1. Reverse Engineering of Point Clouds to Obtain Trimmed NURBS Lavanya Sita Tekumalla Advisor: Prof. Elaine Cohen School of Computing University of Utah Masters Thesis Proposal

2. Motivation Motivation: Digitizing Geometry CAD Modeling Field of Entertainment Aim Reverse Engineering point clouds to obtain Trimmed NURBS http://www.qcinspect.com/rev.htm

3. Problem definition Dealing with Problems associated with point clouds Large data sets Noise Holes and missing data

4. Problem Definition Finding a Suitable Parameterization Minimum distortion Handle holes in the data Intuitive parameterization A rectangular boundary for fitting tensor product surfaces Handling non-rectangular geometry

5. Problem Definition Finding a good fitting strategy Capture detail Knot placement Computation speed Stable

6. Background Moving Least Squares Weighted Least Squares fit Moving least square fit at point (x j, y j ) –The weighting function defined from the point of view of (x j, y j )

7. Background MLS Projection A given point set implicitly defines a surface S A projection procedure F such that S is the set of all points that project onto themselves

8. Previous Work Fitting a network of patches – 1996: Eck et al, M., Hoppe, H. "Automatic reconstruction of B-spline surfaces of arbitrary topological type." – 1999: I.K. Park, I.D. Yun, S.U. Lee, "Constructing NURBS Surface Model from Scattered and Unorganized Range Data“ – 2000: Benjamin F. Gregorski, Bernd Hamann, Kenneth I. Joy, “Reconstruction of B-spline Surfaces from Scattered Data Points”

9. Previous Work Knot placement Non-linear optimization- Free knot problem –Jupp et al –Dierekx –Deboor and Rice Iterative knot insertion and removal –Dierekx –Baussard et al

10. Previous Work Parameterization Projection- Might not be a bijection Curves- chord length parameterization Surfaces

11. Parameterization Convex combination maps Map the boundary vertices to a convex polygon For each interior vertex P i choose a neighborhood N i and positive weights λ j –The parameterization maps P i to U i

12. Previous Work Parameterizing Triangular meshes Convex Combination Maps –Floater Mesh as a Spring System –Hormann et al Harmonic Maps –Eck et al –Floater

13. Previous Work Parameterizing triangular meshes Conformal Maps: Free boundary –Non linear techniques: Hormann et al Sheffer et al –Linear techniques Levoy et al

14. Proposed Research A reverse engineering framework to obtain trimmed NURBS from point clouds Deal with a single NURBS patch Assumption (In the preliminary work): An underlying mesh structure is available.

15. Proposed Research A multistage framework Smoothing for noise removal Hole filling and triangulation of hole Parameterization Extending boundaries- completing rectangular domain Fitting by blending local fits

16. Proposed Research Smoothing Find the local neighborhood of each point Project each point onto the surface obtained using MLS projection procedure Further Proposed Research: Smoothing the boundary curve

17. Preliminary Results Smoothing

18. Proposed Research Hole filling Motivation –Parameterize data –Lack of data – Numerical instabilities –Effect on areas around the hole Issues –Need for a local method –Adequate sampling density

19. Proposed Research Hole Filling For each point in the boundary: Find the local neighborhood Find a local reference plane and a local parameterization by projection Introduce points in the local parameterization Project each point in the parametric domain onto its local least squares surface Triangulate simultaneously(for meshes)

20. Preliminary Results Hole Filling – Curve Example

21. Preliminary Results Hole Filling- Surface

22. Preliminary Results Hole Filling- Mesh

23. Preliminary Results Hole Filling

24. Proposed Research Parameterization Parameterization by harmonic maps Further Proposed Research –Fixing the boundary suitably –Iterative reparameterization based on closest point to the fitted surface –Stretch minimization –Domain specific methods- for circular objects –Meshless parameterization

25. Preliminary Results Parameterization

27. Proposed Research Completing Parametric Domain Intutive parameterization Complete the parametric domain by introducing points in the domain and projecting them onto the surface

28. Completing the Parametric Domain Example Harmonic Map with boundary fixed by projecting the actual boundary Data Added in the parameterization to get a rectangular domain

29. Proposed Research Fitting: Knot placement Hierarchical Domain Decomposition

30. Proposed Research Fitting Blending local fits Moving least squares fit with respect to the mid- point of the patch (x j, y j ) Basis functions: Cubic b-spline bases truncated in a knot interval.

31. Blending Local Fits – Basis Functions

32. The basis functions in the interval t i to t i+1 over an arbitrary (non- uniform) knot vector Blending Local Fits – Basis Functions

33. Blending local Fits-Blending control points

34. Further Proposed Research Fitting Parameters that determine a local fit –Local weighting function –Neighborhood size Constrained fit, given a smooth boundary

35. Further Proposed Research Analysis Hole filling Vs A minimum norm least squares solution with SVD Blending local fits Vs A global least squares fit –Quality of fit –Efficiency of computation –Numerical stability Quantify the quality of fit

36. Preliminary Results

37. Hierarchical Subdivision of Parametric Domain to Decide Knot Placement

38. Summary Preliminary Research Smoothing surfaces by MLS projection Hole filling Parameterization using harmonic maps Hierarchical domain decomposition Fitting by blending local fits

39. Summary Further Proposed Research Smoothing boundary curves Fixing boundary parameterization Fixing boundaries curves Extending boundaries to rectangular domain Avoiding the use of a mesh structure Stretch minimization Iterative reparameterization: Parameter correction Domain specific method for parameterizing circular objects

40. Summary Further Proposed Research Determine the right weighting function and neighborhood size for the MLS process. Compare the fitting process with a minimum norm least squares solution- without filling holes Compare the fitting process with a global least squares fit –Quality of fit –Efficiency of computation –Numerical stability Quantify the quality of fit

41. Questions and Suggestions.