Color Image Segmentation by the Vector-Valued Allen–Cahn Phase-Field Model: A Multigrid Solution

Abstract

We present an efficient numerical solution of a PDE-driven model for color image segmentation and give numerical examples of the results. The method combines the vector-valued Allen-Cahn phase field equation with initial data fitting terms with prescribed interface width and fidelity constants. Efficient numerical solution is achieved using a multigrid splitting of a finite element space, thereby producing an efficient and robust method for the segmentation of large images. We also present the use of adaptive mesh refinement to further speed up the segmentation process.

Preconditioning for Allen–Cahn variational inequalities with non-local constraints

“…This idea has been extended for the vector-valued Allen-Cahn equation [34]. Recently,in [30] this technique is applied to problems in image segmentation. A review on various multigrid methods for obstacle problems can be found in [23].…”

“…The scalar Allen-Cahn equation describes the motion of an interface separating two phases. In practical applications often more than two phases occur [7,18,30] and the phase field concept has been extended to deal with multi phase systems [17]. There a vector-valued order parameter u : Ω → R N is introduced, where each u i describes one phase, i.e.…”

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

“…The Allen-Cahn model (or phase field model) has a variety of applications, e.g. in materials science, image processing, biology and geology, see [6,11,14,23,25,30,39]. In many of these applications more than two phases occur.…”

2009 Index IEEE Transactions on Image Processing Vol. 18

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