Minimizing total variation flow

By Fuensanta Andreu, Coloma Ballester, V. Caselles, José M. Mazón. Differential and Integral Equations, 2000

308 citations recorded by the source corpus.

We prove existence and uniqueness of weak solutions for the minimizing Total Variation flow with initial data in L1 under Neumann boundary conditions. We prove that the HN−1 measure of the boundaries of level sets of the solution decreases with time, as one would expect. We also prove that local maxima (minima) strictly decrease (increase) their level with time. We shall also consider the Dirichlet problem which presents some particular difficulties for general initial data in L1.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6f08e2227f0137d03f25cab3f564ff56d69a3543

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Citations in the corpus

Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing (2003, background): “Existence and uniqueness results for the above minimization problem can be found in [1], [10], [25], [3] (see also [4]).”
Image Decomposition and Restoration Using Total Variation Minimization and the H1 (2003, background): “Existence and uniqueness results of this minimization problem can be found in [1], [7], [20], [2].”
Anzellotti's pairing theory and the Gauss–Green theorem (2017, background): “This difficulty has been overcome in [4,5] through the Anzellotti’s theory of pairings.”
Nonlocal Linear Image Regularization and Supervised Segmentation (2007, methodology): “• The proposed flow performs better (both visually and in terms of SNR) than the original NL-means filter [10] as well as several well-known local PDEbased regularizations [51, 2, 49, 13].”
Variable Exponent, Linear Growth Functionals in Image Restoration (2006, background): “More related work on linear-growth functionals and their flows can be found in [2, 3, 4, 5, 6, 7, 12, 14, 33].”
A nonlocal p-Laplacian evolution equation with Neumann boundary conditions (2008, background): “The main result of [2] is the following: Theorem 3.”
Filling-in by joint interpolation of vector fields and gray levels (2001, background): “…of outside may be constructed as the vector field of directions of the image obtained by regularizing , with in in for (16) As it is shown in [3] , this equation permits a regularization of the vector field of directions of the gradient of , i.e., there is a vector field , such that and .”