1. 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)
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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
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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…
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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
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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
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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
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7. Morse Theory Integral line
follow the direction of max decreasing slope (steepest descent)
go from maximum to mimimum (through saddle)
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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
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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 of the stable and unstable Morse complexes
Critical Net
skeleton of the Morse-Smale complex
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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
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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)
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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
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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
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14. A Boundary-Based Algorithm
Drawing critical lines (step 2)
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15. A Boundary-Based Algorithm
Drawing critical lines (step 2)
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16. A Boundary-Based Algorithm
Drawing critical lines (step 2)
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17. A Boundary-Based Algorithm
Drawing critical lines (step 2)
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18. A Boundary-Based Algorithm
Drawing critical lines (step 2)
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19. A Boundary-Based Algorithm
Drawing critical lines (step 2)
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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
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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
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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
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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
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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)
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25. A Watershed Algorithm
Flooding procedure (step 2)
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26. A Watershed Algorithm
Flooding procedure (step 2)
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27. A Watershed Algorithm
Flooding procedure (step 2)
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28. A Watershed Algorithm
Flooding procedure (step 2)
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29. A Watershed Algorithm
Flooding procedure (step 2)
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30. A Watershed Algorithm
Flooding procedure (step 2)
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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
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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
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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.
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34. The new Algorithm Find minima (step 2)
A vertex is a minimum iff it is classified D
in all its indicent triangles.
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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.
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36. The new Algorithm Can t be included? -- Case 1
If the opposite vertex of t to e is classified D, then do not include t (water flows away from e towards D)
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37. The new Algorithm Can t be included? -- Case 2
If the opposite vertex of t to e is classified S, then include t (water flows away from S towards e)
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38. The new Algorithm
Can t be included? -- Case 2
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39. The new Algorithm
Can t be included? -- Case 2
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40. The new Algorithm
Can t be included? -- Case 2
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41. The new Algorithm
Can t be included? -- Case 2
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42. The new Algorithm
Can t be included? -- Case 2
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43. The new Algorithm
Can t be included? -- Case 2
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44. The new Algorithm
Can t be included? -- Case 2
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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
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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)
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47. The new Algorithm
Can t be included? -- Case 3
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48. The new Algorithm
Can t be included? -- Case 3
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49. The new Algorithm
Can t be included? -- Case 3
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50. The new Algorithm
Can t be included? -- Case 2
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51. The new Algorithm
Can t be included? -- Case 2
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52. The new Algorithm
Can t be included? -- Case 2
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53. The new Algorithm The algorithm has rules to deal with special cases
flat triangles
flat edges
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54. Experimental Comparison
Measure the quality of the results of the new algorithm compared with the other three
STD = new
BND = boundary-based
REG = region-based
WTS = watershed
Evaluate the degree of uncertainty in morphology computation do existing algorithms provide consistent results?
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55. Experimental Comparison
Data sets (TINs with different sizes)
EGGS synthetic terrain built by sampling a combination of 2 planes and Gaussian surfaces
MARCY part of a real terrain (from US Geological Survey), perturbed to remove flat edges
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56. Experimental Comparison
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57. Experimental Comparison
New STD algorithm vs. BND, REG, WTS
Differently classified triangles
STD tends to be closer to watershed
Difference with others can be up to 7% -- 10%
STD vs BND STD vs REG STD vs WTS
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58. Experimental Comparison
New STD algorithm vs. BND, REG, WTS
Differently classified triangles
STD tends to be closer to watershed
Difference with others can be up to 7% -- 10%
STD vs BND STD vs REG STD vs WTS
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59. Experimental Comparison
New STD algorithm vs. BND, REG, WTS
Differently
classified
triangles
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60. Experimental Comparison
Uncertainty
Quantity of terrain surface whose classification is uncertain
(assigned to different basins)
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61. Experimental Comparison
Uncertainty
Quantity of terrain surface whose classification is uncertain
(assigned to different basins)
Up to 10% of the total TIN surface receives four different classifications
Difficult to judge which algorithm is correct
Ground truth is only available for continuous and differentiable functions
Existing methods only approximate the Morse (or Morse-Smale) theory
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62. Conclusions
New STD algorithm to compute the stable Morse complex on a TIN
simple
no floating point computations
management of flat edges and triangles
intuitive results
Uncertainty in morphology computation
STD method suitable for 3D extension
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63. Thanks to European Network of Excellence Aim@Shape
National Science Foundation
MIUR-FIRB project SHALOM
MIUR-PRIN project on Multi-resolution modeling of scalar fields and digital shapes
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