Conjugate Gradient in Noisy Photometric Stereo

Abstract

This paper discusses the problem of reconstructing the Lambertian surface from noisy three-light source Photometric Stereo. In the continuous image setting the shape recovery process is divided into two steps: an algebraic one (gradient computation) and analytical one (gradient integration). The digitized case with added noise has it discrete analogue in which also perturbed gradient from three noisy images is first computed. Generically such non-integrable vector field is subsequently rectified to the ”closest” integrable one. Finally, numerical integration scheme yields the unknown surface. The process of vector field rectification is reduced to the corresponding linear optimization task of very high dimension (comparable with the image resolution). Standard methods based on matrix pseudo-inversion suffer from heavy computation due to the necessity of large matrix inversion. A possible alternative is to set up an iterative scheme based on local snapshots’ optimizations (e.g. 2D-Leap-Frog). Another approach which is proposed in this paper is solving the above global optimization scheme by Conjugate Gradient with no inversion of matrices of large dimension. The experimental results from this paper show that the application of Conjugate Gradient forms a computationally feasible alternative in denoising Photometric Stereo.