Surface reconstruction via Helmholtz reciprocity with a single image pair

This paper proposes a method for three-dimensional reconstruction of surfaces that takes advantage of the symmetry resulting from alternating the positions of a camera and a light source. This set up allows for use of the Helmholtz reciprocity principle to recover the shape of smooth surfaces with arbitrary bidirectional reflectance distribution functions without requiring the presence of texture, as well as for exploiting mutual occlusions between images. Different from previous methods, the technique works with as few as one reciprocal pair, and recovers surface depth and orientation simultaneously by finding the global minimum of an error function via dynamic programming. Since the error is a function not just of depth but of surface orientation as well, the reconstruction is subject to tighter geometric constraints. Given a current estimate of surface geometry and intensity measurements in one image, Helmholtz reciprocity is used to predict the pixel intensity values of the second image. The dynamic program finds the reconstruction that minimizes the total difference between the predicted and measured intensity values. This approach allows for the reconstruction of surfaces displaying specularities and regions of high curvature, which is a challenge commonly encountered in the optical inspection of industrial parts. Results with real data show the quality of the reconstruction obtained with the proposed algorithm.