# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021016

## Inbetweening auto-animation via Fokker-Planck 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

* Corresponding author: Guangzhen Jin

Received  May 2020 Revised  October 2020 Published  February 2021

Fund Project: J.-G. Liu was supported in part by the National Science Foundation (NSF) under award DMS-1812573. G. Jin was supported in part by the the Natural Science Foundation of Guangdong Province under award 2019A1515011487, the China Postdoctoral Science Foundation under award 2020M681269 and the Fundamental Research Funds for the Central Universities under award 20184200031610059

We propose an equilibrium-driven 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 Fokker-Planck 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 (COVID-19) pneumonia invading/fading process, and (III) continental evolution process are computed efficiently.

Citation: Yuan Gao, Guangzhen Jin, Jian-Guo Liu. Inbetweening auto-animation via Fokker-Planck dynamics and thresholding. Inverse Problems & Imaging, doi: 10.3934/ipi.2021016
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The semilog plot of temporal evolution of relative root mean square errors for the RGB facial aging transformation with parameters $\Delta t = 0.01, T = 100$ and $\Delta x = \Delta y = 10^{-4}$. The red, green and blue lines represent the relative errors of the corresponding color modes
Facial aging transformation from initial to equilibrium with parameters $\Delta t = 0.01, T = 100$ and $\Delta x = \Delta y = 10^{-4}$. The updated results after time step $40,100$, $200$, 400, $1000, 2000$, $4000$, $10000$ are shown and compared to the initial and equilibrium images
]. The left column of figures illustrates the evading of pneumonia from January 23th to February 2nd and the right column illustrates the fading away of pneumonia from February 2nd to February 15th after treatment. Red circles indicate the significant COVID-19 pneumonia invading areas and blue circles indicate the significant pneumonia fading away areas">Figure 3.  Chest CT images of the critically severe COVID-19 patient [6]. The left column of figures illustrates the evading of pneumonia from January 23th to February 2nd and the right column illustrates the fading away of pneumonia from February 2nd to February 15th after treatment. Red circles indicate the significant COVID-19 pneumonia invading areas and blue circles indicate the significant pneumonia fading away areas
The semilog plot of temporal evolution of the relative root mean square errors for COVID-19 pneumonia invading and fading away on CT scan images with the parameters $\Delta t = 0.01, T = 100$ and $\Delta x = \Delta y = 10^{-4}$. (up) The error evolution for the pneumonia invading process simulation. (down) The error evolution for the pneumonia fading away process simulation
The simulated COVID-19 pneumonia invading and fading away process on CT scan images with the parameters $\Delta t = 0.01, T = 100$ and $\Delta x = \Delta y = 10^{-4}$. Results after the step 20, 50, 100, 200, 500, 1000, 2000, 5000 are illustrated and compared to the initial and equilibrium scan images. The white part inside the lungs shown on images indicates the evidence of pneumonia. (up) The pneumonia invades into the patient's lungs caused by COVID-19. (down) The pneumonia fades away from the lungs after a stem cell treatment is applied to the patient. Red circles indicate the significant COVID-19 pneumonia invading areas and blue circles indicate the significant pneumonia fading away areas
The unit sphere and the Voronoi cells on it. There are totally 3000 cells on the sphere. The black dots indicate locations of the point clouds on the sphere. The polygons with black edges are the Voronoi polygons generated with CVT algorithm
The semilog plot of temporal evolution of relative root mean square errors for the continental evolution process with thresholding for shape dynamics with $\Delta t = 0.05$ and the total number of the thresholding adjustments is $N_t = 50$. The linear iterations before the $(k+1)$th thresholding adjustment is $2k$. The red line indicates the error of simulations with only the linear Fokker-Planck algorithm while the blue line is the error of the linear algorithm combined with the thresholding scheme. The black dotted lines indicate the time steps when the thresholding adjustments are applied
The evolutions of continental movements on the unit sphere with the parameter $\Delta t = 0.05$ and the total number of the thresholding adjustments is $N_t = 50$. The continental evolution on the sphere after the 2th, 5th, 10th, 30th thresholding adjustment are illustrated and compared with the initial and equilibrium states. The black dots and polygons in each subplot illustrate the point clouds and the Voronoi cells, respectively. The orange and blue patch indicate the land and ocean, respectively. 'TH' is short for 'thresholding step'. The formation of the Antarctic is revealed at the bottom (southern part) of the globe (black arrow in TH 5). Note that the globes are shown in the same view angle so the Antarctic continental is out of view in the last two subplots
Comparison of the time steps needed to reach the torenlence between linear method and threshold method
 Tolerance $10^{-3}$ $10^{-4}$ $10^{-5}$ $10^{-6}$ Steps (Linear) 1737 4026 6304 8581 Steps (Threshold) 960 960 960 960
 Tolerance $10^{-3}$ $10^{-4}$ $10^{-5}$ $10^{-6}$ Steps (Linear) 1737 4026 6304 8581 Steps (Threshold) 960 960 960 960
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