Stability of absolute depth reconstruction from deflectometric measurement data

Until recently, the problem of reconstructing specular surface shapes from deflectometric registration data was considered inherently ambiguous, and thus requiring additional information in order to uniquely determine the absolute surface position. In 2013, Liu, Hartley and Salzmann suggested a solution to the reconstruction problem which employed the first-order derivatives of the registration data in order to recover the absolute depth of the surface along each camera ray. In this work, we demonstrate an alternative derivation of equivalent results, leading to more computationally efficient and tractable expressions. Re-formulated in terms of normal vector field, our results provide a natural regularization that together with or without external regularization data could be easily used within the existing reconstruction algorithms. We further elaborate on the stability and the uniqueness of the solution. In particular, we find conditions when a shape cannot be uniquely recovered and identify two equations that characterize the families of such shapes.