scholarly journals PASSING TO THE LIMIT IN MAXIMAL SLOPE CURVES: FROM A REGULARIZED PERONA–MALIK EQUATION TO THE TOTAL VARIATION FLOW

2012 ◽  
Vol 22 (08) ◽  
pp. 1250017 ◽  
Author(s):  
MARIA COLOMBO ◽  
MASSIMO GOBBINO
Keyword(s):  
Total Variation ◽  
General Result ◽  
General Principle ◽  
Time Scale ◽  
Space Dimension ◽  
Gradient Flow ◽  
Convergence Result ◽  
Gradient Flows ◽  
Slow Time ◽  
Total Variation Flow

We prove that solutions of a mildly regularized Perona–Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.

scholarly journals The convergence of a numerical method for total variation flow

2021 ◽  
Vol 15 ◽  
pp. 174830262110113
Author(s):  
Qianying Hong ◽  
Ming-jun Lai ◽  
Jingyue Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Total Variation ◽  
Iterative Algorithm ◽  
Finite Difference Scheme ◽  
Gradient Flow ◽  
Time Dependent ◽  
Total Variation Flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

scholarly journals On the total variation Wasserstein gradient flow and the TV-JKO scheme

2019 ◽  
Vol 25 ◽  
pp. 42
Author(s):  
Guillaume Carlier ◽  
Clarice Poon
Keyword(s):  
Total Variation ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Minimum Principle ◽  
Gradient Flow ◽  
Convergence Result ◽  
Symmetric Case ◽  
Qualitative Properties ◽  
Time Step ◽  
Dimension One

We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a form of maximum principle and in some cases, a minimum principle as well). Finally, we establish a convergence result as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation, under the additional assumption that the density remains bounded away from zero, this lower bound is shown in dimension one and in the radially symmetric case.

scholarly journals Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

2021 ◽  
Author(s):  
Antonio Esposito ◽  
Francesco S. Patacchini ◽  
André Schlichting ◽  
Dejan Slepčev
Keyword(s):  
Gradient Flow ◽  
Wasserstein Distance ◽  
Convergence Result ◽  
Existence Theory ◽  
Nonlocal Interaction ◽  
Interaction Energies ◽  
Gradient Flows ◽  
Empirical Measures ◽  
Interaction Equation ◽  
The Continuum

AbstractWe consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

scholarly journals Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

2020 ◽  
Author(s):  
Bart M. N. Smets ◽  
Jim Portegies ◽  
Etienne St-Onge ◽  
Remco Duits
Keyword(s):  
Total Variation ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Arbitrary Dimension ◽  
Gradient Flow ◽  
Image Data ◽  
Numerical Approach ◽  
Nonlinear Pdes ◽  
Total Variation Flow ◽  
Pde Framework

Abstract Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

scholarly journals Understanding the Overdispersed Molecular Clock

Genetics ◽  
2000 ◽  
Vol 154 (3) ◽  
pp. 1403-1417 ◽  
Author(s):  
David J Cutler
Keyword(s):  
Molecular Evolution ◽  
Molecular Clock ◽  
Time Scale ◽  
Population Parameter ◽  
Deleterious Mutations ◽  
Slow Time ◽  
Slow Time Scale ◽  
Key Population ◽  
Protein Encoding ◽  
Rates Of Molecular Evolution

Abstract Rates of molecular evolution at some protein-encoding loci are more irregular than expected under a simple neutral model of molecular evolution. This pattern of excessive irregularity in protein substitutions is often called the “overdispersed molecular clock” and is characterized by an index of dispersion, R(T) > 1. Assuming infinite sites, no recombination model of the gene R(T) is given for a general stationary model of molecular evolution. R(T) is shown to be affected by only three things: fluctuations that occur on a very slow time scale, advantageous or deleterious mutations, and interactions between mutations. In the absence of interactions, advantageous mutations are shown to lower R(T); deleterious mutations are shown to raise it. Previously described models for the overdispersed molecular clock are analyzed in terms of this work as are a few very simple new models. A model of deleterious mutations is shown to be sufficient to explain the observed values of R(T). Our current best estimates of R(T) suggest that either most mutations are deleterious or some key population parameter changes on a very slow time scale. No other interpretations seem plausible. Finally, a comment is made on how R(T) might be used to distinguish selective sweeps from background selection.

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