Dense specular shape from multiple specular flows

The inference of specular (mirror-like) shape is a particularly difficult problem because an image of a specular object is nothing but a distortion of the surrounding environment. Consequently, when the environment is unknown, such an image would seem to convey little information about the shape itself. It has recently been suggested (Adato et al., ICCV 2007) that observations of relative motion between a specular object and its environment can dramatically simplify the inference problem and allow one to recover shape without explicit knowledge of the environment content. However, this approach requires solving a non-linear PDE (the 'shape from specular flow equation') and analytic solutions are only known to exist for very constrained motions. In this paper, we consider the recovery of shape from specular flow under general motions. We show that while the 'shape from specular flow' PDE for a single motion is non-linear, we can combine observations of multiple specular flows from distinct relative motions to yield a linear set of equations. We derive necessary conditions for this procedure, discuss several numerical issues with their solution, and validate our results quantitatively using image data.