1 / 63

Extracting Terrain Morphology A New Algorithm and a Comparative Evaluation

Extracting Terrain Morphology A New Algorithm and a Comparative Evaluation. Paola Magillo, Emanuele Danovaro, Leila De Floriani, Laura Papaleo, Maria Vitali Department of Computer Science University of Genova, Genova (Italy). Terrain Morphology. Terrain Height function z = f(x,y)

fraley
Download Presentation

Extracting Terrain Morphology A New Algorithm and a Comparative Evaluation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Extracting Terrain MorphologyA New Algorithm and a Comparative Evaluation Paola Magillo, Emanuele Danovaro, Leila De Floriani, Laura Papaleo, Maria Vitali Department of Computer Science University of Genova, Genova (Italy) [GRAPP 07]

  2. Terrain Morphology • Terrain • Height function z = f(x,y) • Function known at a finite set of points • Approximated through a digital model • Regular Square Grid (RSG) need regularly spaced data • Triangulated Irregular Network (TIN) allow for irregularly spaced data and terrain features [GRAPP 07]

  3. Terrain Morphology • Geometric description of terrain shape • very detailed • huge and unstructured • Morphological description of terrain shape • higher-level of abstraction • compact • support for knowledge-based analysis • Applications in Geographic Information Systems, Virtual Reality, • Enterteinment… [GRAPP 07]

  4. Terrain Morphology • Decomposition of the terrain into basins bounded by ridges, peaks, passes • Decomposition of the terrain into mountains bounded by valleys, pits, passes • Overlay of the two decompositions • Theoretical definition exists only in the continuum • --- Morse theory [GRAPP 07]

  5. Morse Theory • Function z = f(x,y) is continuous and differentiable • Critical point • first derivative is zero • (tangent plane is horizontal) • Maximum = peak • Minimum = pit • Saddle = pass [GRAPP 07]

  6. Morse Theory Function z = f(x,y) is a Morse function (determinant of the Hessian matrix of the second derivatives at a critical point is non-zero) --- The critical points are isolated OK NO [GRAPP 07]

  7. Morse Theory • Integral line • follow the direction of max decreasing slope (steepest descent) • go from maximum to mimimum (through saddle) [GRAPP 07]

  8. Morse Theory • All integral lines converging to a minimum m = • Stable cell of m (basin) • --------- Stable Morse complex • All integral lines emanating from a maximum M = • Unstable cell of M (mountain) • --------- Unstable Morse complex [GRAPP 07]

  9. Morse Theory Function z = f(x,y) is a Morse-Smale function (the boundary of a stable cell and an unstable cell, if not disjoint, cross at saddles) • Morse-Smale complex overlay ofthe stable and unstable Morse complexes • Critical Net skeleton of the Morse-Smale complex [GRAPP 07]

  10. From contimuum to discretum Algorithms approximate Morse (or Morse-Smale) complex on a digital terrain model • simulate the Morse (Morse-Smale) complex for a piecewise linear function --- If no two adjacent vertices have same height, then critical points are isolated [GRAPP 07]

  11. From contimuum to discretum Existing algorithms can be classified based on • Input - Regular Square Grid (RSG) - Triangulated Irregular Network (TIN) • Output - Stable Morse complex - Unstable Morse complex - Morse-Smale complex • Technique - boundary-based = compute critical net - region-based = compute the regions - watershed (region-based) [GRAPP 07]

  12. A Boundary-Based Algorithm Implementation of [Edelsbrunner et al. ACM-CG 2001, Takahashi et al. Computer Graphics Forum 1995] on TINs. Compute the critical net ---- the Morse-Smale complex (stable or unstable Morse complex as sub-products). Need a TIN without flat edges or triangles. 1) Extract critical points 2) Draw the critical lines starting from each saddle [GRAPP 07]

  13. A Boundary-Based Algorithm Drawing critical lines (step 2) From each saddle • two critical lines that follow the steepest descent until they reach a minimum • two critical lines that follow the steepest ascent until they reach a maximum [GRAPP 07]

  14. A Boundary-Based Algorithm Drawing critical lines (step 2) [GRAPP 07]

  15. A Boundary-Based Algorithm Drawing critical lines (step 2) [GRAPP 07]

  16. A Boundary-Based Algorithm Drawing critical lines (step 2) [GRAPP 07]

  17. A Boundary-Based Algorithm Drawing critical lines (step 2) [GRAPP 07]

  18. A Boundary-Based Algorithm Drawing critical lines (step 2) [GRAPP 07]

  19. A Boundary-Based Algorithm Drawing critical lines (step 2) [GRAPP 07]

  20. A Region-Based Algorithm Presented in [Danovaro et al. ACM-GIS 2003]. Compute the stable Morse complex (and the unstable one). Need a TIN without flat edges or triangles. 1) Extract minima and maxima 2) Grow the basins starting from each minimum [GRAPP 07]

