An algebraic approach to shape-from-image problems

摘要

This paper presents a new method for recovering three-dimensional shapes of polyhedral objects from their single-view images. The problem of recovery is formulated in a constrained optimization problem, in which the constraints reflect the assumption that the scene is composed of polyhedral objects, and the objective function to be minimized is a weighted sum of quadric errors of surface information such as shading and texture. For practical purpose it is decomposed into the two more tractable problems: a linear programming problem and an unconstrained optimization problem. In the present method the global constraints placed by the polyhedron assumption are represented in terms of linear algebra, whereas similar constraints have usually been represented in terms of a gradient space. Moreover, superstrictness of the constraints can be circumvented by a new concept ‘position-free incidence structure’. For this reason the present method has several advantages: it can recover the polyhedral shape even if image data are incorrect due to vertex-position errors, it can deal with perspective projection as well as orthographic projection, the number of variables in the optimization problem is very small (three or a little greater than three), and any kinds of surface information can be incorporated in a unifying manner.