Second order local analysis for 3D reconstruction of specular surfaces

We analyze the problem of recovering the shape of a mirror surface. A calibrated scene composed of lines passing through a point is assumed. The lines are reflected by the mirror surface onto the image plane of a calibrated camera, where the intersection, orientation and curvature of such reflections are measured. The relationship between the local geometry of the surface around the point of reflection and the measurements is analyzed. We extend the analysis in Savarese and Perona. (2001, 2002), where we recovered positions and normals and second order local geometry of a specular surface up to one unknown parameter. We show that, provided that we work in a neighborhood of a surface whose third order surface terms can be neglected, the second order parameter ambiguity can be solved by equating the curvatures observed for the reflected lines with those computed from analytical differentiation followed by a perspective projection.