在图像分割中的Ambrosio-Tortorelli近似中正则化参数的选择
Selection of the Regularization Parameter in the Ambrosio-Tortorelli Approximation of the Mumford-Shah Functional for Image Segmentation
AI总结：
本文研究了Ambrosio-Tortorelli函数在图像分割中的正则化参数选择问题，通过渐近分析揭示了正则化参数对分割行为的影响，并提出了一种参数选择策略和解的缩放方法，以提高分割效果。
摘要：
Ambrosio-Tortorelli函数是Mumford-Shah函数的一种相场近似，已被广泛用于图像分割。该近似具有易于实现、保持分割能力以及Γ收敛到Mumford-Shah函数的优点。然而，在实际计算中发现，Ambrosio-Tortorelli函数的分割能力会随着参数的不同而显著变化，并且在某些情况下甚至无法Γ收敛到原始函数。本文对Ambrosio-Tortorelli函数的梯度流方程进行了渐近分析，证明了在小但有限的正则化参数值下，函数具有不同的分割行为，当参数趋于零时，分割能力会丧失。这与现有观察以及本文中的数值示例一致。基于分析，提出了一种正则化参数选择策略和解的缩放过程。数值结果表明，这些方法能够有效提高Ambrosio-Tortorelli函数在真实图像中的分割效果。
英文摘要：
The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah functional that has been widely used for image segmentation. The approximation has the advantages of being easy to implement, maintaining the segmentation ability, and $Γ$-converging to the Mumford-Shah functional. However, it has been observed in actual computation that the segmentation ability of the Ambrosio-Tortorelli functional varies significantly with different values of the parameter and it even fails to $Γ$-converge to the original functional for some cases. In this paper we present an asymptotic analysis on the gradient flow equation of the Ambrosio-Tortorelli functional and show that the functional can have different segmentation behavior for small but finite values of the regularization parameter and eventually loses its segmentation ability as the parameter goes to zero when the input image is treated as a continuous function. This is consistent with the existing observation as well as the numerical examples presented in this work. A selection strategy for the regularization parameter and a scaling procedure for the solution are devised based on the analysis. Numerical results show that they lead to good segmentation of the Ambrosio-Tortorelli functional for real images.
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