Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow
DCDS-S Home
Flux balanced approximation with least-squares gradient for diffusion equation on polyhedral mesh
March 2021, 14(3): 881-891. doi: 10.3934/dcdss.2020233
A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies
Maika Goto 1, , Kazunori Kuwana 1, , Yasuhide Uegata 2,, and Shigetoshi Yazaki 3,
Faculty of Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa-shi, Yamagata 992-8510, Japan
Graduate School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan
School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan
* Corresponding author. E-mail address : uegata@meiji.ac.jp (Yasuhide Uegata)
Received January 2019 Revised October 2019 Published March 2021 Early access December 2019
We propose a simple and accurate procedure how to extract the values of model parameters in a flame/smoldering evolution equation from 2D movie images of real experiments. The procedure includes a novel method of image segmentation, which can detect an expanding smoldering front as a plane polygonal curve. The evolution equation is equivalent to the so-called Kuramoto-Sivashinsky (KS) equation in a certain scale. Our results suggest a valid range of parameters in the KS equation as well as the validity of the KS equation itself.
Keywords: Image segmentation, step-function adjusted method, Kuramoto-Sivashinsky, smoldering combustion, image intensity function, evolving Jordan curve.
Mathematics Subject Classification: Primary: 35R37, 68U10, 80A25; Secondary: 53C80.
Citation: Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233
M. Beneš, M. Kimura, P. Pauš, D. Ševčovič, T. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bulletin of the Institute of Mathematics, Academia Sinica New Series, 3 (2008), 509-523. Google Scholar
C. L. Epstein and M. Gage, The curve shortening flow, Wave Motion: Theory, Modelling, and Computation (Berkeley, Calif., 1986) Mathematical Sciences Research Institute Publications, Springer, New York, 7 (1987), 15-59. doi: 10.1007/978-1-4613-9583-6_2. Google Scholar
M. L. Frankel and G. I. Sivashinsky, On the nonlinear thermal diffusive theory of curved flames, Journal de Physique, 48 (1987), 25-28. doi: 10.1051/jphys:0198700480102500. Google Scholar
M. Goto, K. Kuwana and S. Yazaki, A simple and fast numerical method for solving flame/smoldering evolution equations, JSIAM Letter, 10 (2018), 49-52. doi: 10.14495/jsiaml.10.49. Google Scholar
M. Goto, K. Kuwana, G. Kushida and S. Yazaki, Experimental and theoretical study on near-floor flame spread along a thin solid, Proceedings of the Combustion Institute, 37 (2019), 3783-3791. doi: 10.1016/j.proci.2018.06.001. Google Scholar
M. Kass, A. Witkin and D. Terzopulos, Snakes: Active contour models, Int. J. Computer Vision, 1 (1988), 321-331. doi: 10.1007/BF00133570. Google Scholar
Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progress of Theoretical Physics, 55 (1976), 356-369. doi: 10.1143/PTP.55.356. Google Scholar
K. Mikula and D. Ševčovič, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Math. Methods Appl. Sci., 27 (2004), 1545-1565. doi: 10.1002/mma.514. Google Scholar
K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Comput. Vis. Sci., 6 (2004), 211-225. doi: 10.1007/s00791-004-0131-6. Google Scholar
D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity, Japan J. Indust. Appl. Math., 28 (2011), 413-442. doi: 10.1007/s13160-011-0046-9. Google Scholar
D. Ševčovič and S. Yazaki, On a gradient flow of plane curves minimizing the anisoperimetric ratio, IAENG International J. Appl. Math., 43 (2013), 160-171. Google Scholar
G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames. I. Derivation of basic equations, Acta Astronautica, 4 (1977), 1177-1206. doi: 10.1016/0094-5765(77)90096-0. Google Scholar
N. M. Zaitoun and M. J. Aqel, Survey on image segmentation techniques, Procedia Computer Science, 65 (2015), 797-806. doi: 10.1016/j.procs.2015.09.027. Google Scholar
5]">Figure 1. The photographs depict snapshots from an experimental movie of spreading flame/smoldering front along a sheet of paper placed near the floor at 200th, 400th, 1000th, 1600th, $ \cdots $, 4000th frames at the rate of 30 fps. Experiments were performed by the same method as [5]
4] in detail)">Figure 2. The left figure depicts numerical solutions to (1), with the normal velocity (2) in which the parameters are given by the right table and $ W $ is chosen for controlling the grid-point spacing to be uniform (see section 3). The solution curves evolve from inside to outside. The initial curve is a circle with the diameter $ R = R_\mathrm{ini} $ with 10% noise (see [4] in detail)
Figure 3. (a) Jordan curve $ \Gamma $ (b) Jordan polygonal curve $ \mathcal{P} $
