1. ACM-GIS 2003 1 Morphology-Driven Simplification and Multiresolution Modeling of Terrains Emanuele Danovaro, Leila De Floriani, Paola Magillo, Mohammed Mostefa Mesmoudi, Enrico Puppo Department of Computer Science University of Genova, Genova (Italy) University of Genova, Genova (Italy)

2. ACM-GIS 2003 2 Morphology-Driven Simplification and Multiresolution Modeling of Terrains Terrain: scalar field z = f(x,y)  known at a set of points  represented as a Triangulated Irregular Network (TIN) Morphological features:  maxima (peaks)  minima (pits)  saddles (passes)

3. ACM-GIS 2003 3 Morphology-Driven Simplification and Multiresolution Modeling of Terrains Accurate sampling ----- too many triangles in a TIN  Simplification ------ vertex decimation  Feature-preserving simplification

4. ACM-GIS 2003 4 Morphology-Driven Simplification and Multiresolution Modeling of Terrains Feature-preserving simplification  Features preserved in a multiresolution model  Extract TINs at variable resolution with correct morphological structure

5. ACM-GIS 2003 5 Contribution  Extension of Morse theory (scalar field topology) to TINs ----- algorithm  Feature-preserving TIN decimation  Application to multiresolution modeling

6. ACM-GIS 2003 6 Morse Theory z = f(x,y) continuous and differentiable  Critical point First derivative is zero: maximummaximum minimumminimum saddlesaddle Critical points of a Morse function are isolated

7. ACM-GIS 2003 7 Morse Theory  Isoline: locus of points where f(x,y)=k  Integral line: follows the direction of max decreasing slope ----- steepest descent

8. ACM-GIS 2003 8 Morse Theory  All integral lines converging to a minimum p ----- Stable Cell of p  All integral lines emanating from a maximum p ----- Unstable Cell of p  Stable and Unstable Morse-Smale decomposition  Overlay ----- Critical Net

9. ACM-GIS 2003 9 Extension of Morse-Smale Theory to TINs  z = f(x,y) is piecewise linear  Not differentiable at triangle edges  Isolines and integral lines are polylines

10. ACM-GIS 2003 10 Extension of Morse-Smale Theory to TINs  Each triangle t has its gradient ----- direction of steepest descent  Each edge e of t is: an Exit if grad(f) and norm(e) form an angle <90an Exit if grad(f) and norm(e) form an angle <90 grad(f) and norm(e) form an angle >90an Entrance if grad(f) and norm(e) form an angle >90  Best Exit ----- minimum angle

11. ACM-GIS 2003 11 Extension of Morse-Smale Theory to TINs split merge 5 possible configurations for a triangle: maximum minimum turn

12. ACM-GIS 2003 12 Extension of Morse-Smale Theory to TINs valley ridge flow through 2 possible configurations for an edge:

13. ACM-GIS 2003 13 Extension of Morse-Smale Theory to TINs  Critical lines should sometimes split triangles  Convention: force critical lines to be TIN edges  Triangles are assigned to one cell based on their best exit

14. ACM-GIS 2003 14 Extension of Morse-Smale Theory to TINs Constructive definition of Morse-Smale decomposition for TINs ------ Algorithm:  Compute unstable decomposition  Compute stable decomposition (symmetric)  Overlay ---> Critical Net

15. ACM-GIS 2003 15 Algorithm for the unstable decomposition  Progressively assign triangles to an unstable cell and mark them  Stop when all triangles have been marked  Unstable cells of higher maxima are processed first  Maintain a list of all vertices of unmarked triangles sorted by decreasing elevation (first = global maximum)

16. ACM-GIS 2003 16 Algorithm for the unstable decomposition Loop until all triangles have been marked:  take the next vertex v from the list  traverse and mark a strip of triangles following an integral line descending from v if v is a maximum --- seed of its unstable cellif v is a maximum --- seed of its unstable cell otherwise --- extend an adjacent unstable cellotherwise --- extend an adjacent unstable cell

17. ACM-GIS 2003 17 Algorithm for the unstable decomposition Build the strip for v :  I nclude unmarked triangles incident in v  Loop: select the best exit e from the strip to an unmarked triangle tselect the best exit e from the strip to an unmarked triangle t if e is an entrance of t, then include tif e is an entrance of t, then include t

18. ACM-GIS 2003 18 Topology-Based Simplification Constrained TIN:  some edges are marked as constraints  the edges of the critical net Vertex decimation in a constrained TIN:  removable and non-removable vertices

19. ACM-GIS 2003 19 Topology-Based Simplification Iterative algorithm:  Loop until all removable vertices have been removed: remove one removable vertexremove one removable vertex the one causing the least error increasethe one causing the least error increaseError:  max vertical distance between a data point and the triangulated surface

20. ACM-GIS 2003 20 Multiresolution Multi-Triangulation  A coarse Base TIN  A partially ordered set of modifications which progressively refine it into a detailed Reference TIN

21. ACM-GIS 2003 21 Multiresolution Modification:  Two local TINs at lower and higher resolution  “Higher” replaces “lower” Partial order:  M1 < M2 if some triangle of High(M1) is in Low(M2)  M2 can be applied only after M1

22. ACM-GIS 2003 22 Multiresolution Extraction of a TIN at variable resolution  Threshold condition: is this triangle refined enough?  Apply all local modifications necessary to achieve the threshold conditionachieve the threshold condition respect the partial orderrespect the partial order

23. ACM-GIS 2003 23 Topology-Based Multiresolution Building an MT from the decimation process:  The final simplified TIN ---- the base TIN  Each removed vertex ---- a modification reinserting the vertex  Partial order recovered from deleted and added triangles  Constraint edges are marked in the MT

24. ACM-GIS 2003 24 Topology-Based Multiresolution Extracted TINs maintain the topological structure Error = 0 12800 triangles 318 edges Error <= 0.15 2260 triangles 171 edges

25. ACM-GIS 2003 25 Topology-Based Multiresolution Extracted TINs maintain the topological structure Error = 0 12800 triangles 318 edges Error <= 1.77 114 triangles 20 edges

26. ACM-GIS 2003 26 Topology-Based Multiresolution Extracted TINs maintain the topological structure Error = 0 12800 triangles 318 edges Error = any 20 triangles 12 edges

27. ACM-GIS 2003 27 Thanks to  EC Project MINGLE (Multiresolution in Geometric Modelling)  MIUR Project MacroGEO (Algorithmic and Computational Methods for Geometric Object Representation)  EC Project ARROV (Augmented Reality for Remotely Operated Vehicles…)