1. Multi-Scale Dual Morse Complexes for Representing Terrain Morphology E. Danovaro Free University of Bolzano, Italy L. De Floriani University of Genova, Italy and University of Maryland, MD, USA M. Vitali, P. Magillo University of Genova, Italy

2. ACM-GIS’072 Overview Introduction: motivations and goals Terrain morphology: model and representation Multi-scale terrain morphology Results Conclusions

3. ACM-GIS’073 Modeling Terrain Geometry Terrain (scalar field) Terrain (scalar field) – function z = f(x,y) – known at a set of points Represented as a Triangulated Irregular Network (TIN) Represented as a Triangulated Irregular Network (TIN) – 2D triangle mesh – Piecewise linear interpolation TIN represents terrain geometry TIN represents terrain geometry

4. ACM-GIS’074 From Geometry to Morphology Geometric models (as TINs) give accurate representations But too verbose, especially for large data sets basin peak pass valley Morphological models can give a more effective overview of terrain shape Important for terrain analysis and exploration

5. ACM-GIS’075 From Geometry to Morphology Start from geometry Understand and represent morphological structure basins mountains Face complexity by using a multi-scale model

6. ACM-GIS’076 Multi-Scale Models Implicitly encode a range of representations at variable scale multi-scale model Allow extracting representations according to user requirements

7. ACM-GIS’077 Overview Introduction: motivations and goals Terrain morphology: model and representation Multi-scale terrain morphology Results Conclusions

8. ACM-GIS’078 Morse and Morse-Smale Theory Mathematical theory for modeling morphology of continuous and differentiable scalar fields Version for discrete case (TINs) [ICIAP’07] Morse function iff its critical points are non-degenerate (=isolated): no horizontal lines or areas Critical point iff null gradient (=horizontal tangent plane): maximum, minimum, saddle

9. ACM-GIS’079 Morse and Morse-Smale Theory Local neighborhood of a critical point How many terrain sectors lie above/below the tangent plane? – Totally above = minimum – Totally below = maximum – Mixed = simple saddle or multiple saddle

10. ACM-GIS’0710 Morse and Morse-Smale Theory Integral line: maximal path which is everywhere tangent to the gradient – Minimum = only incoming integral lines – Maximum = only outgoing integral lines – Saddle = both

11. ACM-GIS’0711 Stable and Unstable Morse Complexes Union of incoming integral lines of a minimum = stable cell (basin) Union of outgoing integral lines of a maximum = unstable cell (mountain)

12. ACM-GIS’0712 Stable and Unstable Morse Complexes  Stable Morse complex and Unstable Morse complex basins, ridges, peaksmountains, valleys, pits

13. ACM-GIS’0713 Morse-Smale Functions Morse-Smale function iff boundaries of stable and unstable Morse complexes intersect only at saddle points

14. ACM-GIS’0714 Morse-Smale Functions Morse-Smale complex = overlay of stable and unstable Morse complexes

15. ACM-GIS’0715 Morse-Smale Complex Each saddle is connected with two maxima and two minima Terrain is decomposed into quadrangles bounded by saddle-minimum-saddle- maximum

16. ACM-GIS’0716 Stable and Unstable Complexes are Dual Stable region (basin around minimum) = unstable vertex (minimum) Stable edge (ridge through saddle) = unstable edge (valley through saddle) Stable vertex (maximum) = unstable region (basin around minimum)

17. ACM-GIS’0717 Morphological Structure In [ICIAP’07] we proposed a multi-scale topological model for TINs that assumes: – Morse-Smale function – Simple saddles Here, we extend to: – Morse, but not Morse-Smale, function – Multiple saddles

18. ACM-GIS’0718 Morse (not Smale) Complexes Stable and unstable Morse complexes may intersect in edges, which connect saddle points k=2 Macro-saddle = maximal connected set of edges where both endpoints are saddles A macro-saddle contains k saddles, k>1

19. ACM-GIS’0719 Morse (not Smale) Complexes We can “shrink a macro-saddle to a point” Obtain the same properties as a normal saddle

20. ACM-GIS’0720 Encoding Morphological Structure Two-level representation of the overlay of the stable and unstable Morse complexes: Top-level = description of “Morse-Smale complex” by considering macro-saddles instead of individual saddles Lower-level = description of each macro-saddle

21. ACM-GIS’0721 Top-level data structure Minimum  radially sorted list of adjacent macro-saddles Maximum  radially sorted list of adjacent macro-saddles Macro-saddle  radially sorted list of adjacent maxima and minima

22. ACM-GIS’0722 Lower-level data structure for a macro-saddle Graph describing relations of individual saddles and minima within macro-saddle Graph describing relations of individual saddles and maxima within macro-saddle

23. ACM-GIS’0723 Overview Introduction: motivations and goals Terrain morphology: model and representation Multi-scale terrain morphology Results Conclusions