FIGURE 1">Figure 4. The upper-left figure depicts selected segmentation curves at frames: 400, 1000, 1600, 2200, 2800, 3400, 4000, summarizing the front evolution in the other photographs from left to right, upper to lower. The blue curve in each photograph is a segmentation curve, and the background vague region is the same as that in FIGURE 1
Figure 5. (Left) The total length of front $ \tilde{L} [\mathrm{mm}] $ vs. the actual time $ [\mathrm{second}] $. (Right) The enclosed area $ \tilde{A} [\mathrm{mm}^2] $ vs. the actual time $ [\mathrm{second}] $. Blue points indicate the actual values and red curves are the graphs of (23) and (24), respectively
Figure 6. (Left) $ V^{(0)} $ vs. time, (Right) $ \alpha_\mathrm{eff} $ vs. time
Table 1. Discretizations of length, normal/tangent vector, and normal velocity
$ r_i=\|\boldsymbol{x}_i-\boldsymbol{x}_{i-1}\| $ : The length of $ \mathcal{P}_i $
$ L= \sum\limits_{i=1}^Nr_i $ : The total length of $ \mathcal{P} $
$ \boldsymbol{t}_i=(\boldsymbol{x}_i-\boldsymbol{x}_{i-1})/r_i $ : The unit tangent vector on $ \mathcal{P}_i $
$ \boldsymbol{n}_i=-\boldsymbol{t}_i^\bot $ : The outward unit normal vector on $ \mathcal{P}_i $
$ v_i $ : A given representative normal velocity on $ \mathcal{P}_i $
$ \phi_i=\mathrm{sgn}(D_i)\arccos(\boldsymbol{t}_i\cdot\boldsymbol{t}_{i+1}) $ : The angle between the adjacent edges $ \mathcal{P}_{i} $ and $ \mathcal{P}_{i+1} $ where $ D_i=\det(\boldsymbol{t}_i, \boldsymbol{t}_{i+1}) $
$ \boldsymbol{T}_i=(\boldsymbol{t}_i+\boldsymbol{t}_{i+1})/(2\mathsf{cos}_i) $ : The unit tangent vector at $ \boldsymbol{x}_i $ where $ \mathsf{cos}_i=\cos(\phi_i/2)=\|(\boldsymbol{t}_i+\boldsymbol{t}_{i+1})/2\| $
$ \boldsymbol{N}_i=(\boldsymbol{n}_i+\boldsymbol{n}_{i+1})/(2\mathsf{cos}_i) $ : The outward unit normal vector at $ \boldsymbol{x}_i $
$ V_i=(v_i+v_{i+1})/(2\mathsf{cos}_i) $ : The normal velocity at $ \boldsymbol{x}_i $
Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386
Kiah Wah Ong. Dynamic transitions of generalized Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1225-1236. doi: 10.3934/dcdsb.2016.21.1225
Yu-Lin Chang, Chin-Yu Yang. Some useful inequalities via trace function method in Euclidean Jordan algebras. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 39-48. doi: 10.3934/naco.2014.4.39
Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021098
Milena Stanislavova, Atanas Stefanov. Effective estimates of the higher Sobolev norms for the Kuramoto-Sivashinsky equation. Conference Publications, 2009, 2009 (Special) : 729-738. doi: 10.3934/proc.2009.2009.729
Jared C. Bronski, Razvan C. Fetecau, Thomas N. Gambill. A note on a non-local Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 701-707. doi: 10.3934/dcds.2007.18.701
Peng Gao. Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5649-5684. doi: 10.3934/dcds.2018247
Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91
D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557
Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95
Yuncherl Choi, Jongmin Han, Chun-Hsiung Hsia. Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1933-1957. doi: 10.3934/dcdsb.2015.20.1933
L. Dieci, M. S Jolly, Ricardo Rosa, E. S. Van Vleck. Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 555-580. doi: 10.3934/dcdsb.2008.9.555
Peng Gao. Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2020, 9 (1) : 181-191. doi: 10.3934/eect.2020002
Fan Jia, Xue-Cheng Tai, Jun Liu. Nonlocal regularized CNN for image segmentation. Inverse Problems & Imaging, 2020, 14 (5) : 891-911. doi: 10.3934/ipi.2020041
Ye Yuan, Yan Ren, Xiaodong Liu, Jing Wang. Approach to image segmentation based on interval neutrosophic set. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 1-11. doi: 10.3934/naco.2019028
Dominique Zosso, Jing An, James Stevick, Nicholas Takaki, Morgan Weiss, Liane S. Slaughter, Huan H. Cao, Paul S. Weiss, Andrea L. Bertozzi. Image segmentation with dynamic artifacts detection and bias correction. Inverse Problems & Imaging, 2017, 11 (3) : 577-600. doi: 10.3934/ipi.2017027
Matthew S. Keegan, Berta Sandberg, Tony F. Chan. A multiphase logic framework for multichannel image segmentation. Inverse Problems & Imaging, 2012, 6 (1) : 95-110. doi: 10.3934/ipi.2012.6.95
Yuantian Xia, Juxiang Zhou, Tianwei Xu, Wei Gao. An improved deep convolutional neural network model with kernel loss function in image classification. Mathematical Foundations of Computing, 2020, 3 (1) : 51-64. doi: 10.3934/mfc.2020005
Aslihan Demirkaya. The existence of a global attractor for a Kuramoto-Sivashinsky type equation in 2D. Conference Publications, 2009, 2009 (Special) : 198-207. doi: 10.3934/proc.2009.2009.198
Peng Gao. Null controllability with constraints on the state for the 1-D Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2015, 4 (3) : 281-296. doi: 10.3934/eect.2015.4.281
Maika Goto Kazunori Kuwana Yasuhide Uegata Shigetoshi Yazaki