Abstract
We consider a gradient flow of the total variation in a negative Sobolev space H^{-s},(0le s le 1) under the periodic boundary condition. If s=0, the flow is nothing but the classical total variation flow. If s=1, this is the fourth order total variation flow. We consider a convex variational problem which gives an implicit-time discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to solve this minimizing problem numerically and discuss convergence of a forward–backward splitting scheme. Results of several numerical experiments are given.
Highlights
We consider a gradient flow of the total variation in a negative Sobolev space H −s (0 ≤ s ≤ 1) under the periodic boundary condition
If s = 0, the flow is nothing but the classical total variation flow
During the last two decades, total variation models have become very popular in image processing and analysis
Summary
During the last two decades, total variation models have become very popular in image processing and analysis This interest has been initiated by the seminal paper of Rudin et al [34], where the authors proposed to minimize the functional. An application of derived scheme will allow us to perform numerical experiments to observe evolution of solutions to the Eq (1.5) and their characteristic features with respect to different values of the exponent s Such a close look at this problem may be the basis for further study on the considered evolution equation and its applications. 10, we present results of numerical experiments to illustrate evolution of solutions to considered total variation flows, showing their characteristic features with respect to different values of index s ∈ [0, 1]
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