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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.

References

  1. Avron, H., Sharf, A., Greif, C., Cohen-Or, D.: \(l_1\)-sparse reconstruction of sharp point set surfaces. ACM Trans. Graph. 29(5), 1–12 (2010)

    Article  Google Scholar 

  2. Belyaev, A., Ohtake, Y.: A comparison of mesh smoothing methods. In: Israel-Korea Bi-national conference on geometric modeling and computer graphics, vol. 2 (2003)

  3. Bredies, K., Holler, M., Storath, M., Weinmann, A.: Total generalized variation for manifold-valued data. SIAM J. Imag. Sci. 11(3), 1785–1848 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imag. Sci. 3(3), 492–526 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bredies, K., Sun, H.: A proximal point analysis of the preconditioned alternating direction method of multipliers. J. Optim. Theory Appl. 173(3), 878–907 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bredies, K., Valkonen, T.: Inverse problems with second-order total generalized variation constraints. Preprint arXiv:2005.09725 (2020)

  7. Brito-Loeza, C., Chen, K.: On high-order denoising models and fast algorithms for vector-valued images. IEEE Trans. Image Process. 19(6), 1518–1527 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Caselles, V., Chambolle, A., Novaga, M.: The discontinuity set of solutions of the \({\rm TV}\) denoising problem and some extensions. Multiscale Model. Simul. 6(3), 879–894 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chambolle, A., Caselles, V., Cremers, D., Novaga, M., Pock, T.: An introduction to total variation for image analysis. In: Theoretical Foundations and Numerical Methods for Sparse Recovery, pp. 263–340. de Gruyter (2010)

  10. Chambolle, A., Lions, P.L.: Image recovery via total variation minimization and related problems. Numer. Math. 76(2), 167–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chan, T., Marquina, A., Mulet, P.: High-order total variation-based image restoration. SIAM J. Sci. Comput. 22(2), 503–516 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chan, T.F., Esedoglu, S., Park, F.: A fourth order dual method for staircase reduction in texture extraction and image restoration problems. In: IEEE International Conference on Image Processing, pp. 4137–4140 (2010)

  13. Chan, T.F., Golub, G.H., Mulet, P.: A nonlinear primal-dual method for total variation-based image restoration. SIAM J. Sci. Comput. 20(6), 1964–1977 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chan, T.F., Wong, C.K.: Total variation blind deconvolution. IEEE Trans. Image Process. 7(3), 370–375 (1998)

    Article  Google Scholar 

  15. Condat, L.: Discrete total variation: New definition and minimization. SIAM J. Imag. Sci. 10(3), 1258–1290 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. De Goes, F., Butts, A., Desbrun, M.: Discrete differential operators on polygonal meshes. ACM Trans. Graph. 39(4), 110:1-110:14 (2020)

    Google Scholar 

  17. Deng, W., Yin, W.: On the global and linear convergence of the generalized alternating direction method of multipliers. J. Sci. Comput. 66(3), 889–916 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators in nD (2000)

  19. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. SIAM (1999)

  20. Glowinski, R., Le Tallec, P.: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM (1989)

  21. He, L., Schaefer, S.: Mesh denoising via \({\rm L}_0\) minimization. ACM Trans. Graph. 32(4), 1–8 (2013)

    MATH  Google Scholar 

  22. Herrmann, M., Herzog, R., Kröner, H., Schmidt, S., Vidal, J.: Analysis and an interior-point approach for \({\rm TV}\) image reconstruction problems on smooth surfaces. SIAM J. Imag. Sci. 11(2), 889–922 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Huska, M., Lanza, A., Morigi, S., Sgallari, F.: Convex non-convex segmentation of scalar fields over arbitrary triangulated surfaces. J. Comput. Appl. Math. 349, 438–451 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kang, M., Kang, M., Jung, M.: Total generalized variation based denoising models for ultrasound images. J. Sci. Comput. 72(1), 172–197 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kimmel, R.: Intrinsic scale space for images on surfaces: \({\rm T}\)he geodesic curvature flow. In: International Conference on Scale-Space Theories in Computer Vision, pp. 212–223. Springer (1997)

  26. Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation \(({\rm TGV})\) for \({\rm MRI}\). Magn. Reson. Med. 65(2), 480–491 (2011)

