Least Squares Surface Reconstruction from Measured Gradient Fields

Abstract

This paper presents a new method for the reconstruction of a surface from its x and y gradient field, measured, for example, via Photometric Stereo. The new algorithm produces the unique discrete surface whose gradients are equal to the measured gradients in the global vertical-distance least-squares sense. We show that it has been erroneously believed that this problem has been solved before via the solution of a Poisson equation. The numerical behaviour of the algorithm allows for reliable surface reconstruction on exceedingly large scales, e.g., full digital images; moreover, the algorithm is direct, i.e., non-iterative. We demonstrate the algorithm with synthetic data as well as real data obtained via photometric stereo. The algorithm does not exhibit a low-frequency bias and is not unrealistically constrained to arbitrary boundary conditions as in previous solutions. In fact, it is the first algorithm which can reconstruct a surface of polynomial degree two or higher exactly. It is hence the first viable algorithm for online industrial inspection where real defects (as opposed to phantom defects) must be identified in a robust manner.