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

Motivation

Motivation: Digitizing Geometry• CAD Modeling• Field of Entertainment

Aim• Reverse Engineering point clouds

to obtain Trimmed NURBS http://www.qcinspect.com/rev.htm

Problem definition

Dealing with Problems associated with point clouds

• Large data sets• Noise• Holes and missing data

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

Problem Definition

Finding a good fitting strategy• Capture detail• Knot placement• Computation speed• Stable

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)

2( ( ) ) ( )i i j

i

Min f x y w i

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

( ) ( ( ))F x F F x

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”

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

Previous WorkParameterization

• Projection- Might not be a bijection• Curves- chord length parameterization• Surfaces

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

, 1i

i j

j N

,

i

i i j j

j N

U U

Previous WorkParameterizing Triangular meshes• Convex Combination Maps

– Floater• Mesh as a Spring System

– Hormann et al• Harmonic Maps

– Eck et al– Floater

Previous WorkParameterizing triangular meshes• Conformal Maps: Free boundary

– Non linear techniques: • Hormann et al• Sheffer et al

– Linear techniques• Levoy et al

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.

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

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

Preliminary ResultsSmoothing

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

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)

Preliminary ResultsHole Filling – Curve Example

Preliminary ResultsHole Filling- Surface

Preliminary ResultsHole Filling- Mesh

Preliminary ResultsHole Filling

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

Preliminary Results Parameterization

Preliminary Results Parameterization

Proposed ResearchCompleting Parametric Domain

•Intutive parameterization

•Complete the parametric domain by introducing points in the domain and projecting them onto the surface

Completing the Parametric DomainExample

Harmonic Map with boundary fixed by projecting the actual

boundary

Data Added in the parameterization to get a

rectangular domain

Proposed ResearchFitting: Knot placementHierarchical Domain Decomposition

Proposed Research Fitting

• 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.

2( ( ) ) ( )i i j

i

Min f x y w i

Blending Local Fits – Basis Functions

3

, 33 3 3

2 21 1 3

2 1 1 1 1 3 2 1

1 2

2 1 2

( )(0,3)( )( )( )

( ) ( ) ( ) ( )(2,3)( )( )( ) ( )( )( )

( )( )( ) ( )( )(

ii

i i i i i i

i i i i

i i i i i i i i i i i i

i i i

i i i i

t tf Bt t t t t t

t t t t t t t tft t t t t t t t t t t tt t t t t t

t t t t

2, 31

31

3, 31 2 1 1

1, 3

)( )(3,3)

( )( )( )(1,3) 1 ( (0,3) (2,3) (3,3))

ii i

ii

i i i i i i

i

Bt t

t tf Bt t t t t t

f f f f B

The basis functions in the interval ti to ti+1 over an arbitrary (non-uniform) knot vector

Blending Local Fits – Basis Functions

Blending local Fits-Blending control points

Further Proposed ResearchFitting

• Parameters that determine a local fit– Local weighting function– Neighborhood size

• Constrained fit, given a smooth boundary

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

Preliminary Results

Preliminary Results

Hierarchical Subdivision of Parametric Domain to Decide Knot Placement

SummaryPreliminary Research

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

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

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

Questions and Suggestions.