Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours

Egil Bae; Xue-Cheng Tai; Wei Zhu

doi:10.3934/ipi.2017001

Article Contents

2017, Volume 11, Issue 1: 1-23. Doi: 10.3934/ipi.2017001

This issue Previous Article Next Article A source time reversal method for seismicity induced by mining

Augmented Lagrangian method for an Euler's elastica based segmentation model that promotes convex contours

1.
Norwegian Defence Research Establishment, P.O. Box 25, NO-2027 Kjeller, Norway
2.
Department of Mathematics, University of Bergen, P.O. Box 7803, NO-5020 Bergen, Norway
3.
Department of Mathematics, University of Alabama, Box 870350, Tuscaloosa, AL 35487, USA

* Corresponding author: Wei Zhu
* Corresponding author: Wei Zhu

Received: September 2015

Revised: April 2016

Published: March 2018

The first author is supported by the Norwegian Research Council eVita project 214889

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Abstract

In this paper, we propose an image segmentation model where an $L^1$ variant of the Euler's elastica energy is used as boundary regularization. An interesting feature of this model lies in its preference for convex segmentation contours. However, due to the high order and non-differentiability of Euler's elastica energy, it is nontrivial to minimize the associated functional. As in recent work on the ordinary $L^2$-Euler's elastica model in imaging, we propose using an augmented Lagrangian method to tackle the minimization problem. Specifically, we design a novel augmented Lagrangian functional that deals with the mean curvature term differently than in previous works. The new treatment reduces the number of Lagrange multipliers employed, and more importantly, it helps represent the curvature more effectively and faithfully. Numerical experiments validate the efficiency of the proposed augmented Lagrangian method and also demonstrate new features of this particular segmentation model, such as shape driven and data driven properties.

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Mathematics Subject Classification: Primary: 68U10, 65K10; Secondary: 49M05.

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Figure 1. Two vectors $\nu$, $\mathbf{a}$ and a possible unit vector $\mathbf{b}$ that maximizes the function $\psi(\mathbf{b})$.

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Figure 2. The original image (the first row) and the segmentation results (the second row) with different curvature parameter $b$. Two features can be observed from these results: 1) data driven property: as the parameter $b$ increases, objects of relatively small size will be omitted in the final segmentation; 2) shape driven property: with the same parameter $b$, among those objects with equal areas, the one without a convex shape is taken out from the final segmentation (see the staircase-like shape). In this experiment, the other parameters are: $a=10^{-4},r_{1}=50,r_{2}=20,r_{3}=5$

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Figure 3. The original image and the segmentation result. The result demonstrates that the invisible part of the hat can be restored by the proposed segmentation model with a relatively large curvature parameter $b$. In this experiment, the parameters are $a=10^{-3},b=80, r_{1}=60,r_{2}=40,r_{3}=10$

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Figure 4. The original image and the segmentation result. The result shows that only the ''cap" part of the mushroom is captured while the ''stem" is omitted in order to form a convex segmentation contour by the proposed segmentation model with a relatively large curvature parameter $b$. In this experiment, the parameters are $a=10^{-4},b=18, r_{1}=20,r_{2}=10,r_{3}=5$

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Figure 5. The effect of the curvature parameter $b$ on the final segmentation results. One may observe that the smaller the parameter $b$ is chosen, the more salient the indentation of the hat would be. In these two experiments, all the other parameters are the same as those in Fig. 3

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Figure 6. The effect of the curvature parameter $b$ on the final segmentation results. One can see that once the parameter $b$ is chosen smaller, more part of the stem will be allowed in the final segmentation. In these two experiments, all the other parameters are the same as those in Fig. 4

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Figure 7. The plots of relative residuals (Eq. 43), relative errors in Lagrange multipliers (Eq. 44), and relative error in $u^{k}$ (Eq. 45) for the two examples ''hat" (Left column) and ''mushroom" (Right column). These plots demonstrate the convergence of the iteration process

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Table 1. Augmented Lagrangian method for the $ECV$-$L^{1}$ segmentation model

1. Initialization: $\phi^{0}$, $q^{0}$, $\mathbf{p}^{0}$, $\mathbf{n}^{0}$, and $\mathbf{\lambda}_{1}^{0}$, $\lambda_{2}^{0}$, $\mathbf{\lambda}_{3}^{0}$.

