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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 LeeCAGD 2000
Least squares
2{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 rr H
w 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 Y
SD 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 curve
Hongwei Lin, Wei Chen, Guojin Wang
The 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
d
rN
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 fitting
Ardeshir GoshtasbyACM Transactions on Graphics, 1
9:185--203, 2000
,Inverse distance between ( , ,) and ( ) :i ix y x y
2 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 u
du
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 bending
error 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 term
2Min ( ) || ( ) || ,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 minimization
Wenping Wang, Helmut Pottmann, Yang LiuToG 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 D
P 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 t
X P t2
, || ( ) ||PD k k ke P t X
Tangent distance minimization
2, || ( ( ) ) ||T
TD k k k ke P t X N
,
1
2TD TD k sk
f e f
Squared distance minimization
Pottman 2003
2 2( , )d
g x y x yd
0( , ) ( , )X x y N X
2 2| |ˆ ( , )
| |
dg x y x y
d
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
dP t X T
d
e P t X N d
P t X N d
Squared distance minimization
1 21
1
2
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 computation
2 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 ofthe 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
( )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
1
1
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 i
i ii i i i
X P t P tt t
X 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 t
X 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 curves
Zhouwang 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( ) ( ) 0
1 0T
f X
P X f X
( )( , ( )) min || ||
Y V fd P V f P Y
0
0
f f g
X 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