A new median formula with applications to PDE based denoising

We develop a simple algorithm for finding the minimizer of the function E(x) = P n i=1 w i |xa i | + F (x), when the w i are nonnegative and F is strictly convex. If F is also differentiable and F is bijective, we obtain an explicit formula in terms of a median. This enables us to obtain approximate solutions to certain important variational problems arising in image denoising. We also present a generalization with E(x) = J(x) + F (x) for J(x) a convex piecewise differentiable function with a finite number of nondifferentiable points.