Second-order edge-penalization in the Ambrosio-Tortorelli functional. (English) Zbl 1332.49016
Summary: We propose and study two variants of the Ambrosio-Tortorelli functional where the first-order penalization of the edge variable \(v\) is replaced by a second-order term depending on the Hessian or on the Laplacian of \(v\), respectively. We show that both the variants above provide an elliptic approximation of the Mumford-Shah functional in the sense of \(\Gamma\)-convergence. In particular, the variant with the Laplacian penalization can be implemented numerically without any difficulties compared to the standard Ambrosio-Tortorelli functional. The computational results indicate several additional advantages. First of all, the diffuse approximation of the edge contours appears smoother and clearer for the minimizers of the second-order functional. Moreover, the convergence of alternating minimization algorithms seems improved for the new functional. We also illustrate the findings with several computational results.
MSC:
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 49M30 | Other numerical methods in calculus of variations (MSC2010) |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
| 68U10 | Computing methodologies for image processing |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
Keywords:
Ambrosio-Tortorelli approximation; free-discontinuity problems; \(\Gamma\)-convergence; variational image segmentationReferences:
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