Discrete multiscale vector field decomposition
ACM SIGGRAPH 2003 Papers
Modeling Textures with Total Variation Minimization and Oscillating Patterns in Image Processing
Journal of Scientific Computing
Image Decomposition into a Bounded Variation Component and an Oscillating Component
Journal of Mathematical Imaging and Vision
Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
The Split Bregman Method for L1-Regularized Problems
SIAM Journal on Imaging Sciences
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In this paper, we address the issue of decomposing a given real-textured image into a cartoon/geometric part and an oscillatory/texture part. The cartoon component is modeled by a function of bounded variation, while, motivated by the works of Meyer [Y. Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, vol. 22 of University Lecture Series, AMS, 2001], we propose to model the oscillating component v by a function of the space G of oscillating functions, which is, in some sense, the dual space of BV(@W). To overcome the issue related to the definition of the G-norm, we introduce auxiliary variables that naturally emerge from the Helmholtz-Hodge decomposition for smooth fields, which yields to the minimization of the L^~-norm of the gradients of the new unknowns. This constrained minimization problem is transformed into a series of unconstrained problems by means of Bregman Iteration. We prove the existence of minimizers for the involved subproblems. Then a gradient descent method is selected to solve each subproblem, becoming related, in the case of the auxiliary functions, to the infinity Laplacian. Existence/Uniqueness as well as regularity results of the viscosity solutions of the PDE introduced are proved.