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

*(English)*Zbl 1416.94013Summary: 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.

##### MSC:

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

65K10 | Numerical optimization and variational techniques |

49M05 | Numerical methods based on necessary conditions |

##### Keywords:

Euler’s elastica; augmented Lagrangian method; image segmentation; convex contour; variational model
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\textit{E. Bae} et al., Inverse Probl. Imaging 11, No. 1, 1--23 (2017; Zbl 1416.94013)

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