Semi-automatic segmentation of NATURA 2000 habitats in Sentinel-2 satellite images by evolving open curves

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doi: 10.3934/dcdss.2020231

Semi-automatic segmentation of NATURA 2000 habitats in Sentinel-2 satellite images by evolving open curves

Karol Mikula ^1,2,, , Jozef Urbán ^1,2, , Michal Kollár ^1,2, , Martin Ambroz ^1,2, , Ivan Jarolímek ^3, , Jozef Šibík ^3, and Mária Šibíková ^3,

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

^* Corresponding author

Received January 2019 Revised September 2019 Published December 2019

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

In this paper we introduce mathematical model and real-time numerical method for segmentation of Natura 2000 habitats in satellite images by evolving open planar curves. These curves in the Lagrangian formulation are driven by a suitable velocity vector field, projected to the curve normal. Besides the vector field, the evolving curve is influenced also by the local curvature representing a smoothing term. The model is numerically solved using the flowing finite volume method discretizing the arising intrinsic partial differential equation with Dirichlet boundary conditions. The time discretization is chosen as an explicit due to the ability of real-time edge tracking. We present the results of semi-automatic segmentation of various areas across Slovakia, from the riparian forests to mountainous areas with scrub pine. The numerical results were compared to habitat boundaries tracked by GPS device in the field by using the mean and maximal Hausdorff distances as criterion.

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. Semi-automatic segmentation of NATURA 2000 habitats in Sentinel-2 satellite images by evolving open curves. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020231

References:

[1]	M. Ambroz, M. Balažovjech, M. Medla and K. Mikula, Numerical modeling of wildland surface fire propagation by evolving surface curves, Advances in Computational Mathematics, 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, Proceedings of ALGORITMY, (2012), 42-52. Google Scholar
[3]	V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1997), 61-79. doi: 10.1109/ICCV.1995.466871. Google Scholar
[4]	S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Jr., Conformal curvature flows: From phase transitions to active vision, Arch. Rational Mech. Anal., 134 (1996), 275-301. doi: 10.1007/BF00379537. Google Scholar
[5]	M. Kolář, M. Beneš and D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane, Mathematics and Computers in Simulation, 126 (2016), 1-13. doi: 10.1016/j.matcom.2016.02.004. Google Scholar
[6]	Z. Krivá, K. Mikula, M. Peyriéras, B. Rizzi, A. Sarti and O. Stašová, 3D early embryogenesis image filtering by nonlinear partial differential equations, Medical Image Analysis, 14 (2010), 510-526. Google Scholar
[7]	K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Computing and Visualization in Science, 6 (2004), 211-225. doi: 10.1007/s00791-004-0131-6. Google Scholar
[8]	K. Mikula and D. Ševčovič, Evolution of curves on a surface driven by the geodesic curvature and external force, Applicable Analysis, 85 (2006), 345-362. doi: 10.1080/00036810500333604. Google Scholar
[9]	K. Mikula, D. Ševčovič and M. Balažovjech, A simple, fast and stabilized flowing finite volume method for solving general curve evolution equations, Communications in Computational Physics, 7 (2010), 195-211. doi: 10.4208/cicp.2009.08.169. Google Scholar
[10]	P. Pauš, M. Beneš, M. Kolář and J. Kratochvíl, Dynamics of dislocations described as evolving curves interacting with obstacles, Modelling and Simulation in Materials Science and Engineering, 24 (2016), 34 pp. Google Scholar
[11]	P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, Proceedings of the IEEE Society Workshop on Computer Vision, 12 (1987), 629-639. doi: 10.1109/34.56205. Google Scholar
[12]	Hausdorff distance, 20 12 2018, https://en.wikipedia.org/wiki/Hausdorff_distance. Google Scholar

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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, Advances in Computational Mathematics, 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, Proceedings of ALGORITMY, (2012), 42-52. Google Scholar
[3]	V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, International Journal of Computer Vision, 22 (1997), 61-79. doi: 10.1109/ICCV.1995.466871. Google Scholar
[4]	S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Jr., Conformal curvature flows: From phase transitions to active vision, Arch. Rational Mech. Anal., 134 (1996), 275-301. doi: 10.1007/BF00379537. Google Scholar
[5]	M. Kolář, M. Beneš and D. Ševčovič, Computational analysis of the conserved curvature driven flow for open curves in the plane, Mathematics and Computers in Simulation, 126 (2016), 1-13. doi: 10.1016/j.matcom.2016.02.004. Google Scholar
[6]	Z. Krivá, K. Mikula, M. Peyriéras, B. Rizzi, A. Sarti and O. Stašová, 3D early embryogenesis image filtering by nonlinear partial differential equations, Medical Image Analysis, 14 (2010), 510-526. Google Scholar
[7]	K. Mikula and D. Ševčovič, Computational and qualitative aspects of evolution of curves driven by curvature and external force, Computing and Visualization in Science, 6 (2004), 211-225. doi: 10.1007/s00791-004-0131-6. Google Scholar
[8]	K. Mikula and D. Ševčovič, Evolution of curves on a surface driven by the geodesic curvature and external force, Applicable Analysis, 85 (2006), 345-362. doi: 10.1080/00036810500333604. Google Scholar
[9]	K. Mikula, D. Ševčovič and M. Balažovjech, A simple, fast and stabilized flowing finite volume method for solving general curve evolution equations, Communications in Computational Physics, 7 (2010), 195-211. doi: 10.4208/cicp.2009.08.169. Google Scholar
[10]	P. Pauš, M. Beneš, M. Kolář and J. Kratochvíl, Dynamics of dislocations described as evolving curves interacting with obstacles, Modelling and Simulation in Materials Science and Engineering, 24 (2016), 34 pp. Google Scholar
[11]	P. Perona and J. Malik, Scale space and edge detection using anisotropic diffusion, Proceedings of the IEEE Society Workshop on Computer Vision, 12 (1987), 629-639. doi: 10.1109/34.56205. Google Scholar
[12]	Hausdorff distance, 20 12 2018, https://en.wikipedia.org/wiki/Hausdorff_distance. Google Scholar

