Curve fitting to point clouds

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Curve fitting to point clouds

• Reporter Lincong Fang
• Oct 18, 2006

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Curve fitting

• The data points are ordered.

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Curve fitting to point clouds

• The data points are unorganized.

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Applications

• Some applications
• Reverse engineering
• Curve design
• Surface reconstruction
• Etc.

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Approaches overview

• Preprocess the point clouds
• Thin the point clouds (Levin98, Lee00)
• Point clusters (Lin 04)
• Map into a digital image (Goshtasby00)
• Mathematical model
• Parameteric curves
• Implicit curves
• Other methods

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Curve Reconstruction from unorganized points

• In-Kwon Lee
• CAGD 2000

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Least squares
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Moving Least Squares
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Moving Least Squares
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Moving Least Squares
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Moving Least Squares
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The choice of H
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Improved Moving Least Squares
Delaunay triangulation
Euclidean minimum spanning tree (EMSP)
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Correlation
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Correlation
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Refining
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Compare with and without EMST
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Ordering points
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Example
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Approaches overview

• Preprocess the point clouds
• Thin the point clouds (Levin98, Lee00)
• Point clusters (Lin 04)
• Map into a digital image (Goshtasby00)
• Mathematical model
• Parameteric curves
• Implicit curves
• Other methods

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Curve reconstruction based on interval B-spline
curve

• Hongwei Lin, Wei Chen,
• Guojin Wang
• The Visual Computer,21(6), 418-427,2005

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Overview
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Shape-based joining scheme
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Sequence Joining Method
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Boundary sequence
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Example
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Example
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Approaches overview

• Preprocess the point clouds
• Thin the point clouds (Levin98, Lee00)
• Point clusters (Lin 04)
• Map into a digital image (Goshtasby00)
• Mathematical model
• Parameteric curves
• Implicit curves
• Other methods

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Grouping and parameterizing irregularly spaced
points for curve fitting

• Ardeshir Goshtasby
• ACM Transactions on Graphics, 19185--203, 2000

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Minor and major ridges
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Map into a digital image
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Minor and major ridges
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Example
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Example
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Approaches overview

• Preprocess the point clouds
• Thin the point clouds (Levin98, Lee00)
• Point clusters (Lin 04)
• Map into a digital image (Goshtasby00)
• Mathematical model
• Parameteric curves
• Implicit curves
• Other methods

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Multidimensional curve fitting to unorganized
data points by nonlinear minimization

• Lian Fang, David C Gossard
• CAD 95

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Physical analogy
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Error term
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Example
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Fitting B-spline curves to point clouds by
curvature-based squared distance minimization

• Wenping Wang, Helmut Pottmann, Yang Liu
• ToG 2006

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Point distance minimization
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Tangent distance minimization
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Squared distance minimization
Pottman 2003
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Squared distance minimization
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Squared distance minimization
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Comparison

• PDM has slow convergence
• TDM has fast but unstable convergence
• SDM yields a more balanced performance between
  efficiency and stability

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Comparison
The fitting curve generated by PDM, TDM, SDM in
50 iterations
Initial curve
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Foot point computation
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Open curves
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Initial curves and control points

• Specify by user
• Compute a quadtree partition of
• the data points
• Automatic or specify by user, and adjustment
  (Yang 2004)

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Example
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Reconstructing B-spline curves from point
cloudsA tangential flow approach using least
squares minimization

• Yang Liu, Huaiping Yang, Wenping Wang
• Shape Modeling and Applications, 2005
  International Conference

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Input

• Unacceptable point clouds.

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Data Analysis
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Initialization and approximation
Random point S
B-spline curve
Fitting line L
I
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Growing
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Knot insertion

• All points are handled, add a knot where the
  maximum error occurs
• Else insert a knot and redistribute all the knots
  and make them equally spaced

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Finding projection points
Sharp corners
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Filtering points
T
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Other cases
Less control points
EMST with wrong topology
Very sharp corner
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Example
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Example
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Approaches overview

• Preprocess the point clouds
• Thin the point clouds (Levin98, Lee00)
• Point clusters (Lin 04)
• Map into a digital image (Goshtasby00)
• Mathematical model
• Parameteric curves
• Implicit curves
• Other methods

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Fitting unorganized point clouds with active
implicit B-spline curves

• Zhouwang Yang, Jiansong Deng, Falai Chen
• Visual Computer 2005

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Example
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Approaches overview

• Preprocess the point clouds
• Thin the point clouds (Levin98, Lee00)
• Point clusters (Lin 04)
• Map into a digital image (Goshtasby00)
• Mathematical model
• Parameteric curves
• Implicit curves
• Other methods

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Conclusion

• Complex topology
• Digital image
• Implicit curves
• Tangential flow
• Initial curves
• Parameteric curves
• Implicit curves

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Problems and future work

• Knot insertion
• Foot point compute
• Singular points
• Surface reconstruction

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Thank you!