Fast ADMM ℓ₁ minimization by applying SMW formula and multi-row simultaneous estimation for Light Transport Matrix acquisition

The Light Transport Matrix (LTM) is a fundamental expression of the light propagation of the projector-camera system. The matrix includes all the characteristics of light rays transferred from the projector to the camera, and it is used for scene relighting, understanding the light path, and 3D measurement. Especially, LTM enables robust 3D measurement even if the scene includes metallic or semi-transparent objects; thus it is already used for robot vision. The LTM is often estimated by ℓ₁ minimization because the LTM has a huge number of elements. ℓ₁ minimization methods, which utilize the Alternating Direction Method of Multipliers (ADMM), can reduce the number of observations. In addition, a powerful extended ADMM ℓ₁ minimization method named Saturation ADMM, which can estimate the LTM under saturated conditions, also exists. In the study presented in this paper, we reduce the computational cost of ADMM ℓ₁ minimization by applying the Sherman-Morrison-Woodbury (SMW) formula. Furthermore, we propose "multi-row simultaneous LTM estimation, " which is a new method to improve the computational efficiency. The contribution of this paper is to propose the use of these two methods to speed up LTM estimation and demonstrate that our methods reduce the computational cost in theory and the calculation time in practice. Experiments indicate that our method accelerates ADMM ℓ₁ minimization by up to 4.64 times, and Saturation ADMM ℓ₁ minimization by up to 2.54 times compared to the original methods.