner are computed from their orthographic projection. Then, we study the constraints on the image of a rectangular polyhedron as a whole. Our analysis is based on the observation that two rectangular corners are either identical or mirror images of each other with respect to an appropriately placed mirror, and all the mappings between different types of rectangular corners form a group of transformations. Next, we turn to perspective projection. We make use of the fact that as far as the 3D interpretation of a corner is concerned, the distinction between orthographic and perspective projections disappears if the camera is rotated in such a way that the vertex is moved to the center of the image plane. Then, we study the constraints imposed by vanishing points and vanishing lines, which are typical of perspective projection. The duality of points and lines introduced in Chap. 4 plays an essential role.
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