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March  2021, 14(3): 1017-1032. doi: 10.3934/dcdss.2020348

## An automated segmentation of NATURA 2000 habitats from Sentinel-2 optical data

 1 Department of Mathematics, Slovak University of Technology, Radlinského 11,810 05 Bratislava, Slovakia, Algoritmy:SK, s.r.o., Šulekova 6,811 06 Bratislava, Slovakia 2 Institute of Botany, Slovak Academy of Sciences, Dúbravská cesta 9,845 23 Bratislava, Slovakia

* Corresponding author

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

Fund Project: This work was supported by projects APVV-16-0431, APVV-15-0522, VEGA 1/0608/15 and ESA Contract No. 4000122575/17/NL/SC

In this paper, we present a mathematical model and numerical method designed for the segmentation of satellite images, namely to obtain in an automated way borders of Natura 2000 habitats from Sentinel-2 optical data. The segmentation model is based on the evolving closed plane curve approach in the Lagrangian formulation including the efficient treatment of topological changes. The model contains the term expanding the curve in its outer normal direction up to the region of habitat boundary edges, the term attracting the curve accurately to the edges and the smoothing term given by the influence of local curvature. For the numerical solution, we use the flowing finite volume method discretizing the arising advection-diffusion intrinsic partial differential equation including the asymptotically uniform tangential redistribution of curve grid points. We present segmentation results for satellite data from a selected area of Western Slovakia (Záhorie) where the so-called riparian forests represent the important European Natura 2000 habitat. The automatic segmentation results are compared with the semi-automatic segmentation performed by the botany expert and with the GPS tracks obtained in the field. The comparisons show the ability of our numerical model to segment the habitat areas with the accuracy comparable to the pixel resolution of the Sentinel-2 optical data.

Citation: Karol Mikula, Jozef Urbán, Michal Kollár, Martin Ambroz, Ivan Jarolímek, Jozef Šibík, Mária Šibíková. An automated segmentation of NATURA 2000 habitats from Sentinel-2 optical data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1017-1032. doi: 10.3934/dcdss.2020348
##### References:
 [1] M. Ambroz, M. Balažovjech, M. Medla and K. Mikula, Numerical modeling of wildland surface fire propagation by evolving surface curves, Adv. Comput. Math., 45 (2019), 1067-1103.  doi: 10.1007/s10444-018-9650-4. [2] M. Balažovjech, K. Mikula, M. Petrášová and J. Urbán, Lagrangean method with topological changes for numerical modelling of forest fire propagation, 19th Conference on Scientific Computing, Slovakia, 2012. [3] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Internat. J. Comput. Vision, 22 (1997), 61-79.  doi: 10.1109/ICCV.1995.466871. [4] E. Faure et al., A workflow to process 3D+time microscopy images of developing organisms and reconstruct their cell lineage, Nat. Commun., 7 (2016). doi: 10.1038/ncomms9674. [5] M.A. Finney, et al., FARSITE: Fire Area Simulator–model development and evaluation, U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station, Ogden, UT, 1998. doi: 10.2737/RMRS-RP-4. [6] T.Y. Hou, J. Lowengrub and M. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys., 114 (1994), 312-338.  doi: 10.1006/jcph.1994.1170. [7] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Conformal curvature flows: From phase transitions to active vision, Arch. Rational Mech. Anal., 134 (1996), 275-301.  doi: 10.1007/BF00379537. [8] M. Kimura, Numerical analysis for moving boundary problems using the boundary tracking method, Japan J. Indust. Appl. Math., 14 (1997), 373-398.  doi: 10.1007/BF03167390. [9] K. Mikula and M. Ohlberger, Inflow-implicit/outflow-explicit scheme for solving advection equations, in Finite Volumes for Complex Applications VI. Problems & Perspectives. Volume 1, 2, Springer Proc. Math., 4, Springer, Heidelberg, 2011,683–691. doi: 10.1007/978-3-642-20671-9_72. [10] K. Mikula, M. Ohlberger and J. Urbán, Inflow-implicit/outflow-explicit finite volume methods for solving advection equations, Appl. Numer. Math., 85 (2014), 16-37.  doi: 10.1016/j.apnum.2014.06.002. [11] K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), 1473-1501.  doi: 10.1137/S0036139999359288. [12] 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. [13] K. Mikula, D. Ševčovič and M. Balažovjech, A simple, fast and stabilized flowing finite volume method for solving general curve evolution equations, Commun. Comput. Phys., 7 (2010), 195-211.  doi: 10.4208/cicp.2009.08.169. [14] K. Mikula and J. Urbán, New fast and stable Lagrangean method for image segmentation, 5th International Congress on Image and Signal Processing, Chongqing, China, 2012. doi: 10.1109/CISP.2012.6469852. [15] K. Mikula, et al., Report on semi-automatic segmentation methods and software tool for static data, ESA PECS project NaturaSat Deliverable 2.1, 2018. [16] G. Nakamura and R. Potthast, Inverse Modeling, IOP Expanding Physics, IOP Publishing, Bristol, 2015, 2053–2563. doi: 10.1088/978-0-7503-1218-9. [17] P. Pauš, M. Beneš, Algorithm for topological changes of parametrically described curves, Proceedings of ALGORITMY, 2009,176–184. [18] A. Sarti, R. Malladi and J. A. Sethian:, Subjective surfaces: A method for completing missing boundaries, Proc. Natl. Acad. Sci. USA, 97 (2000), 6258-6263.  doi: 10.1073/pnas.110135797.

