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Approximation of functional depending on jumps by elliptic functional via t‐convergence
Abstract
We show how it is possible to approximate the Mumford‐Shah (see [29]) image segmentation functional magnified image by elliptic functionals defined on Sobolev spaces. The heuristic idea is to consider functionals 𝒢h(u, z) with z ranging between 0 and 1 and related to the set K. The minimizing zh are near to 1 in a neighborhood of the set K, and far from the neighborhood they are very small. The neighborhood shrinks as h → + ∞. For a similar approach to the problem compare Kulkarni; see [25]. The approximation of 𝒢h to 𝒢 takes place in a variational sense, the De Giorgi F‐convergence.
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Marr, D., & Hildreth, E. (1980) · Proceedings of the Royal Society of London. Series B. Biological Sciences · doi:10.1098/rspb.1980.0020 · Cited by 6,510
Cited by
A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model
Vese, L. A., & Chan, T. F. (2002) · International Journal of Computer Vision · doi:10.1023/a:1020874308076 · Cited by 2,384
Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models
Chan, T. F., Esedoglu, S., & Nikolova, M. (2006) · SIAM Journal on Applied Mathematics · doi:10.1137/040615286 · Cited by 1,049
Nguyen, T. T., Yvonnet, J., Zhu, Q.-Z., Bornert, M., & Chateau, C. (2016) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2015.10.007 · Cited by 359
Hofacker, M., & Miehe, C. (2013) · International Journal for Numerical Methods in Engineering · doi:10.1002/nme.4387 · Cited by 306
A parallel generalized relaxation method for high-performance image segmentation on GPUs
D’Ambra, P., & Filippone, S. (2016) · Journal of Computational and Applied Mathematics · doi:10.1016/j.cam.2015.04.035 · Cited by 19
Metadata
- Authors
- Ambrosio, L., & Tortorelli, V. M.
- Venue
- Communications on Pure and Applied Mathematics
- Publisher
- Wiley
- Published
- 1990-12
- Volume
- 43
- Issue
- 8
- Pages
- 999-1036
- ISSN
- 0010-3640 (ISSN-L)
- 1097-0312
- OpenAlex
- W2155280009
- Cited by
- 1,471
- References
- 1
- FWCI
- 5.76
- Type
- journal article
- Language
- en
- Open access
- closed
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- Ambrosio, L., & Tortorelli, V. M. (1990). Approximation of functional depending on jumps by elliptic functional via t‐convergence. Communications on Pure and Applied Mathematics. https://doi.org/10.1002/cpa.3160430805
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