European Journal of Applied Mathematics ( IF 1.9 ) Pub Date : 2021-04-14 , DOI: 10.1017/s0956792521000061 J.M BUDD 1 , Y. VAN GENNIP 1
An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.
中文翻译:
图上基于质量守恒扩散的动力学
图像分割、半监督学习和一般分类问题中的一种新兴技术涉及使用在有限图上定义的相分离流。该技术在 Bertozzi 和 Flenner (2012, Multiscale Modeling and Simulation 10 (3), 1090-1118) 中率先使用,在图上使用了 Allen-Cahn 流,然后在 Merkurjev 等人中得到了扩展。(2013, SIAM J. Imaging Sci. 6 (4), 1903–1930) 在图上使用 Merriman-Bence-Osher (MBO) 方案。在作者之前的工作中,Budd 和 Van Gennip (2020, SIAM J. Math. Anal. 52(5), 4101-4139),我们给出了使用 MBO 方案代替 Allen-Cahn 流的理论理由,表明 MBO 方案是 Allen- 的“半离散”数值方案的特例。卡恩流。在本文中,我们扩展了这项早期工作,表明通过半离散方案的这种链接对于传递到质量守恒情况是稳健的。灵感来自 Rubinstein 和 Sternberg (1992, IMA J. Appl. Math. 48, 249–264),我们在图上定义了一个质量守恒的 Allen-Cahn 方程。然后,在凸优化工具的帮助下,我们证明了我们早期的机器可以用于推导图上的质量守恒 MBO 方案,作为质量守恒 Allen-Cahn 的半离散方案的特例。我们对这种流动和方案进行了理论分析,证明了流动的存在性和唯一性以及方案的收敛性等各种期望性质,并表明半离散方案产生了质量守恒 MBO 方案的选择函数.