24. ACM-GIS’0724 Multi-Scale (Geometric) Model Built off-line Built off-line through a simplification process ncode Encode all simplification steps multi-scale model simplification query Queried on-line to extract variable-resolution representations User gives resolution requirements

25. ACM-GIS’0725 Multi-Scale (Geometric) Model The most simplified TIN The most simplified TIN A partially ordered set of refinement steps which reverse the simplification process and progressively refine it into the full-scale TIN A partially ordered set of refinement steps which reverse the simplification process and progressively refine it into the full-scale TIN

26. ACM-GIS’0726 Extraction of a TIN at variable scale: Perform a subset of refinements which is consistent with the partial order Perform a subset of refinements which is consistent with the partial order Driven by user requirements Driven by user requirements Multi-Scale (Geometric) Model

27. ACM-GIS’0727 Extract TINs at variable resolution with consistent morphology Extract TINs at variable resolution with consistent morphology Preserve the structure of the dual Morse complex Preserve the structure of the dual Morse complex Simplification of geometry only Simplification of geometry only Morphology-preserving geometric simplification [ACM-GIS’03]

28. ACM-GIS’0728 Morphology Simplification Morphology is often too complex, especially for large terrains Morphology is often too complex, especially for large terrains Many meaningless features due to noise Many meaningless features due to noise Simplification to remove undesired features Simplification to remove undesired features Simplify the dual Morse complexes Simplify the dual Morse complexes

29. ACM-GIS’0729 Morphology simplification for Morse-Smale complex [ICIAP’07] Simplification operators: minimum-saddle-minimum (collapse to 1 minimum) maximum-saddle-maximum (collapse to 1 maximum)

30. ACM-GIS’0730 Morphology simplification for dual Morse Complexes Each operator becomes three: Minimum-saddle-minimum – for an isolated (= non-macro) simple saddle – (unchanged) remove saddle Minimum-saddle h -minimum – for an isolated (= non-macro) multiple saddle – decrease multiplicity of saddle to h-1 Minimum-k-saddle-minimum – for an macro-saddle containing k simple saddles – decrease cardinality of macro-saddle to k-1

31. ACM-GIS’0731 Morphology simplification for dual Morse Complexes Minimum-saddle h -minimum – for an isolated (= non-macro) multiple saddle – collapse 2 consecutive minima and reduce multiplicity of saddle

32. ACM-GIS’0732 Morphology simplification for dual Morse Complexes Minimum-k-saddle-minimum – for an macro-saddle containing k simple saddles – collapse 2 minima and the simple saddle connecting them to 1 minimum, and reduce cardinality of macro-saddle

33. ACM-GIS’0733 Multi-Scale Morphological Model Each simplification step affects a limited number of entities (vertices, edges, regions) in the dual Morse complexes (Detailed description of such entities is in the paper) It can be described as one set of entities replaced by another set of entities: u=(u+, u-) Each simplification step can be reversed into a refinement step: rev(u) = (u-,u+)

34. ACM-GIS’0734 Multi-Scale Morphological Model The multi-scale models is composed of: Coarse model (final result of simplification) Refinement steps that allow reconstructing the initial full-scale model (inverse of performed simplification steps)

35. ACM-GIS’0735 Multi-Scale Morphological Model Partial order based on dependency relation – A refinement step u2 = (u2-,u2+) depends on u1 = (u1-,u1+) if u2 removes some element created by u1, I.e., u1+ and u2- have non-empty intersection Represented as a directed Acyclic Graph (DAG)

36. ACM-GIS’0736 Multi-Scale Morphological Model Each modification has a persistence value (approximation induced by morphology simplification) User gives a persistence threshold over the whole domain, or in a region of interest Extract the coarsest representation fitting the threshold

37. ACM-GIS’0737 Overview Introduction: motivations and goals Terrain morphology: model and representation Multi-scale terrain morphology Results Conclusions

38. ACM-GIS’0738 Experimental Results Multi-scale TIN (geometry) Multi-scale dual Morse complexes (morphology) Synchronization  simultaneous extraction of matching geometry and morphology for user criteria

39. ACM-GIS’0739 Elba island at uniform resolution Full resolution: 785 minima 509 maxima (75k triangles) Persistence = 3m: 157 minima 199 maxima (13k triangles)

40. ACM-GIS’0740 Elba island with focus on a region Full resolution: 158 minima 114 maxima (9.9k triangles) Persistence = 3m: 52 minima 69 maxima (2.7k triangles)

41. ACM-GIS’0741 Overview Introduction: motivations and goals Terrain morphology: model and representation Multi-scale terrain morphology Results Conclusions

42. ACM-GIS’0742 Conclusions Model and data structure to represent terrain morphology of Morse (but not Morse-Smale) functions Multi-Scale version of such model and construction through morphological simplification Extraction of variable-scale morphology representations Matching extracted morphology and geometry (not in the paper)

43. ACM-GIS’0743 End of talk….. Thank you! Questions?