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##### References:
 [1] M. Ambroz, M. Balažovjech, M. Medla and K. Mikula, Numerical modeling of wildland surface fire propagation by evolving surface curves, Adv. Comput. Math., 45 (2019), 1067-1103.  doi: 10.1007/s10444-018-9650-4. [2] M. Balažovjech, K. Mikula, M. Petrášová and J. Urbán, Lagrangean method with topological changes for numerical modelling of forest fire propagation, 19th Conference on Scientific Computing, Slovakia, 2012. [3] V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, Internat. J. Comput. Vision, 22 (1997), 61-79.  doi: 10.1109/ICCV.1995.466871. [4] E. Faure et al., A workflow to process 3D+time microscopy images of developing organisms and reconstruct their cell lineage, Nat. Commun., 7 (2016). doi: 10.1038/ncomms9674. [5] M.A. Finney, et al., FARSITE: Fire Area Simulator–model development and evaluation, U.S. Department of Agriculture, Forest Service, Rocky Mountain Research Station, Ogden, UT, 1998. doi: 10.2737/RMRS-RP-4. [6] T.Y. Hou, J. Lowengrub and M. Shelley, Removing the stiffness from interfacial flows with surface tension, J. Comput. Phys., 114 (1994), 312-338.  doi: 10.1006/jcph.1994.1170. [7] S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Conformal curvature flows: From phase transitions to active vision, Arch. Rational Mech. Anal., 134 (1996), 275-301.  doi: 10.1007/BF00379537. [8] M. Kimura, Numerical analysis for moving boundary problems using the boundary tracking method, Japan J. Indust. Appl. Math., 14 (1997), 373-398.  doi: 10.1007/BF03167390. [9] K. Mikula and M. Ohlberger, Inflow-implicit/outflow-explicit scheme for solving advection equations, in Finite Volumes for Complex Applications VI. Problems & Perspectives. Volume 1, 2, Springer Proc. Math., 4, Springer, Heidelberg, 2011,683–691. doi: 10.1007/978-3-642-20671-9_72. [10] K. Mikula, M. Ohlberger and J. Urbán, Inflow-implicit/outflow-explicit finite volume methods for solving advection equations, Appl. Numer. Math., 85 (2014), 16-37.  doi: 10.1016/j.apnum.2014.06.002. [11] K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy, SIAM J. Appl. Math., 61 (2001), 1473-1501.  doi: 10.1137/S0036139999359288. [12] 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. [13] K. Mikula, D. Ševčovič and M. Balažovjech, A simple, fast and stabilized flowing finite volume method for solving general curve evolution equations, Commun. Comput. Phys., 7 (2010), 195-211.  doi: 10.4208/cicp.2009.08.169. [14] K. Mikula and J. Urbán, New fast and stable Lagrangean method for image segmentation, 5th International Congress on Image and Signal Processing, Chongqing, China, 2012. doi: 10.1109/CISP.2012.6469852. [15] K. Mikula, et al., Report on semi-automatic segmentation methods and software tool for static data, ESA PECS project NaturaSat Deliverable 2.1, 2018. [16] G. Nakamura and R. Potthast, Inverse Modeling, IOP Expanding Physics, IOP Publishing, Bristol, 2015, 2053–2563. doi: 10.1088/978-0-7503-1218-9. [17] P. Pauš, M. Beneš, Algorithm for topological changes of parametrically described curves, Proceedings of ALGORITMY, 2009,176–184. [18] A. Sarti, R. Malladi and J. A. Sethian:, Subjective surfaces: A method for completing missing boundaries, Proc. Natl. Acad. Sci. USA, 97 (2000), 6258-6263.  doi: 10.1073/pnas.110135797.
