Abstract
In image processing, nonlocal operators define a new type of functionals that extend the ability of classical PDE-based algorithms in handling textures and repetitive structures. We showed that in the particular space with a fuzzy partition, the nonlocal Laplacean and partial derivatives can be represented by the \(F^0\)- and \(F^1\)-transforms. For the restoration problem specified by relatively large damaged areas, we propose a new total variation model with the F-transform-based nonlocal operators. We show that the proposed model together with the corresponding algorithm increase the quality of a (usually considered) patch-based searching algorithm.
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Notes
- 1.
Speaking formally, the Dirac’s delta is not a function, but a generalized function or a linear functional. Therefore, it makes sense to use it only with respect to its action on some function.
- 2.
Every function \(a_H(k\cdot h-t)\) is considered as a membership function of a fuzzy set, so that the whole collection \(\mathscr {A}_{h,H}\) is said to be a fuzzy partition. Each element in \(\mathscr {A}_{h,H}\) is a fuzzy set with a bounded support.
- 3.
Actually, \(\tilde{d}\) is not a distance in the pure sense of this notion. It does not fulfill the basic requirements such as “identity of indiscernibles” or “triangle inequality”.
- 4.
The term “big enough” means that a damaged area of this size cannot be reconstructed by any noise-removing technique.
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Acknowledgements
The SW and original idea to connect the filling order strategy with the technique of F-transforms belong to Dr. Pavel Vlašánek. This work was partially supported by the project LQ1602 IT4Innovations excellence in science. The implementation of the F-transform technique is available as a part of the OpenCV framework. The module fuzzy is included in opencv_contrib and available at https://github.com/itseez/opencv_ contrib.
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Perfilieva, I. (2021). F-Transform Representation of Nonlocal Operators with Applications to Image Restoration. In: Lesot, MJ., Marsala, C. (eds) Fuzzy Approaches for Soft Computing and Approximate Reasoning: Theories and Applications. Studies in Fuzziness and Soft Computing, vol 394. Springer, Cham. https://doi.org/10.1007/978-3-030-54341-9_19
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