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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

 1 Faculty of Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa-shi, Yamagata 992-8510, Japan 2 Graduate School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan 3 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.

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
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##### References:
]">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]
] 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)
(a) Jordan curve $\Gamma$      (b) Jordan polygonal curve $\mathcal{P}$
">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
(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
(Left) $V^{(0)}$ vs. time, (Right) $\alpha_\mathrm{eff}$ vs. time
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 $\boldsymbol{x}_i$
 $r_i=\|\boldsymbol{x}_i-\boldsymbol{x}_{i-1}\|$ : 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 $\boldsymbol{x}_i$
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