American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Semi-automatic segmentation of NATURA 2000 habitats in Sentinel-2 satellite images by evolving open curves
  • DCDS-S Home
  • This Issue
  • Next Article
    Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems
March  2021, 14(3): 1017-1032. doi: 10.3934/dcdss.2020348

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

Karol Mikula 1,, , Jozef Urbán 1, , Michal Kollár 1, , Martin Ambroz 1, , Ivan Jarolímek 2, , Jozef Šibík 2, and  Mária Šibíková 2,

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  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.

Keywords: Image segmentation, curve evolution, numerical method, Natura 2000, satellite images, Sentinel-2.
Mathematics Subject Classification: Primary: 35R01, 65M08; Secondary: 35R37, 92F05.
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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

P. Pauš, M. Beneš, Algorithm for topological changes of parametrically described curves, Proceedings of ALGORITMY, 2009,176–184. Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

P. Pauš, M. Beneš, Algorithm for topological changes of parametrically described curves, Proceedings of ALGORITMY, 2009,176–184. Google Scholar

[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.  Google Scholar

Figure 1.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Closed planar curve discretization (left) corresponding to the uniform discretization of the unit circle (right)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  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]
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  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)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  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)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  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)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[2]

Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
[3]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321
[4]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405
[5]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[6]

Jia Cai, Guanglong Xu, Zhensheng Hu. Sketch-based image retrieval via CAT loss with elastic net regularization. Mathematical Foundations of Computing, 2020, 3 (4) : 219-227. doi: 10.3934/mfc.2020013
[7]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[8]

Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
[9]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[10]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[11]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017
[12]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018
[13]

Petra Csomós, Hermann Mena. Fourier-splitting method for solving hyperbolic LQR problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 17-46. doi: 10.3934/naco.2018002
[14]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263
[15]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[16]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[17]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030
[18]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

[19]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[20]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (68)
  • HTML views (317)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences