6533b81ffe1ef96bd127849f

RESEARCH PRODUCT

The Neumann Problem for the Total Variation Flow

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This chapter is devoted to prove existence and uniqueness of solutions for the minimizing total variation flow with Neumann boundary conditions, namely $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = div\left( {\frac{{Du}} {{\left| {Du} \right|}}} \right) in Q = (0,\infty ) \times \Omega , \hfill \\ \frac{{\partial u}} {{\partial \eta }} = 0 on S = (0,\infty ) \times \partial \Omega , \hfill \\ u(0,x) = u_0 (x) in x \in \Omega , \hfill \\ \end{gathered} \right. $$ (2.1) where Ω is a bounded set in ℝ N with Lipschitz continuous boundary ∂ Ω and u0 ∈ L1(Ω). As we saw in the previous chapter, this partial differential equation appears when one uses the steepest descent method to minimize the total variation, a method introduced by L. Rudin, S. Osher and E. Fatemi ([174], [175]) in the context of image denoising and reconstruction. Then solving (2.1) amounts to regularizing or, in other words, to filtering the initial datum u0. This filtering process has less destructive effect on the edges than filtering with a Gaussian, i.e., than solving the heat equation with initial condition u0. In this context the given image Condition u0is a function defined on a bounded, smooth or piecewise smooth open subset Ωof ℝ N ; typically, Ω will be a rectangle in ℝ2. As argued in [7], the choice of Neumann boundary conditions is a natural choice in image processing. It corresponds to the reflection of the picture across the boundary and has the advantage of not imposing any value on the boundary and not creating edges on it. When dealing with the deconvolution or reconstruction problem one minimizes thetotal variation functional, i.e., the functional $$ {\smallint _\Omega }\left| {Du} \right| $$ (2.2) under some constraints which model the process of image acquisition, including blur and noise ([146], [174], [175], [71], [72], [64], [201], [202]).