Seeing in 3D is a fundamental problem for any organism or device that has to operate in the real world. Answering questions such as “how far away is that?” or “can we fit through that opening?” requires perceiving and making judgments about the size of objects in three dimensions. So how do we see in three dimensions? Given a sufficiently accurate model of the world and its illumination, complex but accurate models exist for generating the pattern of illumination that will strike the retina or cameras of an active agent (see Foley et al., 1995). The inverse problem, how to build a three-dimensional representation from such two-dimensional patterns of light impinging on our retinas or the cameras of a robot, is considerably more complex.

In fact, the problem of perceiving 3D shape and layout is a classic example of an ill-posed and underconstrained inverse problem. It is an underconstrained problem because a unique solution is not obtainable from the visual input. Even when two views are present (with the slightly differing viewpoints of each eye), the images do not necessarily contain all the information required to reconstruct the three-dimensional structure of a viewed scene. It is an illposed problem because small changes in the input can lead to significant changes in the output: that is, reconstruction is very vulnerable to noise in the input signal. The problem of constructing the three-dimensional structure of the viewed scene is an extremely difficult and usually impossible problem to solve uniquely.