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Large region inpainting by reweighted regularized methods
Inbetweening autoanimation via FokkerPlanck dynamics and thresholding
1.  Department of Mathematics, Duke University, Durham, NC 27708, USA 
2.  College of Marine Ecology and Environment, Shanghai Ocean University, China 
3.  Southern marine science and engineering Guangdong laboratory, Zhuhai, China 
4.  Department of Mathematics and Department of Physics, Duke University, Durham, NC 27708, USA 
We propose an equilibriumdriven deformation algorithm (EDDA) to simulate the inbetweening transformations starting from an initial image to an equilibrium image, which covers images varying from a greyscale type to a colorful type on planes or manifolds. The algorithm is based on the FokkerPlanck dynamics on manifold, which automatically incorporates the manifold structure suggested by dataset and satisfies positivity, unconditional stability, mass conservation law and exponentially convergence. The thresholding scheme is adapted for the sharp interface dynamics and is used to achieve the finite time convergence. Using EDDA, three challenging examples, (I) facial aging process, (II) coronavirus disease 2019 (COVID19) pneumonia invading/fading process, and (III) continental evolution process are computed efficiently.
References:
[1] 
M. Cong, M. Bao, Jane L. E., K. S. Bhat and R. Fedkiw, Fully automatic generation of anatomical face simulation models, In Proceedings of the 14th ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2015), 175–183. doi: 10.1145/2786784.2786786. Google Scholar 
[2] 
Q. Du, M. D. Gunzburger and L. Ju, Voronoibased finite volume methods, optimal voronoi meshes, and pdes on the sphere, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 39333957. doi: 10.1016/S00457825(03)003943. Google Scholar 
[3] 
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, NorthHolland, Amsterdam, 2000, 713–1020. doi: 10.1086/phos.67.4.188705. Google Scholar 
[4] 
Y. Gao, T. Li, X. Li and J.G. Liu, Transition path theory for langevin dynamics on manifold: optimal control and datadriven solver, arXiv: 2010.09988, 2020. Google Scholar 
[5] 
Y. Gao, J.G. Liu and N. Wu, Datadriven efficient solvers and predictions of conformational transitions for langevin dynamics on manifold in high dimensions, arXiv: 2005.12787, 2020. Google Scholar 
[6] 
Z. Leng, R. Zhu, W. Hou, Y. Feng, Y. Yang, Q. Han, G. Shan, F. Meng, D. Du, S. Wang, J. Fan, W. Wang, L. Deng, H. Shi, H.Li, Z. Hu, F. Zhang, J. Gao, H. Liu, X. Li, Y. Zhao, K. Yin, X. He, Z. Gao, Y. Wang, B. Yang, R. Jin, I. Stambler, L. W. Lim, H. Su, A. Moskalev, A. Cano, S. Chakrabarti, K.. Min, G. EllisonHughes, C. Caruso, K. Jin and R. C. Zhao, Transplantation of ACE2mesenchymal stem cells improves the outcome of patients with COVID19 pneumonia, Aging and disease, 11 (2020), 216. Google Scholar 
[7] 
The Sydney Morning Herald, July 4, 2014, 'Can your face reveal how long you'll live?', http://www.dailylife.com.au/healthandfitness/dlfitness/canyourfacerevealhowlongyoulllive201407083bjsk.html. Google Scholar 
[8] 
R. J. Renka, Algorithm 772: Stripack: Delaunay triangulation and voronoi diagram on the surface of a sphere, ACM Transactions on Mathematical Software (TOMS), 23 (1997), 416434. doi: 10.1145/275323.275329. Google Scholar 
[9] 
M. Zollhöfer, J. Thies, P. Garrido, D. Bradley, T. Beeler, P. Pérez, M. Stamminger, M. Nießner and C. Theobalt, State of the art on monocular 3d face reconstruction, tracking, and applications, In Computer Graphics Forum, volume 37. Wiley Online Library, 2018, 523–550. Google Scholar 
show all references
References:
[1] 
M. Cong, M. Bao, Jane L. E., K. S. Bhat and R. Fedkiw, Fully automatic generation of anatomical face simulation models, In Proceedings of the 14th ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2015), 175–183. doi: 10.1145/2786784.2786786. Google Scholar 
[2] 
Q. Du, M. D. Gunzburger and L. Ju, Voronoibased finite volume methods, optimal voronoi meshes, and pdes on the sphere, Computer Methods in Applied Mechanics and Engineering, 192 (2003), 39333957. doi: 10.1016/S00457825(03)003943. Google Scholar 
[3] 
R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, NorthHolland, Amsterdam, 2000, 713–1020. doi: 10.1086/phos.67.4.188705. Google Scholar 
[4] 
Y. Gao, T. Li, X. Li and J.G. Liu, Transition path theory for langevin dynamics on manifold: optimal control and datadriven solver, arXiv: 2010.09988, 2020. Google Scholar 
[5] 
Y. Gao, J.G. Liu and N. Wu, Datadriven efficient solvers and predictions of conformational transitions for langevin dynamics on manifold in high dimensions, arXiv: 2005.12787, 2020. Google Scholar 
[6] 
Z. Leng, R. Zhu, W. Hou, Y. Feng, Y. Yang, Q. Han, G. Shan, F. Meng, D. Du, S. Wang, J. Fan, W. Wang, L. Deng, H. Shi, H.Li, Z. Hu, F. Zhang, J. Gao, H. Liu, X. Li, Y. Zhao, K. Yin, X. He, Z. Gao, Y. Wang, B. Yang, R. Jin, I. Stambler, L. W. Lim, H. Su, A. Moskalev, A. Cano, S. Chakrabarti, K.. Min, G. EllisonHughes, C. Caruso, K. Jin and R. C. Zhao, Transplantation of ACE2mesenchymal stem cells improves the outcome of patients with COVID19 pneumonia, Aging and disease, 11 (2020), 216. Google Scholar 
[7] 
The Sydney Morning Herald, July 4, 2014, 'Can your face reveal how long you'll live?', http://www.dailylife.com.au/healthandfitness/dlfitness/canyourfacerevealhowlongyoulllive201407083bjsk.html. Google Scholar 
[8] 
R. J. Renka, Algorithm 772: Stripack: Delaunay triangulation and voronoi diagram on the surface of a sphere, ACM Transactions on Mathematical Software (TOMS), 23 (1997), 416434. doi: 10.1145/275323.275329. Google Scholar 
[9] 
M. Zollhöfer, J. Thies, P. Garrido, D. Bradley, T. Beeler, P. Pérez, M. Stamminger, M. Nießner and C. Theobalt, State of the art on monocular 3d face reconstruction, tracking, and applications, In Computer Graphics Forum, volume 37. Wiley Online Library, 2018, 523–550. Google Scholar 
Tolerance  
Steps (Linear)  1737  4026  6304  8581 
Steps (Threshold)  960  960  960  960 
Tolerance  
Steps (Linear)  1737  4026  6304  8581 
Steps (Threshold)  960  960  960  960 
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