TY - JOUR
T1 - Selection of Regularization Parameter in the Ambrosio-Tortorelli Approximation of the Mumford-Shah Functional for Image Segmentation
AU - Yufei Yu & Weizhang Huang
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 211
EP - 234
PY - 2018
DA - 2018/11
SN - 11
DO - http://doi.org/10.4208/nmtma.OA-2017-0074
UR - https://global-sci.org/intro/article_detail/nmtma/12427.html
KW - 
AB - <p style="text-align: justify;">The Ambrosio-Tortorelli functional is a phase-field approximation of the Mumford-Shah
functional that has been widely used for image segmentation. It has the
advantages of being easy to implement, maintaining the segmentation ability, and Γ-converging
to the Mumford-Shah functional as the regularization parameter goes to zero.
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 behaviors 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 treatd
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.</p>