What does a single light-ray reveal about a transparent object?

We address the following problem in refractive shape estimation: given a single light-ray correspondence, what shape information of the transparent object is revealed along the path of the light-ray, assuming that the light-ray refracts twice. We answer this question in the form of two depth-normal ambiguities. First, specifying the surface normal at which refraction occurs constrains the depth to a unique value. Second, specifying the depth at which refraction occurs constrains the surface normal to lie on a 1D curve. These two depth-normal ambiguities are fundamental to shape estimation of transparent objects and can be used to derive additional properties. For example, we show that correspondences from three light-rays passing through a point are needed to correctly estimate its surface normal. Another contribution of this work is that we can reduce the number of views required to reconstruct an object by enforcing shape models. We demonstrate this property on real data where we reconstruct shape of an object, with light-rays observed from a single view, by enforcing a locally planar shape model.