# American Institute of Mathematical Sciences

March  2021, 14(3): 851-863. doi: 10.3934/dcdss.2020347

## A new numerical method for level set motion in normal direction used in optical flow estimation

 Faculty of Civil Engineering, Slovak University of Technology, Department of Mathematics and Descriptive Geometry, Radlinského 11,810 05 Bratislava, Slovak Republic

Received  December 2018 Revised  December 2019 Published  March 2021 Early access  May 2020

Fund Project: Work supported by grants VEGA 1/0728/15, APVV-15-0522 and APVV-16-0431. The authors are grateful for a support of company Tatramed in Bratislava, Slovakia

We present a new numerical method for the solution of level set advection equation describing a motion in normal direction for which the speed is given by the sign function of the difference of two given functions. Taking one function as the initial condition, the solution evolves towards the second given function. One of possible applications is an optical flow estimation to find a deformation between two images in a video sequence. The new numerical method is based on a bilinear interpolation of discrete values as used for the representation of images. Under natural assumptions, it ensures a monotone decrease of the absolute difference between the numerical solution and the target function, and it handles properly the discontinuity in the speed due to the dependence on the sign function. To find the deformation between two functions (or images), the backward tracking of characteristics is used. Two numerical experiments are presented, one with an exact solution to show an experimental order of convergence and one based on two images of lungs to illustrate a possible application of the method for the optical flow estimation.

Citation: Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347
##### References:

show all references

##### References:
The example with exact solution: the function $F$ (left), the function $G$ (middle), and the deformation $-\vec{D}^{ex}$ (right)
Comparison of the exact deformation $\vec{D}^{ex}$ (the blue arrows) with the numerical one $\vec{D}$ (the red arrows) for the discretization steps $h = 0.1$ (top left), $h = 0.05$ (top right), $h = 0.025$ (bottom left), and $h = 0.0125$ (bottom right)
The images of lungs scan: the source image $F$ (left), the target image $G$ (middle), the difference image $|G-F|$ (right)
The plot and the table of the normalized norm $e^n$ in (36)
The image given by the values $F(x_{ij}-\vec{D}_{ij})$ (left) and the difference image given by the values $|G_{ij}-F(x_{ij}-\vec{D}_{ij})|$ (right)
The plot of deformation $\vec{D}$ for the example with the images of lungs
The plot of deformation $\vec{D}$ for the example with the images of lungs. Only arrows in the points where $|G_{ij}-F_{ij}| > E_{crit}$ are plotted
The error norm (35) and the corresponding $EOC$ for the example with exact solution
 $I$ $N$ $E_{RT}$ EOC $E_{CTU}$ EOC 10 1 0.00703 - 0.00312 - 20 2 0.00401 0.81 0.00171 0.87 40 4 0.00235 0.77 0.000893 0.94 80 8 0.00160 0.55 0.000458 0.96 160 16 0.00281 -0.81 0.000232 0.98
 $I$ $N$ $E_{RT}$ EOC $E_{CTU}$ EOC 10 1 0.00703 - 0.00312 - 20 2 0.00401 0.81 0.00171 0.87 40 4 0.00235 0.77 0.000893 0.94 80 8 0.00160 0.55 0.000458 0.96 160 16 0.00281 -0.81 0.000232 0.98
The $\text{L}_1$-norms and the experimental rates of convergence for the example with exact solution
 $I$ $N$ $E_L$ EOC $E_D$ EOC 10 1 0.003120 - 0.004433 - 20 2 0.001307 1.2556 0.002379 0.8985 40 4 0.000528 1.3075 0.001259 0.9175 80 8 0.000220 1.2667 0.000659 0.9339 160 16 0.000096 1.1947 0.000339 0.9590
 $I$ $N$ $E_L$ EOC $E_D$ EOC 10 1 0.003120 - 0.004433 - 20 2 0.001307 1.2556 0.002379 0.8985 40 4 0.000528 1.3075 0.001259 0.9175 80 8 0.000220 1.2667 0.000659 0.9339 160 16 0.000096 1.1947 0.000339 0.9590
 [1] Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 [2] Yoshikazu Giga, Hiroyoshi Mitake, Hung V. Tran. Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3983-3999. doi: 10.3934/dcdsb.2019228 [3] Peter Frolkovič, Karol Mikula, Jozef Urbán. Distance function and extension in normal direction for implicitly defined interfaces. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 871-880. doi: 10.3934/dcdss.2015.8.871 [4] Lukas F. Lang, Otmar Scherzer. Optical flow on evolving sphere-like surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 305-338. doi: 10.3934/ipi.2017015 [5] Thomas Schuster, Joachim Weickert. On the application of projection methods for computing optical flow fields. Inverse Problems & Imaging, 2007, 1 (4) : 673-690. doi: 10.3934/ipi.2007.1.673 [6] Aniello Raffaele Patrone, Otmar Scherzer. On a spatial-temporal decomposition of optical flow. Inverse Problems & Imaging, 2017, 11 (4) : 761-781. doi: 10.3934/ipi.2017036 [7] Abraão D. C. Nascimento, Leandro C. Rêgo, Raphaela L. B. A. Nascimento. Compound truncated Poisson normal distribution: Mathematical properties and Moment estimation. Inverse Problems & Imaging, 2019, 13 (4) : 787-803. doi: 10.3934/ipi.2019036 [8] Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561 [9] Tom Goldstein, Xavier Bresson, Stan Osher. Global minimization of Markov random fields with applications to optical flow. Inverse Problems & Imaging, 2012, 6 (4) : 623-644. doi: 10.3934/ipi.2012.6.623 [10] Andrei Fursikov. Stabilization of the simplest normal parabolic equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1815-1854. doi: 10.3934/cpaa.2014.13.1815 [11] Bin Dong, Aichi Chien, Yu Mao, Jian Ye, Fernando Vinuela, Stanley Osher. Level set based brain aneurysm capturing in 3D. Inverse Problems & Imaging, 2010, 4 (2) : 241-255. doi: 10.3934/ipi.2010.4.241 [12] Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems & Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029 [13] Andreas Widder. On the usefulness of set-membership estimation in the epidemiology of infectious diseases. Mathematical Biosciences & Engineering, 2018, 15 (1) : 141-152. doi: 10.3934/mbe.2018006 [14] Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic & Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237 [15] Yuan Li, Lei Yan, Lingbo Wang, Wei Hou. Estimation of normal distribution parameters and its application to carbonation depth of concrete girder bridges. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1091-1100. doi: 10.3934/dcdss.2019075 [16] Wenqing Hu, Chris Junchi Li. A convergence analysis of the perturbed compositional gradient flow: Averaging principle and normal deviations. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4951-4977. doi: 10.3934/dcds.2018216 [17] Andrei Fursikov. The simplest semilinear parabolic equation of normal type. Mathematical Control & Related Fields, 2012, 2 (2) : 141-170. doi: 10.3934/mcrf.2012.2.141 [18] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [19] Alexandre M. Bayen, Hélène Frankowska, Jean-Patrick Lebacque, Benedetto Piccoli, H. Michael Zhang. Special issue on Mathematics of Traffic Flow Modeling, Estimation and Control. Networks & Heterogeneous Media, 2013, 8 (3) : i-ii. doi: 10.3934/nhm.2013.8.3i [20] Bertram Düring, Ansgar Jüngel, Lara Trussardi. A kinetic equation for economic value estimation with irrationality and herding. Kinetic & Related Models, 2017, 10 (1) : 239-261. doi: 10.3934/krm.2017010

2020 Impact Factor: 2.425