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Reverse Engineering of Point Clouds to Obtain Trimmed NURBS

Reverse Engineering of Point Clouds to Obtain Trimmed NURBS. Masters Thesis Proposal. Lavanya Sita Tekumalla. School of Computing University of Utah. Advisor: Prof. Elaine Cohen. Motivation. Motivation: Digitizing Geometry CAD Modeling Field of Entertainment Aim

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Reverse Engineering of Point Clouds to Obtain Trimmed NURBS

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

  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 (xj , yj) • The weighting function defined from the point of view of (xj , yj)

  7. BackgroundMLS 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 WorkKnot 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 WorkParameterization • Projection- Might not be a bijection • Curves- chord length parameterization • Surfaces

  11. ParameterizationConvex combination maps • Map the boundary vertices to a convex polygon • For each interior vertex Pi choose a neighborhood Ni and positive weights λj • The parameterization maps Pi to Ui

  12. Previous WorkParameterizing Triangular meshes • Convex Combination Maps • Floater • Mesh as a Spring System • Hormann et al • Harmonic Maps • Eck et al • Floater

  13. Previous WorkParameterizing 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 ResearchSmoothing • 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 ResultsSmoothing

  18. Proposed ResearchHole 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 ResearchHole 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 ResultsHole Filling – Curve Example

  21. Preliminary ResultsHole Filling- Surface

  22. Preliminary ResultsHole Filling- Mesh

  23. Preliminary ResultsHole Filling

  24. Proposed ResearchParameterization • 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

  26. Preliminary Results Parameterization

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

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

  29. Proposed ResearchFitting: Knot placement Hierarchical Domain Decomposition

  30. Proposed ResearchFitting • Blending local fits • Moving least squares fit with respect to the mid-point of the patch (xj , yj) • Basis functions: Cubic b-spline bases truncated in a knot interval.

  31. Blending Local Fits – Basis Functions

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

  33. Blending local Fits-Blending control points

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

  35. Further Proposed ResearchAnalysis • 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. Preliminary Results Hierarchical Subdivision of Parametric Domain to Decide Knot Placement

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

  39. SummaryFurther 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. SummaryFurther 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.

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