  21. A Region-Based Algorithm Basin growing procedure (step 2) Consider the gradient of each triangle and the angles between the gradient and the normal vector at each edge • edge with largest angle = entrance • edge with smallest angle = exit [GRAPP 07]

  22. A Region-Based Algorithm Basin growing procedure (step 2) From each minimum • initial basin = its incident triangles • iteratively consider a triangle sharing an edge with an already included triangle • if such edge is exit for the new triangle and entrance for the old one, then include [GRAPP 07]

  23. A Watershed Algorithm Based on simulated immersion, implementation of [Vincent and Soille IEEE Trans. Pattern Analysis and Machine Intelligence 1991]. Compute the stable Morse complex (and the unstable one). Accept a TIN with flat edges and triangles. 1) Sort vertices according to increasing height value 2) Assign every vertex to a basin through flooding (vertices on the boundary of two basins remain unclassified -- watershed vertices) 3) Assign every triangle to a basin based on its vertices [GRAPP 07]

  24. A Watershed Algorithm Flooding procedure (step 2) At each iteration, consider a height value h (initially, h = minimum height) Consider vertices p with height = h that are neighbors of a vertex labeled during previous iteration • if all neighbors of p are in the same basin, or are watershed points, then assign p to that basin • otherwise, p is a watershed point Vertices with height = h not assigned to any basin are minima (a new basin will start from them) [GRAPP 07]

  25. A Watershed Algorithm Flooding procedure (step 2) [GRAPP 07]

  26. A Watershed Algorithm Flooding procedure (step 2) [GRAPP 07]

  27. A Watershed Algorithm Flooding procedure (step 2) [GRAPP 07]

  28. A Watershed Algorithm Flooding procedure (step 2) [GRAPP 07]

  29. A Watershed Algorithm Flooding procedure (step 2) [GRAPP 07]

  30. A Watershed Algorithm Flooding procedure (step 2) [GRAPP 07]

  31. A Watershed Algorithm Triangle classification (step 3) • If all vertices of a triangle t, that are not watershed points, belong to the same basin, then t belongs to that basin • If they belong to different basins, then assign t to the lowest basin [GRAPP 07]

  32. The new Algorithm Region-based. We call it the STD algorithm. Accept a TIN with flat edges and triangles. 1) Classify the three vertices of each triangle 2) Find minima 3) Grow the basins starting from each minimum [GRAPP 07]

  33. The new Algorithm Vertex classification (step 1) • highest vertex = S (source) • middle vertex = T (through) • lowest vertex = D (drain) Water flows from S to D through T. [GRAPP 07]

  34. The new Algorithm Find minima (step 2) A vertex is a minimum iff it is classified D in all its indicent triangles. [GRAPP 07]

  35. The new Algorithm Grow basins starting from the minima (step 3) Let m be a minimum. • Initial basin of m = its incident triangles • Iteratively examine a triangle t externally adjacent to the current basin, sharing an edge e Can t be included in the current basin? --- three cases. [GRAPP 07]

  36. The new Algorithm Can t be included? -- Case 1 • If the opposite vertex of t to e is classified D, then do not includet (water flows away from e towards D) [GRAPP 07]

  37. The new Algorithm Can t be included? -- Case 2 • If the opposite vertex of t to e is classified S, then includet (water flows away from S towards e) [GRAPP 07]

  38. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

  39. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

  40. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

  41. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

  42. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

  43. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

  44. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

  45. The new Algorithm Can t be included? -- Case 3 • If the opposite vertex of t to e is classified T, then examine the fan of triangles having their D vertex in the D vertex of t [GRAPP 07]

  46. The new Algorithm Can t be included? -- Case 3 • Split the fan in two parts at its radial edge of maximum slope • Include the part of fan adjacent to e (if not empty) [GRAPP 07]

  47. The new Algorithm Can t be included? -- Case 3 [GRAPP 07]

  48. The new Algorithm Can t be included? -- Case 3 [GRAPP 07]

  49. The new Algorithm Can t be included? -- Case 3 [GRAPP 07]

  50. The new Algorithm Can t be included? -- Case 2 [GRAPP 07]

More Related