0000000000280365
AUTHOR
Fuensanta Andreu-vaillo
Total Variation Based Image Restoration
For the purpose of image restoration the process of image formation can be modeled in a first approximation by the formula [207] $$ {u_d} = Q\{ II(k*u) + n\} , $$ (1.1) where u represents the photonic flux k is the point spread function of the optical-captor joint apparatus П is a sampling operator, i.e., a Dirac comb supported by the centers of the matrix of digital sensors, n represents a random perturbation due to photonic or electronic noise, and Qis a uniform quantization operator mapping ℝ to a discrete interval of values, typically [0, 255].
The Neumann Problem for the Total Variation Flow
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 …
Asymptotic Behaviour and Qualitative Properties of Solutions
The purpose of this chapter is to give some qualitative properties of the flow $$ frac{{\partial u}}{{\partial t}} = div\left( {\frac{{Du}}{{\left| {Du} \right|}}} \right) in\;]0,\infty [ \times {\mathbb{R}^N} $$ (4.1) .
Parabolic Equations Minimizing Linear Growth Functionals: L1-Theory
Let Ω be a bounded set in ℝN with boundary of class C1. We are interested in the problem $$ \left\{ \begin{gathered} \frac{{\partial u}} {{\partial t}} = diva\left( {x,Du} \right)in Q = \left( {0,\infty } \right) \times \Omega , \hfill \\ u\left( {t,x} \right) = \phi \left( x \right)on S = \left( {0,\infty } \right) \times \partial \Omega , \hfill \\ u\left( {0,x} \right) = u_0 \left( x \right)in x \in \Omega \hfill \\ \end{gathered} \right. $$ (1) where ϕ ∈ L1(∂Ω), u0 ∈ L2(Ω) and a(x, ξ) = ∇ξ f(x, ξ, f being a function with linear growth in ‖ξ‖ as ‖ξ‖ → ∞. One of the classical examples is the nonparametric area integrand for which \( f(x,\xi ) = \sqrt {1 + \left\| \xi \right\|^2 } \). Prob…