Figure 1. An example of artificial greyscale 2D image (left) and smoothed image (right)

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Figure 2. 3D graph of image intensity function $ I^\sigma(\mathbf{x}) $ (left) and graph of the image intensity gradient norm $ \left|\nabla I^\sigma(\mathbf{x})\right| $ (right), for the smoothed image in Fig. 1 right

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Figure 3. 3D graph of the edge detector $ g(\left|\nabla I^\sigma(\mathbf{x})\right|) $ for the smoothed image in Fig. 1 right

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Figure 4. A visualization of the vector field $ \mathbf{v}\left(\mathbf{x}\right) $ for image in Fig. 1 right. We see that arrows points to the edge in the image from both sides

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Figure 11. A discrete segmentation curve evolving to habitat boundary in a real 3-band Sentinel-2 optical image. The green color shows trajectories of moving discrete curve points and blue points represents the result of segmentation of this particular section of the habitat border

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Figure 5. An open curve discretization (left) corresponding to the uniform discretization of parameter $ u\in \left[0,1\right] $ (right)

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Figure 6. Trajectories of points of a discrete segmentation curve (red) evolved in the vector field $ \mathbf{v} $ driven to the image edge. The final state of discrete segmentation curve is given by green points localized on the image edge

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Figure 7. Trajectories of points of a discrete segmentation curve (red) evolved in the vector field $ \mathbf{v} $ and their final position (green) visualized over the original image. One can see a problem of non-uniform distribution of points on evolving discrete segmentation curve due to non-controlled tangential velocities in the vector field $ \mathbf{v} $

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Figure 8. Trajectories of points of a discrete segmentation curve (red) evolved in the vector field $ \mathbf{v} $ and their final position (green) visualized over the original image. An improved distribution of the curve grid points after removing the tangential component of the velocity vector field $ \mathbf{v} $ is obvious

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Figure 9. Trajectories of points of a discrete segmentation curve (red) evolved in the vector field $ \mathbf{v} $ and their final position (green) visualized over the original image. Top: the curve evolution by (5) in a less smoothed image when problem of crossing, accumulating and not moving points may arise; bottom: such undesired behaviour is removed by employing the local curvature influence into the model (7)

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Figure 10. Visualization of the curve discretization [1] curve grid points (red), discrete curve segments (different colors) and the midpoints (black). 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} $

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Figure 12. An example of the semi-automatic segmentation showing consecutive building of the segmentation curve (yellow), the final result is on the bottom right

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Figure 13. Semi-automatic segmentation (yellow) and GPS track (light-blue) with almost exact overlap. The maximal Haussdorff distance is 62.1m and the mean Hausdorff distance is 14.0m in this case, which means that we obtained almost the pixel resolution (10m) accuracy

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Figure 14. An example of a complicated border of the riparian forest. We compare the semi-automatic segmentation (yellow) and GPS track (light-blue). The mean Haussdorff distance is 12.0m and the maximal Haussdorff distance, in this case, is 62.1m, indicating correctly discrepancy in habitat area estimate in the field and by employing the Sentinel-2 optical data

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Figure 15. The locality with the highest, 413.3m, maximal Hausdorff distance between semi-automatically segmented and GPS tracked curves among bushes with Pinus mugo tested areas, here also the mean Hausdorff distance was the highest, 44.8m. On the North-West habitat border, we can see the "ecotone zone" that was included during field tracking (light-blue) and excluded by using the semi-automatic segmentation (yellow)

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Figure 16. The locality dominated by Pinus mugo with the "ecotone zone" that was included during the field tracking (light-blue) and excluded by using the semi-automatic segmentation (yellow). The mean Hausdorff distance is 19.1m and the maximal Hausdorff distance is 171.0m

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2019 Impact Factor: 1.233

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