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Total Generalized Variation for Triangulated Surface Data

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Abstract

By defining spaces and differential operators on mesh surfaces, we extend the total generalized variation (\(\textrm{TGV}\)) model on the 2-dimensional space to triangulated surfaces. Based on the new definition of \(\textrm{TGV}\) model on triangulated surfaces, we introduce total generalized variation restoration optimization problem for data (image pixel/surface normal) restoration over triangulated surfaces. The optimization problem is solved effectively by augmented Lagrangian method (\(\textrm{ALM}\)). Closed form solutions for subproblems of the \(\textrm{ALM}\) method are obtained. Convergence analysis of the \(\textrm{ALM}\) algorithm is presented. Through series of numerical experiments, we show that the \(\textrm{TGV}\) method can alleviate the staircase effect and recover more structures and details. As a result, the \(\textrm{TGV}\) model outperforms several existing models visually and quantitatively. The robustness of the \(\textrm{TGV}\) method is also confirmed numerically.

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Notes

  1. https://software.intel.com/content/www/us/en/develop/tools/math-kernel-library.html.

  2. https://www.tau.ac.il/~stoledo/taucs/.

  3. http://eigen.tuxfamily.org/index.php?title=Main_Page.

  4. https://wang-ps.github.io/denoising.html.

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Acknowledgements

We would like to thank Chunlin Wu for providing their data of [45], the authors of \(\textrm{CNR}\) [40] and \(\textrm{DNF}\) [27] for providing their results, the authors of \(\textrm{RoFi}\) [48] for sharing their codes, and also thank Zhifang Liu for discussing about the properties of TGV. This work was supported by the NSF of China (Nos. 61802279, 61602341) and NSF of Tianjin (Nos. 18JCQNJC00100, 17JCQNJC00600).

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  1. School of Computer Science and Technology, Tiangong University, Tianjin, China

    Huayan Zhang

  2. Department of Mathematics, Rensselaer Polytechnic Institute, Troy, USA

    Zhichao Peng

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Zhang, H., Peng, Z. Total Generalized Variation for Triangulated Surface Data. J Sci Comput 93, 87 (2022). https://doi.org/10.1007/s10915-022-02047-8

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