Extraction of Watersheds from Digital Elevation Models: Mathematical Morphological Approach

1. Extraction of Watersheds from Digital Elevation Models: Mathematical Morphological Approach DineshSathyamoorthy Science and Technology Research Institute for Defence (STRIDE), Ministry of Defence, Malaysia

2. Digital Elevation Models • The Global Digital Elevation Model (GTOPO30) of Great Basin

3. Watersheds

4. Mathematical Morphology • 3D representation of a grayscale image A grayscale image The corresponding surface in 3D Euclidean space

5. Mathematical Morphology • Mathematical morphological operators require two inputs: • The input image A • The kernel B which is used to determine the precise effect if the operator. • The two fundamental morphological operators are • Dilation: A B= {a+b: aєA,bєB} • Erosion: AB= (AcB) c • Erosion and dilation are used to implement opening and closing: • Opening: Erosion followed by dilation • AB= (AB)  B • Closing: Dilation followed by erosion • A B= (AB)  B

6. Mathematical Morphology • Morphological operators • Morphological reconstruction • The geodesic dilation used in the reconstruction process is performed through iteration of elementary geodesic dilations, until stability is achieved. δG(Y) = δ(1)(Y) ο δ(1) (Y) ο δ(1) (Y) …until stability • The elementary dilation process is performed using standard dilation of size one followed by an intersection. δ(1)(Y) = AB ∩ X

7. Mathematical Morphology • Morphological Operators • Ultimate Erosion • Ultimate erosion is used to compute the number of clustered convex objects in an image by detecting their pseudo-centres. • The principle of the algorithm is to successively erode the image until all particles vanish (images Xi), and to reconstruct each eroded image into the erosion of smaller size (images Yi)

8. Mathematical Morphology • Morphological Operators • Ultimate Erosion

9. Mathematical Morphology • Morphological Operators • Hit or miss transform AB • Thinning A-(AB) • Skeletonization

10. Mathematical Morphology • Morphological Operators • Skeletonization • A basic formulation for skeletonization is based on the work of Lantuéjoul. • The skeleton subset is defined by: Sk(A)={(AΘkB)-[(AΘkB) B]} k=0,1,2..K • where K is the largest value of k before the set Sk(A) becomes empty. • The structuring element B is chosen to approximate a circular disc, that is, convex, bounded and symmetric. • The skeleton, S(A) is then the union of the skeleton subsets: • Disadvantage of this method: • does not preserve the topology

11. Mathematical Morphology • Morphological Operators • Skeletonization • Can be done using thinning. • The following structuring elements are used. • At each iteration, the image is first thinned by the left hand structuring element, and then by the right hand one, and then with the remaining six 90° rotations of the two elements. • The process is repeated in cyclic fashion until none of the thinnings produces any further change.

12. Mathematical Morphology • Morphological Operators • Pruning • Thinning and skeletonization algorithms tend to leave parasitic components, known as spurs, that need to be cleaned up by postprocessing. • The spur removal process is known as pruning. • Pruning is done using the following structuring element • The image is pruned by performing thinning on the image using the 45° rotations of the structuring element above.

13. Mathematical Morphology • Morphological Operators • Skeletonization by influence zone • A skeleton by influence zone (SKIZ) is a skeletal structure that divides an image into regions, each of which contains just one of the distinct objects in the image. • The boundaries are drawn such that all points within a particular boundary are closer to the binary object contained within that boundary than to any other. • The SKIZ is also known as the Voronoi diagram. • Calculation of the SKIZ of an image is done by first determining the exoskeleton of the image (the skeleton of the background) and then pruning this until convergence to remove all branches except those forming closed loops, or that intersect the boundary.

14. Watershed Extraction

15. Watershed Extraction • Preprocessing: DEM smoothening using morphological smoothening by reconstruction Drainage network of the DEM of Great Basin Spurious peaks cause the formation of closed loops in the extracted drainage network, while spurious pits cause a portion of the extracted the drainage network to be incomplete and disconnected.

16. Watershed Extraction • Preprocessing: DEM smoothening using morphological smoothening by reconstruction The smoothened DEM Mask of pixels modified by the smoothening process

17. Watershed Extraction • Preprocessing: DEM smoothening using morphological smoothening by reconstruction Drainage network of the smoothened DEM The extracted of drainage network is loopless, complete and connected.

18. Watershed Extraction • Step 1: Pit Extraction The extracted pits

19. Watershed Extraction • Step 2: Skeletonization of extracted and dilated pits SKIZ of extracted pits SKIZ of dilated pits Identification of individual catchments

20. Watershed Extraction • Evaluation of results Catchments extracted using the proposed algorithm Catchments extracted using the algorithm proposed in Jenson and Domingue (1988)

21. Conclusion • In this paper, a mathematical morphological based algorithm to extract watersheds from DEMs was proposed. • The algorithm is able to operate effectively on flat areas in DEMs, and allows for the fast and effective segmentation of DEMs into their constituent catchments • Scope for future work: Mapping of size and shape of catchments extracted from DEMs