Non-local Functionals Related to the Total Variation and Connections with Image Processing
Tóm tắt
We present new results concerning the approximation of the total variation,
$$\int _{\Omega } |\nabla u|$$
, of a function u by non-local, non-convex functionals of the form
$$\begin{aligned} \Lambda _\delta (u) = \int _{\Omega } \int _{\Omega } \frac{\delta \varphi \big ( |u(x) - u(y)|/ \delta \big )}{|x - y|^{d+1}} \, dx \, dy, \end{aligned}$$
as
$$\delta \rightarrow 0$$
, where
$$\Omega $$
is a domain in
$$\mathbb {R}^d$$
and
$$\varphi : [0, + \infty ) \rightarrow [0, + \infty )$$
is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi’s concept of
$$\Gamma $$
-convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.
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