Integrated Parallel 2D-Leap-Frog Algorithm for Noisy Three Image Photometric Stereo

In this paper a feasible computational scheme for reconstructing a smooth Lambertian surface $$S_L$$ from noisy images is discussed. The noiseless case of Photometric Stereo relies on solving image irradiance equations. In fact, the entire shape recovery consists of gradient computation and gradient integration. The presence of added noise re-transforms the latter depending on the adopted model into a high-dimensional linear or non-linear optimization, solvable e.g. by a 2D-Leap-Frog. This algorithm resorts to the overlapping local image snapshot optimizations to reduce a large dimension of the original optimization task. Several practical steps to improve the feasibility of 2D-Leap-Frog are integrated in this work. Namely, an initial guess is obtained from a linear version of denoising Photometric Stereo. A non-integrable vector field estimating the normals to $$S_L$$ is rectified first to yield an initial guess $$S_{L_a}\approx S_L$$ for a non-linear 2D-Leap-Frog. Computationally, the integrability of non-integrable normals is enforced here by Conjugate Gradient which avoids numerous inversions of the large size matrices. In sequel, $$S_{L_a}$$ is fed through to the adjusted version of non-linear 2D-Leap-Frog. Such setting not only improves the recovery of $$S_L$$ from $$S_{L_a}\approx S_L$$ to $$\hat{S}_{L_a}\approx S_L$$ but also it removes potential outliers upon enforcing a continuity on $$\hat{S}_{L_a}$$ occurring in the previous version of 2D-Leap-Frog. In addition, a speed-up of shape reconstruction is achieved with parallelization of non-linear 2D-Leap-Frog applied to the modified cost function. The experiments are performed on images with different resolutions and varying number of kernels. Finally, the comparison tests between standard 2D-Leap-Frog either linear or non-linear and its improved outlier-free version are presented illustrating differences in the quality of the reconstructed surface.