## An optimization-based multilevel algorithm for variational image segmentation models

Abdul K. Jumaat and Ke Chen

### Abstract

Variational active contour models have become very popular in recent years, especially global variational models which segment all objects in an image. Given a set of user-defined prior points, selective variational models aim at selectively segmenting one object only. We are concerned with the fast solution of the latter models. Time marching methods with semi-implicit schemes (gradient descents) or additive operator splitting are used frequently to solve the resulting Euler-Lagrange equations derived from these models. For images of moderate size, such methods are effective. However, to process images of large size, urgent need exists in developing fast iterative solvers. Unfortunately, geometric multigrid methods do not converge satisfactorily for such problems. Here we propose an optimization-based multilevel algorithm for efficiently solving a class of selective segmentation models. It also applies to the solution of global segmentation models. In a level-set function formulation, our first variant of the proposed multilevel algorithm has the expected optimal $O(N\log N)$ efficiency for an image of size $n\times n$ with $N=n^2$. Moreover, modified localized models are proposed to exploit the local nature of the segmentation contours, and consequently, our second variant–-after modifications–-practically achieves super-optimal efficiency $O(\sqrt{N}\log N)$. Numerical results show that a good segmentation quality is obtained, and as expected, excellent efficiency is observed in reducing the computational time.

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### Key words

active contours, image segmentation, level-set function, multilevel, optimization methods, energy minimization

### AMS subject classifications

62H35, 65N22, 65N55, 74G65, 74G75

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