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A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies

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  • 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.

    Mathematics Subject Classification: Primary: 35R37, 68U10, 80A25; Secondary: 53C80.


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  • 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]

    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 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 $
     | Show Table
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  • [1] M. BenešM. KimuraP. PaušD. ŠevčovičT. 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. 
    [2] 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.
    [3] 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.
    [4] M. GotoK. 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.
    [5] M. GotoK. KuwanaG. 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.
    [6] M. KassA. Witkin and D. Terzopulos, Snakes: Active contour models, Int. J. Computer Vision, 1 (1988), 321-331.  doi: 10.1007/BF00133570.
    [7] 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.
    [8] 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.
    [9] 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.
    [10] 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.
    [11] 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. 
    [12] 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.
    [13] 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.
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