$L_1$-Penalized Robust Estimation for a Class of Inverse Problems Arising in Multiview Geometry

We propose a new approach to the problem of robust estimation in multiview geometry. Inspired by recent advances in the sparse recovery problem of statistics, our estimator is defined as a Bayesian maximum a posteriori with multivariate Laplace prior on the vector describing the outliers. This leads to an estimator in which the fidelity to the data is measured by the $L_\infty$-norm while the regularization is done by the $L_1$-norm. The proposed procedure is fairly fast since the outlier removal is done by solving one linear program (LP). An important difference compared to existing algorithms is that for our estimator it is not necessary to specify neither the number nor the proportion of the outliers. The theoretical results, as well as the numerical example reported in this work, confirm the efficiency of the proposed approach.