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 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 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 Gradient-flow techniques for the analysis of numerical schemes for multi-phase mean-curvature flow

2018 ◽  
Vol 3 (1) ◽  
pp. 76-89
Author(s):  
Tim Laux
Keyword(s):  
Mean Curvature ◽  
General Framework ◽  
Mean Curvature Flow ◽  
Gradient Flow ◽  
Curvature Flow ◽  
Numerical Schemes ◽  
Two Phase ◽  
Guiding Principle ◽  
Convergence Results ◽  
Multi Phase

Abstract Several recent convergence results for numerical schemes for mean-curvature flow in particular in the multi-phase case with arbitrary surface tensions are discussed. The guiding principle of all these works is the gradient-flow structure of multi-phase mean-curvature flow which is explained in the general framework. For simplicity, the convergence results are presented in the simpler two-phase case.

The gradient structure of the mean curvature flow

2015 ◽  
Vol 8 (3) ◽  
Author(s):  
Martijn M. Zaal
Keyword(s):  
Metric Space ◽  
Mean Curvature ◽  
Metric Spaces ◽  
Mean Curvature Flow ◽  
Gradient Flow ◽  
Curvature Flow ◽  
Gradient Structure ◽  
Length Space ◽  
Curve Length ◽  
The Mean

AbstractThe concept of curve of maximal slope, a generalized notion of gradient flow, is extended to the setting of length spaces, which includes all metric spaces. This extended definition is used to show that curves of maximal slope in a metric space do not depend on the full metric, but only on the concept of curve length generated by it. Subsequently, it is shown that a length space can be constructed to describe

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 Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

scholarly journals Mean curvature flow with free boundary outside a hypersphere

2017 ◽  
Vol 369 (12) ◽  
pp. 8319-8342 ◽  
Author(s):  
Glen Wheeler ◽  
Valentina-Mira Wheeler
Keyword(s):  
Free Boundary ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow

scholarly journals Ancient solutions for Andrews’ hypersurface flow

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Peng Lu ◽  
Jiuru Zhou
Keyword(s):  
Ricci Flow ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Euclidean Spaces ◽  
Round Cylinder ◽  
Ancient Solutions ◽  
Singular Time

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.

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