Morphology-Driven Simplification and Multiresolution Modeling of Terrains

1. Morphology-Driven Simplificationand 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)

2. Morphology-Driven Simplificationand 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. Morphology-Driven Simplificationand Multiresolution Modeling of Terrains Accurate sampling ----- too many triangles in a TIN Simplification ------ vertex decimation Feature-preserving simplification

4. Morphology-Driven Simplificationand Multiresolution Modeling of Terrains Feature-preserving simplification Features preserved in a multiresolution model Extract TINs at variable resolution with correct morphological structure

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

6. Morse Theory z = f(x,y) continuous and differentiable • Critical point First derivative is zero: • maximum • minimum • saddle Critical points of a Morse function are isolated

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

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. 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. 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 <90 • an Entrance if grad(f) and norm(e) form an angle >90 • Best Exit ----- minimum angle

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

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

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. 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. 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. 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 cell • otherwise --- extend an adjacent unstable cell

17. Algorithm for the unstable decomposition Build the strip for v : • Include unmarked triangles incident in v • Loop: • select the best exit e from the strip to an unmarked triangle t • if e is an entrance of t , then include t

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. Topology-Based Simplification Iterative algorithm: • Loop until all removable vertices have been removed: • remove one removable vertex • the one causing the least error increase Error: • max vertical distance between a data point and the triangulated surface

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

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. Multiresolution Extraction of a TIN at variable resolution • Threshold condition:is this triangle refined enough? • Apply all local modifications necessary to • achieve the threshold condition • respect the partial order

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

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

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

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…)