Meshless wavelets and their application to terrain modeling

1. Meshless wavelets and their application to terrain modeling A DARPA GEO* project Jack Snoeyink, Leonard McMillan, Marc Pollefeys, Wei Wang (UNC-CH) Charles Chui, Wenjie He (UMSL)

2. Outline • Project Team, Motivation, & Objectives • Meshless wavelets • CK Chui: Compactly supported, refinable spline fcns • Y Liu: Order-k Voronoi diagrams & simplex splines • Simplification/compression for applications • Mobility: elevation & slope mapping • Feature identification and matching • Management, Risks & Rewards

3. Team Introduction • U of Missouri, St. Louis • Charles K Chui: wavelets & splines • Wenjie He: splines • UNC Chapel Hill • Jack Snoeyink: computational geometry • Marc Pollyfeys: computer vision • Leonard McMillan: computer graphics • Wei Wang: spatial databases • Yuanxin (Leo) Liu & Henry McEuen

4. Self-evident truths … • Terrain data volumes are increasing. • NIMA: “In only 9 days and 18 hours, SRTM collected elevation data for 80% of the world's landmass to enable the production of DTED Level 2.” • Old data formats were chosen for ease of computation more than completeness of representation. • Consider USGS raster DEM’s use of integer identifiers. • Terrain is irregular and multi-scale; its representation should be, too. • breaklines, multiple sources & sensors, viewer level of interest… • Consistency is a virtue in multi-(use, resolution, sensor, spectral...) • Example of elevation and slope mentioned in BAA • Image compression schemes are designed to look good. • TIFF, JPEG, JPEG2000, … • The GIS industry cannot innovate on data reps. • Backward compatibility trumps even algorithmic improvements • It is a good time to look at new options for terrain representation.

5. Key research question • What compact representations of terrain still support interesting queries? • Elevation + slope for mobility + visibility • Feature identification across imaging modes and viewing conditions for localization, change detection, and terrain construction

6. Bivariate meshless wavelets We propose • a new compact representation for geospatial data that is optimized for specific geometric and image queries. • ``meshless'' bivariate wavelets defined over scattered point sets allow a flexible description since the point set can be specified without connectivity and each point's influence is local, while still supporting the multiscale analysis afforded by wavelets. Objectives • complete the theory of bivariate meshless wavelets • point/knot selection algorithms optimized for specific geometric tasks and data queries • demonstration implementation showing the advantages of our modeling approach.

7. Meshless Wavelet Tight-Frames Charles Chui Wenjie He University of Missouri-St. Louis March 29, 2005 Savannah, Georgia

8. Stationary Wavelets

9. Stationary wavelet notation

10. Definition of stationary wavelet tight-frames A family is a stationary wavelet frame of , if there exist constants such that If , the frame is called a normalized tightframe.

11. Characterization of wavelet tight-frames Theorem.Frazier-Garrigós-Wang-Weiss 1996, Ron-Shen 1997, Chui-Shi 1999. Let . The family is a normalized tight frame of , if and only if and odd.

12. Wavelet tight-frames associated with Multiresolution Analysis (MRA) • Refinable function: • Frame generators: • Two-scale symbols: • Vanishing moments of order K: is divisible by

13. Unitary matrix extension (UEP) for MRA tight frames Let Then is a normalized tight frame.

14. Equivalent matrix formulation on

15. Limitations of UEP Applicable only if For , i.e., cardinal B-spline of order m, at least one of the has only the factor of but not a higher power, (i.e., only one vanishing moment for the corresponding frame generator). on

16. Full characterization of MRA tight frames Oblique Extension Principle (OEP)

17. Minimum-supported VMR functions for cardinal B-splines For achieving vanishing moments for all tight-frame generators with symbols

18. Orders of vanishing moments Each has at least K vanishing moments, i.e. has vanishing moments of order at least K, if and only if

19. Wavelet decomposition and reconstruction Decomposition and perfect reconstruction scheme for computing DFWT

20. FIR schemes New FIR filters for perfect reconstruction from DFWT with higher order of vanishing moments.

21. Existence of perfect reconstruction FIR filters (Chui and He) Suppose that are Laurent polynomials, and that the matrix has full rank for Then there exist such that

22. Non-Stationary Wavelets

23. Non-stationary MRA (NMRA) wavelets Let and be the two-scale matrices of the “refinable” functions and the wavelets , respectively; that is, where

24. Vanishing moment condition • is an approximate dual of order L. • If I is a finite interval, the above condition is equivalent to : the space of all polynomials of degree up to .

25. NMRA wavelet tight-frames VMR matrices are symmetric positive semi-definite banded matrices: • If I is a finite interval, • If I is an infinite interval,

26. NMRA tight-frame conditions (1) For a finite interval I, For an infinite interval I, each is bounded on and (2)

27. Non-stationary filters Non-stationary DFWT decomposition and perfect reconstruction

28. Matrix factorization for stationary tight frames

29. Matrix factorization for non-stationary tight frames where we use the notations and the even rows of the odd rows of

30. FIR filters for non-stationary perfect reconstruction

31. Two-scale matrix • Consider two nested knot vectors we have the refinement equation where the matrix has non-negative entries, with each row summing to 1. • can be derived by a sequence of knot insertions.

32. Interior wavelets with simple knots

33. Boundary wavelets with simple interior knots

34. Interior wavelets with double knots

35. Boundary wavelets with double interior knots

36. Meshless Spline Wavelets

37. Simplex spline D: a bounded convex polygonal domain in T: a knot set in D such that the projection of the set of vertices of simplex to is .

38. Neamtu’s work  on bivariate splines • The space of bivariate polynomials of (total) degree k is locally generated by simplex splines defined on the Delaunay configuration of degree k

39. A multi-level approximation by bivariate B-splines Let be a nested sequence of knot sets. Let denote the Delaunay configuration associated with the knot set . represent bivariate B-splines corresponding to

40. Refinement matrices can be derived by the “knot insertion" identity where and with

41. Tight-frame wavelets with maximum order of vanishing moments • Wavelets • Define operators that associate with some symmetric matrices ’s • Tight wavelet frames

42. Tight frame condition imposed on the nonstationary wavelets and

43. VMR matrices ’sconstruction is the row-vector of approximate duals for , that is, where P is the polar form of

44. k-Voronoi diagrams & simplex spline interpolation

45. k-Voronoi diagrams A set of knots X in 2D A family of (i+3) subsets of X ( features in (i+1)-Voronoi diagram ) A set degree-k of simplex spline basis A set of terrain samples P in 2D Simplex spline surface

46. k-Voronoi diagrams • Definition: A k-Voronoi diagram in 2D partitions the plane into cells such that points in each cell have the same closest k neighbors. Order 1 Order 3

47. k-Voronoi diagrams • Computation - Theory: O(n log(n)) time O(n) space - Practice: O(n) time • Engineering challenges: • speed • memory (streaming ) • robustness ( degeneracy, round-off errors )

48. Simplex spline interpolation • Problem: Given a set of terrain sample points, reconstruct the terrain with simplex splines.

49. Simplex spline interpolation • What knot sets to use?

50. k-Voronoi diagrams A set of knots X in 2D A family of (i+3) subsets of X ( features in (i+1)-Voronoi diagram ) A set degree-k of simplex spline basis A set of terrain samples P in 2D Simplex spline surface