Measuring shape of a mirror with a moving camera

If a moving camera observes a specular surface that reflects some distant environment, the recorded pixel values do not directly characterise the surface. However, the associated optical flow (OF), or specular flow (SF), as it is known in this case, is an (almost) environment-agnostic observable that depends on the camera position and the mirror shape. We present a derivation of the SF expected for a given view ray and mirror form, and establish a simple relation between the SF and the Gaussian curvature of the surface. The quality of the simulated SF is studied using physically-accurate rendering and state-of-the-art OF estimation software. Finally, we suggest and numerically verify a method to reconstruct the surface shapes from the SF, and discuss the possible ambiguities of the reconstruction.