Depth and shape from differential perspective in the presence of bending deformations

It is shown that there exist classes of useful partial solutions to what has become known as the structure from motion problem, that is, the problem of finding the three-dimensional shape as well as the movement in space of objects on the basis of their optical projections in several distinct phases of the movement. The class of solutions treated in this paper is quite different from the usual treatment in that the rigidity condition is relaxed so as to include the important class of bending deformations. For instance, we show a case in which important aspects of shape are obtained for a configuration of seven points, no four of which move in a rigid way, and which can be obtained from only two views. (This violates all requirements of the structure from motion theorem.) We discuss an algorithm that obtains the three-dimensional configuration of a polyhedral vertex up to a relief transformation (a similar transformation to the one that maps object to image space in paraxial optics), in the presence of arbitrary flections of the edges, from the projection and the instantaneous-movement parallax. Such a solution can, for instance, be used to predict side views of the vertex, as is demonstrated for a specific example. The solution can be used in an iterative fashion on triangulations to solve the case of extended curved surfaces suffering arbitrary bendings. The class of bending deformations encompasses all deformations that conserve distances along the surface (but not necessarily through space); thus stretchings are excluded. Practical examples include, e.g., deformations of flexible but nonstretchable shells. Actual numerical examples illustrate the practicability of the method.