A new ADMM algorithm for the Euclidean median and its application to robust patch regression

The Euclidean Median (EM) of a set of points $\Omega$ in an Euclidean space is the point x minimizing the (weighted) sum of the Euclidean distances of x to the points in $\Omega$. While there exits no closed-form expression for the EM, it can nevertheless be computed using iterative methods such as the Wieszfeld algorithm. The EM has classically been used as a robust estimator of centrality for multivariate data. It was recently demonstrated that the EM can be used to perform robust patch-based denoising of images by generalizing the popular Non-Local Means algorithm. In this paper, we propose a novel algorithm for computing the EM (and its box-constrained counterpart) using variable splitting and the method of augmented Lagrangian. The attractive feature of this approach is that the subproblems involved in the ADMM-based optimization of the augmented Lagrangian can be resolved using simple closed-form projections. The proposed ADMM solver is used for robust patch-based image denoising and is shown to exhibit faster convergence compared to an existing solver.