
Previous Article
On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements
 IPI Home
 This Issue

Next Article
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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
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 
[1] 
Filippo Dell'Oro, Olivier Goubet, Youcef Mammeri, Vittorino Pata. A semidiscrete scheme for evolution equations with memory. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 56375658. doi: 10.3934/dcds.2019247 
[2] 
YuanNan Young, Doron Levy. RegistrationBased Morphing of Active Contours for Segmentation of CT Scans. Mathematical Biosciences & Engineering, 2005, 2 (1) : 7996. doi: 10.3934/mbe.2005.2.79 
[3] 
Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 845855. doi: 10.3934/dcds.2006.14.845 
[4] 
Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563588. doi: 10.3934/jmd.2017022 
[5] 
Gabriel Rivière. Remarks on quantum ergodicity. Journal of Modern Dynamics, 2013, 7 (1) : 119133. doi: 10.3934/jmd.2013.7.119 
[6] 
Amy Allwright, Abdon Atangana. Augmented upwind numerical schemes for a fractional advectiondispersion equation in fractured groundwater systems. Discrete and Continuous Dynamical Systems  S, 2020, 13 (3) : 443466. doi: 10.3934/dcdss.2020025 
[7] 
Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331348. doi: 10.3934/jmd.2020012 
[8] 
Karl Grill, Christian Tutschka. Ergodicity of two particles with attractive interaction. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 48314838. doi: 10.3934/dcds.2015.35.4831 
[9] 
Henk Bruin, Gregory Clack. Inducing and unique ergodicity of double rotations. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 41334147. doi: 10.3934/dcds.2012.32.4133 
[10] 
Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287313. doi: 10.3934/jmd.2008.2.287 
[11] 
Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187208. doi: 10.3934/jmd.2008.2.187 
[12] 
Amin Boumenir. Determining the shape of a solid of revolution. Mathematical Control and Related Fields, 2019, 9 (3) : 509515. doi: 10.3934/mcrf.2019023 
[13] 
Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147167. doi: 10.3934/jmd.2015.9.147 
[14] 
Marianne Akian, Stéphane Gaubert, Antoine Hochart. Ergodicity conditions for zerosum games. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 39013931. doi: 10.3934/dcds.2015.35.3901 
[15] 
David Ralston, Serge Troubetzkoy. Ergodicity of certain cocycles over certain interval exchanges. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 25232529. doi: 10.3934/dcds.2013.33.2523 
[16] 
F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for nonuniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 7481. doi: 10.3934/era.2007.14.74 
[17] 
Yunhua Zhou. The local $C^1$density of stable ergodicity. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 26212629. doi: 10.3934/dcds.2013.33.2621 
[18] 
Mark Pollicott. Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 599604. doi: 10.3934/dcds.2002.8.599 
[19] 
Jon Aaronson, Michael Bromberg, Nishant Chandgotia. Rational ergodicity of step function skew products. Journal of Modern Dynamics, 2018, 13: 142. doi: 10.3934/jmd.2018012 
[20] 
Peter Hinow, Ami Radunskaya. Ergodicity and loss of capacity for a random family of concave maps. Discrete and Continuous Dynamical Systems  B, 2016, 21 (7) : 21932210. doi: 10.3934/dcdsb.2016043 
2020 Impact Factor: 1.639
Tools
Metrics
Other articles
by authors
[Back to Top]