First row: the original image $I^0$ and smoothed image $I^{\sigma_0}$. Second row: the visualization of $g(\mathbf{x})$, smoothed edge detector $g_1(\mathbf{x})$ and a zoom of the vector field $-\nabla g_1(\mathbf{x})$ where we see arrows pointing towards the edges in $I^0$. Third row: the functions $H(\mathbf{x})$ using (5) and $g_2(\mathbf{x} )$ evaluated by using the initial circle plotted in the Fourth row, left. Fourth row: the initial segmentation curve placed in $I^0$ and its time evolution until the final segmentation state (bottom right). In the middle image we see that the evolving curve undergoes topological changes which are resolved efficiently
Closed planar curve discretization (left) corresponding to the uniform discretization of the unit circle (right)
Visualization of the curve discretization: curve grid points (red) and their midpoints. Finite volumes $\mathbf{p}_{i-1}, \mathbf{p}_i,$ and $\mathbf{p}_{i+1}$ are highlighted by green, brown and yellow color. Note that $\mathbf{p}_i$ is not a straight line given by $\mathbf{x}_{i-\frac{1}2}$ and $\mathbf{x}_{i+\frac{1}2}$, but a broken line given by $\mathbf{x}_{i-\frac{1}2}$, $\mathbf{x}_{i}$ and $\mathbf{x}_{i+\frac{1}2}$, see also [1]
First row: the original image $I^0$ and smoothed image $I^{\sigma_0}$. Second row: the visualization of the function $g(\mathbf{x})$ and smoothed edge detector $g_1(\mathbf{x})$. Third row: the function $H(\mathbf{x})$ and $g_2(\mathbf{x})$ evaluated by using the initial circle plotted in the Fourth row, left. Fourth row: the initial segmentation curve placed in $I^0$ (bottom left) and its time evolution (bottom middle) until the final segmentation state (bottom right) is reached
First row: the original image $I^0$ and smoothed image $I^{\sigma_0}$. Second row: the visualization of the function $g(\mathbf{x})$ and smoothed edge detector $g_1(\mathbf{x})$. Third row: the function $H(\mathbf{x})$ and $g_2(\mathbf{x})$ evaluated by using the initial circle plotted in the Fourth row, left. Fourth row: the initial segmentation curve placed in $I^0$ (bottom left) and its time evolution (bottom middle) until the final segmentation state (bottom right) is reached
Left: the evolution of the segmentation curve from the initial circle to the final state. Right: the final automatic segmentation (red) together with the result of the semi-automatic segmentation (yellow) and the GPS track (light-blue)
Left: the evolution of the segmentation curve from the initial circle to the final state. Right: the final automatic segmentation (red) together with the result of the semi-automatic segmentation (yellow) and the GPS track (light-blue)
Left: the evolution of the segmentation curve from the initial circle to the final state. Right: the final automatic segmentation (red) together with the result of the semi-automatic segmentation (yellow) and the GPS track (light-blue)
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