Volume 11, Issue 2
Selection of Regularization Parameter in the Ambrosio-Tortorelli Approximation of the Mumford-Shah Functional for Image Segmentation

Yufei Yu & Weizhang Huang

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 211-234.

Published online: 2018-11

[An open-access article; the PDF is free to any online user.]

Preview Full PDF 600 4639
Export citation
  • Abstract

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.

  • Keywords

  • AMS Subject Headings

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{NMTMA-11-211, author = {}, title = {Selection of Regularization Parameter in the Ambrosio-Tortorelli Approximation of the Mumford-Shah Functional for Image Segmentation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {2}, pages = {211--234}, abstract = {

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.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2017-0074}, url = {http://global-sci.org/intro/article_detail/nmtma/12427.html} }
TY - JOUR T1 - Selection of Regularization Parameter in the Ambrosio-Tortorelli Approximation of the Mumford-Shah Functional for Image Segmentation 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 -

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.

Yufei Yu & Weizhang Huang. (2020). Selection of Regularization Parameter in the Ambrosio-Tortorelli Approximation of the Mumford-Shah Functional for Image Segmentation. Numerical Mathematics: Theory, Methods and Applications. 11 (2). 211-234. doi:10.4208/nmtma.OA-2017-0074
Copy to clipboard
The citation has been copied to your clipboard