Shape Recovery from a Monocular Image by Relaxative Application of Orthogonality Hypothesis

Abstract

This paper describes a method of shape recovery from a monocular image. We usually have definite ideas about the three-dimensional shapes from a two-dimensional view though infinite number of shapes can produce the same view. To reduce the number, we must use some assumptions about the world. As the assumptions, we will use geometrical regularities, especially the orthogonality of the directions of the edges. In a cultural environment, we can find many right angles, and in many cases they definitely limit the shapes of the objects. By no means, however, we can perfectly point out the three-dimensional right angles on the two-dimensional image before the shape recovery. Therefore the total process of shape recovery should be considered as a kind of hypothesis-based reasoning. Here, a hypothesis is characterized by the assignments of orthogonality assumption to the angles. First, we define the local energy function that represents the error of an angle from the right angle to evaluate the orthogonality of two edges. Shape recovery under a hypothesis can be performed by minimizing the total of the local energy for each angle that is assumed to be a right angle. Next, we define the evaluation function of the hypothesis with respect to the recovered shape as a function of the above total energy and the number of right angles. As a result, the evaluation function becomes a function of both the shape and the assignments of orthogonality. By relaxative minimization of the function with respect to the shape and the assignments, shape recovery with dynamic assignment of orthogonality assumption can be performed. Some experimental results demonstrate the availability of the proposed method.