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.
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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
Figure 3. 3D graph of the edge detector $ g(\left|\nabla I^\sigma(\mathbf{x})\right|) $ for the smoothed image in Fig. 1 right
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
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} $
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
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)
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} $
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
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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An example of artificial greyscale 2D image (left) and smoothed image (right)
3D graph of image intensity function
3D graph of the edge detector
A visualization of the vector field
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
An open curve discretization (left) corresponding to the uniform discretization of parameter
Trajectories of points of a discrete segmentation curve (red) evolved in the vector field
Trajectories of points of a discrete segmentation curve (red) evolved in the vector field
Trajectories of points of a discrete segmentation curve (red) evolved in the vector field
Trajectories of points of a discrete segmentation curve (red) evolved in the vector field
Visualization of the curve discretization [1] curve grid points (red), discrete curve segments (different colors) and the midpoints (black). Finite volumes
An example of the semi-automatic segmentation showing consecutive building of the segmentation curve (yellow), the final result is on the bottom right
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
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
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)
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