Specular flow and Weingarten map : Technical Report IES-2015-13

If a moving camera observes a specular surface in which some environment is reflected, the pixel values per se do not directly characterise the surface. However, the associated optical flow, or specular flow (SF), as it is called in this situation, is an environment-agnostic observable that depends on the surface position, orientation, and curvature. The derivation of the SF in the limit of an infinitely remote environment has been published earlier by the authors, but in a relatively opaque coordinate-dependent form. In this report, we present a simpler and a more general derivation of the SF as a function of the surface structure, where the crucial part is played by the so-called Weingarten map. This result allows us to formulate the conditions when the SF diverges, and to derive a simple formula to relate the Gaussian curvature of the surface to the SF.