For $k \geq 1$, do the following steps (Step $2\sim 4$):

2. Compute an approximate minimizer $(\phi^{k},q^{k},\mathbf{p}^{k},\mathbf{n}^{k})$ of the augmented Lagrangian functional with the fixed Lagrangian multiplier $\mathbf{\lambda}_{1}^{k-1}$, $\lambda_{2}^{k-1}$, $\mathbf{\lambda}_{3}^{k-1}$:

$(32)\;\;\;\;\;\;\;\;\;\;(\phi^{k},q^{k},\mathbf{p}^{k},\mathbf{n}^{k}) \approx \mbox{argmin } \mathcal{L}(\phi,q,\mathbf{p},\mathbf{n}, \mathbf{\lambda}_{1}^{k-1}, \lambda_{2}^{k-1}, \mathbf{\lambda}_{3}^{k-1}).$

3. Update the Lagrangian multipliers

$\mathbf{\lambda}_{1}^{k} = \mathbf{\lambda}_{1}^{k-1}+r_{1}(\mathbf{p}^{k}-\nabla \phi^{k})$

$\lambda_{2}^{k} = \lambda_{2}^{k-1}+r_{2}(q^{k}-\nabla\cdot \mathbf{n}^{k})$

$\mathbf{\lambda}_{3}^{k} = \mathbf{\lambda}_{3}^{k-1}+r_{3}(|\mathbf{p}|\mathbf{n}^{k}-\mathbf{p}^{k}),$

4. Measure the relative residuals and stop the iteration if they are smaller than a threshold $\epsilon_{r}$.

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Table 2. Alternating minimization method for solving the subproblems.

1. Initialization: $\widetilde{\phi}^{0}=\phi^{k-1}$, $\widetilde{q}^{0}=q^{k-1}$, $\widetilde{\mathbf{p}}^{0}=\mathbf{p}^{k-1}$, and $\widetilde{\mathbf{n}}^{0}=\mathbf{n}^{k-1}$.

2. For fixed Lagrangian multiplier $\mathbf{\lambda}_{1}=\mathbf{\lambda}_{1}^{k-1}$, $\lambda_{2}=\lambda_{2}^{k-1}$, and $\mathbf{\lambda}_{3}=\mathbf{\lambda}_{3}^{k-1}$, solve the following subproblems :

$(33)\;\;\;\;\;\;\;\;\;\;\;\;\widetilde{\phi}^{1} = \mbox{argmin } \mathcal{L}(\phi,\widetilde{q}^{0},\widetilde{\mathbf{p}}^{0},\widetilde{\mathbf{n}}^{0}, \mathbf{\lambda}_{1}, \lambda_{2}, \mathbf{\lambda}_{3}) $

$(34)\;\;\;\;\;\;\;\;\;\;\;\;\widetilde{q}^{1} = \mbox{argmin } \mathcal{L}(\widetilde{\phi}^{1},q,\widetilde{\mathbf{p}}^{0},\widetilde{\mathbf{n}}^{0}, \mathbf{\lambda}_{1}, \lambda_{2}, \mathbf{\lambda}_{3}) $

$(35)\;\;\;\;\;\;\;\;\;\;\;\;\widetilde{\mathbf{p}}^{1} = \mbox{argmin } \mathcal{L}(\widetilde{\phi}^{1},\widetilde{q}^{1},\mathbf{p},\widetilde{\mathbf{n}}^{0}, , \mathbf{\lambda}_{1}, \lambda_{2}, \mathbf{\lambda}_{3}) $

$(36)\;\;\;\;\;\;\;\;\;\;\;\;\widetilde{\mathbf{n}}^{1} = \mbox{argmin } \mathcal{L}(\widetilde{\phi}^{1},\widetilde{q}^{1},\widetilde{\mathbf{p}}^{1},\mathbf{n}, \mathbf{\lambda}_{1}, \lambda_{2}, \mathbf{\lambda}_{3}) $

3. $(\phi^{k},q^{k},\mathbf{p}^{k},\mathbf{n}^{k})=(\widetilde{\phi}^{1},\widetilde{q}^{1},\widetilde{\mathbf{p}}^{1}, \widetilde{\mathbf{n}}^{1})$.

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Table 3. The presentation of the sizes, the numbers of iterations, and the computational time for the two real images.

Image	Size of image	Number of iterations	Time (sec)
Fig. 3	$321\times 481$	1600	472.3
Fig. 4	$481\times 321$	1600	480.8

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