    Article  Google Scholar 

  27. Li, X., Li, R., Zhu, L., Fu, C.W., Heng, P.A.: \({\rm DNF}\)-\({\rm N}\)et: A deep normal filtering network for mesh denoising. IEEE Transactions on Visualization and Computer Graphics preprint (2020)

  28. Liu, L., Zhang, L., Xu, Y., Gotsman, C., Gortler, S.J.: A local/global approach to mesh parameterization. Comput. Graph. Forum 27(5), 1495–1504 (2008)

    Article  Google Scholar 

  29. Ma, J., Chen, C., Li, C., Huang, J.: Infrared and visible image fusion via gradient transfer and total variation minimization. Inf. Fusion 31, 100–109 (2016)

    Article  Google Scholar 

  30. Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Visualization and mathematics III, pp. 35–57. Springer (2003)

  31. Nikolova, M.: Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math. 61(2), 633–658 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ring, W.: Structural properties of solutions to total variation regularization problems. ESAIM: Math. Model. Numer. Anal. 34(4), 799–810 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Romano, Y., Elad, M.: Boosting of image denoising algorithms. SIAM J. Imag. Sci. 8(2), 1187–1219 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Physica D 60(1–4), 259–268 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sander, P.V., Snyder, J., Gortler, S.J., Hoppe, H.: Texture mapping progressive meshes. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, pp. 409–416 (2001)

  36. Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Springer (2009)

  37. Tasdizen, T., Whitaker, R., Burchard, P., Osher, S.: Geometric surface smoothing via anisotropic diffusion of normals. In: IEEE Visualization, pp. 125–132. IEEE (2002)

  38. Valkonen, T., Bredies, K., Knoll, F.: Total generalized variation in diffusion tensor imaging. SIAM J. Imag. Sci. 6(1), 487–525 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wali, S., Zhang, H., Chang, H., Wu, C.: A new adaptive boosting total generalized variation \(({\rm TGV})\) technique for image denoising and inpainting. J. Vis. Commun. Image Represent. 59, 39–51 (2019)

    Article  Google Scholar 

  40. Wang, P.S., Liu, Y., Tong, X.: Mesh denoising via cascaded normal regression. ACM Trans. Graph. (SIGGRAPH Asia) 35(6), 232:1-232:12 (2016)

    Google Scholar 

  41. Wang, Y., Yang, J., Yin, W., Zhang, Y.: A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imag. Sci. 1(3), 248–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wu, C., Deng, J., Chen, F., Tai, X.: Scale-space analysis of discrete filtering over arbitrary triangulated surfaces. SIAM J. Imag. Sci. 2(2), 670–709 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wu, C., Deng, J., Zhu, W., Chen, F.: Inpainting images on implicit surfaces. In: Proc. Pacific Graphics Conf, pp. 142–144 (2005)

  44. Wu, C., Tai, X.C.: Augmented lagrangian method, dual methods, and split bregman iteration for \({\rm ROF}\), vectorial \({\rm TV}\), and high order models. SIAM J. Imag. Sci. 3(3), 300–339 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wu, C., Zhang, J., Duan, Y., Tai, X.C.: Augmented lagrangian method for total variation based image restoration and segmentation over triangulated surfaces. J. Sci. Comput. 50(1), 145–166 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wu, C., Zhang, J., Tai, X.C.: Augmented lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Probl. Imaging 5(1), 237–261 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  47. Xu, L., Yan, Q., Xia, Y., Jia, J.: Structure extraction from texture via relative total variation. ACM Trans. Graph. 31(6), 1–10 (2012)

    Google Scholar 

  48. Yadav, S.K., Reitebuch, U., Polthier, K.: Robust and high fidelity mesh denoising. IEEE Trans. Visual Comput. Graphics 25(6), 2304–2310 (2019)

    Article  Google Scholar 

  49. Zhang, H., Wu, C., Zhang, J., Deng, J.: Variational mesh denoising using total variation and piecewise constant function space. IEEE Trans. Visual Comput. Graphics 21(7), 873–886 (2015)

    Article  Google Scholar 

  50. Zhang, J., Zheng, J., Wu, C., Cai, J.: Variational mesh decomposition. ACM Trans. Graph. 31(3), 1–14 (2012)

    Article  Google Scholar 

  51. Zorin, D., Schröder, P., Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. In: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, pp. 189–192 (1996)

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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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