Curve fitting to point clouds Reporter: Lincong Fang Oct 18, 2006

Curve fitting The data points are ordered.

Curve fitting to point clouds The data points are unorganized.

Applications Some applications: Reverse engineering Curve design Surface reconstruction Etc.

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

Curve Reconstruction from unorganized points In-Kwon Lee CAGD 2000

Least squares

Moving Least Squares

The choice of H

Improved Moving Least Squares Delaunay triangulation Euclidean minimum spanning tree (EMSP)

Correlation

Refining

Compare with and without EMST

Ordering points

Example

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

Curve reconstruction based on interval B-spline curve Hongwei Lin, Wei Chen, Guojin Wang The Visual Computer,21(6), ,2005

Overview

Shape-based joining scheme

Sequence Joining Method

Boundary sequence

Example

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

Grouping and parameterizing irregularly spaced points for curve fitting Ardeshir Goshtasby ACM Transactions on Graphics, 19: , 2000

Minor and major ridges

Map into a digital image

Minor and major ridges

Example

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

Multidimensional curve fitting to unorganized data points by nonlinear minimization Lian Fang, David C Gossard CAD 95

Physical analogy

Error term

Example

Fitting B-spline curves to point clouds by curvature-based squared distance minimization Wenping Wang, Helmut Pottmann, Yang Liu ToG 2006

Point distance minimization

Tangent distance minimization

Squared distance minimization Pottman 2003

Squared distance minimization

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

Comparison Initial curve The fitting curve generated by PDM, TDM, SDM in 50 iterations

Foot point computation

Open curves

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)

Example

Reconstructing B-spline curves from point clouds — A tangential flow approach using least squares minimization Yang Liu, Huaiping Yang, Wenping Wang Shape Modeling and Applications, 2005 International Conference

Input Unacceptable point clouds.

Data Analysis

Initialization and approximation Random point S I Fitting line L B-spline curve

Growing

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

Finding projection points Sharp corners

Filtering points T

Other cases Less control points EMST with wrong topologyVery sharp corner

Example

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

Fitting unorganized point clouds with active implicit B-spline curves Zhouwang Yang, Jiansong Deng, Falai Chen Visual Computer 2005

Example

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

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

Problems and future work Knot insertion Foot point compute Singular points Surface reconstruction