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

  • * Corresponding author: Sören Bartels

    * Corresponding author: Sören Bartels 
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  • 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.

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

    Citation:

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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 $\delta=h/\alpha$

    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 $\delta=h/\alpha$

    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)

    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-splitHeron-penaltyHeron-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$
     | Show Table
    DownLoad: CSV

    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-splitHeron-penaltyHeron-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$
     | Show Table
    DownLoad: CSV
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