Curve fitting to point clouds

Reporter: Lincong FangOct 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 LeeCAGD 2000

Least squares2{P ( , ) | 1,..., }i i iS x y i N

2

1

Min ( )N

l i ii

D ax b y

A regression line,L:y=ax+b

Moving Least Squares

2

1

Min ( )i i

N

l ii

D ax b y w

2 /r H

iw e2

*|| P P ||ir

: prescribed real constantH

*

*

For a point P ,a local regression line, L : y ax b

Moving Least Squares*

*

New origin: P axis: parallel to Lx

ˆ ˆ ˆ ˆ{P ( , ) | 1,..., }: transformed point seti i iS x y i N 2

* *ˆˆ ˆ ˆQ : for P :y ax bx c

2 2

1

ˆ ˆ ˆMin ( )N

q i ii ii

D ax bx c y w

*P̂

(0, )c

Moving Least Squares

2 2

1

ˆ ˆ ˆMin ( )N

q i ii ii

D ax bx c y w

3 2

3 22 3 1, if , 0 if ,

(wyvill et al.,1986)

i

r r r Hw H Hr H

Moving Least Squares

The choice of H

Improved Moving Least Squares

Delaunay triangulationEuclidean minimumspanning tree (EMSP)

Correlation

Covariance:( , ) [( ( ))( ( ))]

= ( ) ( ) ( )Cov X Y E X E X Y E Y

E XY E X E Y

( , )( , )( ) ( )

Cov X YX YSD X SD Y

, : random variablesX Y

Correlation

Refining

*P||P Q ||

|A|

*j

jA

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 curveHongwei Lin, Wei Chen, Guojin WangThe Visual Computer,21(6), 418-427,2005

Overview

max : the largest edge length of adjecent

edges of in (Delaunay triangulation)pd

p M

max

max

Average sampling radius:

pp S

dr

N

Shape-based joining scheme

Sequence Joining Method

Boundary sequence

Example

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 fittingArdeshir Goshtasby

ACM Transactions on Graphics, 19:185--203, 2000

,Inverse distance between ( , ,) and ( ) :i ix y x y2 2 1/ 2[( ) ( ) ]i ix x y y

2 2 2 1/ 2

1

A surface

( , ) [( ) ( ) ]N

i ii

g x y x x y y r

Minor and major ridges

Map into a digital image

Minor and major ridges

Example

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 GossardCAD 95

2

1 1

Min

( ) ( ) ( ) || || || ( ) ||mM N

m i imCm i

d w uE w u P w udu

1 2( ) ( ( ), ( ),..., ( ))dw u x u x u x u

Physical analogy2 2 2

N2

i=1

( ) ( || ' || || '' || || ''' || )

+ ( , ( ))

C

i i

E w w w w du

D P w u

2

2

|| ' || : strain energy of stretching

|| '' || : strain energy of bendingerror term: strain energy stored in springs

w

w

1

( ) ( )n

Ti i

i

w u q u Q

: a set of vectorsiq

( ) : a set of functionsi u

1 2[ , ,..., ]TnQ q q q

1 2[ , ,..., ]Tn

Error term2Min ( ) || ( ) || ,0G u P w u u h

( ) : ( ) 0g u G u

,0 0

( ) ( )M M

ii j j M

i j

g u a u g B t

Example

Fitting B-spline curves to point clouds by curvature-based squared distance minimizationWenping Wang, Helmut Pottmann, Yang Liu

ToG 2006

2

1

1Min ( ) ( ( ), )2

n

k sk

f t d P t X f

1

( ) ( )m

i ii

P t PB t

( ( ), ) min || ( ) ||k ktd P t X P t X

: regularization termsf

,1

( ) ( ) : current fitting curvem

c i c ii

P t B t P

1 2

,1

( , ,..., )

( ) ( )( ) : updated curves

m

c

m

i c i ii

D D D DP P D

P t B t P D

Point distance minimization

21ˆ || ( ) ||2 k k s

k

f P t X f ( ) : foot point

of on ( )c k

k c

P tX P t

2, || ( ) ||PD k k ke P t X

Tangent distance minimization

2, || ( ( ) ) ||T

TD k k k ke P t X N

,12TD TD k s

k

f e f

Squared distance minimization

Pottman 2003

2 2( , ) dg x y x yd

0( , ) ( , )X x y N X

2 2| |ˆ ( , )| |

dg x y x yd

Squared distance minimization

/( ) max{0, /( )}d d d d

2

2,

2

[( ( ) ) ]

[( ( ) ) ] ,if 0

[( ( ) ) ] , if 0

Tk k k

TSD k k k k

Tk k k

d P t X Td

e P t X N d

P t X N d

Squared distance minimization

1 21

12

n

kk

f e F F

2

1

22

|| '( ) ||

|| ''( ) ||

F P t dt

F P t dt

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 computation2 1 1 2 1 2( ) /( )kt d t d t d d

Open curves

,0 ,0 ,0cos (1 cos )outer PD SDe e e

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 minimizationYang Liu, Huaiping Yang, Wenping Wang

Shape Modeling and Applications, 2005 International Conference

( )k k kV X P t 2 2 2( , ( )) ( ) ( )T T

k k k k k kd X P t V T V N

2 2

0

1 ( ( ) ( ) )1

11

NT T

k k k k kk

s

F V T V NN

fn

21 12

0 0|| '( ) || (1 ) || ''( ) ||sf P t dt P t

Input

Unacceptable point clouds.

Data Analysis

mUniform cells{B ,m=0,...,M}

Thickness mw

Initialization and approximation

Random point S

10S il wI

Fitting line LB-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*

2

( ( )) '( )( ( )) "( ) '( )

i i ii i

i i i i

X P t P tt tX P t P t P t

Sharp corners

Filtering points

*

* *8

* *

satisfies

|| ( ) || / 2

| ( ( )) '( ) | 10|| ( ) |||| '( ) ||

will not involved

i

i i i

i i i

i i i

X

X P t w

X P t P tX P t P t

T

*( )iP t

iX

Other cases

Less control pointsEMST with wrong topologyVery sharp corner

Example

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 curvesZhouwang Yang, Jiansong Deng, Falai Chen

Visual Computer 2005

,

( , ) ( ) ( )rs r sr s

f x y c M x N y

2

Implicit B-spline curve

( ) {( , ) | ( , ) 0}V f x y f x y

2

1

Min ( ) ( , ( )) ( )M

i Ti

R f d P V f wEng f

2 2 2( ) ( )T xx xy yyEng f f f f dxdy

( )( , ( )) min || ||

Y V fd P V f P Y

( ) 0 0 1

( ) ( ) 01 0

T

f X

P X f X

( )( , ( )) min || ||

Y V fd P V f P Y

0

0

f f gX X X

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