Iterative finite element solution of a constrained total variation regularized model problem

Sören Bartels; Marijo Milicevic

doi:10.3934/dcdss.2017066

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2017, Volume 10, Issue 6: 1207-1232. Doi: 10.3934/dcdss.2017066

This issue Previous Article Tomáš Roubíček celebrates his sixtieth anniversary Next Article Stress-diffusive regularizations of non-dissipative rate-type materials

Iterative finite element solution of a constrained total variation regularized model problem

Sören Bartels^, and
Marijo Milicevic

Department of Applied Mathematics, Mathematical Institute, University of Freiburg, Hermann-Herder-Str. 9,79104 Freiburg i. Br., Germany

^* Corresponding author: Sören Bartels
^* Corresponding author: Sören Bartels

Received: April 30, 2016

Revised: October 31, 2016

Published: December 2017

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Abstract

The discretization of a bilaterally constrained total variation minimization problem with conforming low order finite elements is analyzed and three iterative schemes are proposed which differ in the treatment of the non-differentiable terms. Unconditional stability and convergence of the algorithms is addressed, an application to piecewise constant image segmentation is presented and numerical experiments are shown.

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Mathematics Subject Classification: Primary: 65K15, 49M27; Secondary: 94A08.

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Figure 1. $L^2$-error between approximate minimizer $\tilde{u}_h$ of $E$ (generated by the split-split method) and iterates $u_h^j$, $s_h^j$ of the Heron-penalty method, $h=\sqrt{2}2^{-6}$, $\varepsilon=h$, for Example 1. Note that the iterates $u_h^j$ serve as good approximations of $\tilde{u}_h$ when $\delta=h/\alpha$

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Figure 2. $L^2$-error between approximate minimizer $\tilde{u}_h$ of $E$ (generated by the split-split method) and iterates $u_h^j$, $s_h^j$ of the Heron-penalty method, $h=\sqrt{2}2^{-6}$, $\varepsilon=h$, for Example 2. Again, the iterates $u_h^j$ approximate $\tilde{u}_h$ properly when $\delta=h/\alpha$

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Figure 3. Original image and outputs of Algorithm 5 using the split-split, Heron-penalty and Heron-split method in step (2), respectively (horizontal white lines are due to image conversion)

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Table 1. Iteration numbers with (5), (6), (7) for Example 1 and $\sigma_1=h^{-3/2}$ for the split-split method and $\tau=1$ for the Heron-penalty and the Heron-split method

	Split-split			Heron-penalty		Heron-split
	$\sigma_2$			$\delta$		$\sigma$
	$1$	$\alpha$	$1/h$	$h/\alpha$	$h$	$1$	$\alpha$	$1/h$
$\sqrt{2}/2^5$	$344$	$39$	$37$	$89$	$42$	$892$	$52$	$57$
$\sqrt{2}/2^6$	$289$	$79$	$79$	$143$	$55$	$576$	$77$	$77$
$\sqrt{2}/2^7$	$283$	$77$	$78$	$211$	$69$	$473$	$94$	$94$
$\sqrt{2}/2^8$	$287$	$115$	$120$	$426$	$101$	$373$	$148$	$151$

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Table 2. Iteration numbers with (5), (6), (7) for Example 2 and $\sigma_1=h^{-3/2}$ for the split-split method and $\tau=1$ for the Heron-penalty and the Heron-split method

	Split-split			Heron-penalty		Heron-split
	$\sigma_2$			$\delta$		$\sigma$
	$1$	$\alpha$	$1/h$	$h/\alpha$	$h$	$1$	$\alpha$	$1/h$
$\sqrt{2}/2^5$	$2745$	$10$	$124$	$74$	$24$	$4592$	$15$	$189$
$\sqrt{2}/2^6$	$2808$	$15$	$65$	$156$	$27$	$3586$	$21$	$72$
$\sqrt{2}/2^7$	$2847$	$23$	$35$	$298$	$35$	$3133$	$25$	$42$
$\sqrt{2}/2^8$	$-$	$29$	$30$	$572$	$43$	$-$	$33$	$37$

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