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We develop a simple algorithm for finding the minimizer of the function E(x)= ∑n i=1wi |x−ai|+F (x), when the wi 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.

A new median formula with applications to PDE based denoising