curve fitting to point clouds reporter: lincong fang oct 18, 2006
TRANSCRIPT
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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 LeeCAGD 2000
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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
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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
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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
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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
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Moving Least Squares
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The choice of H
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Improved Moving Least Squares
Delaunay triangulationEuclidean minimumspanning tree (EMSP)
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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
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Correlation
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Refining
*P||P Q ||
|A|
*j
jA
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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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max : the largest edge length of adjecent
edges of in (Delaunay triangulation)
pd
p M
max
max
Average sampling radius:
pp S
d
rN
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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 GoshtasbyACM Transactions on Graphics, 1
9:185--203, 2000
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,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
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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 GossardCAD 95
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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
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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
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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
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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
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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 LiuToG 2006
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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
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,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
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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
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Tangent distance minimization
2, || ( ( ) ) ||T
TD k k k ke P t X N
,
1
2TD TD k sk
f e f
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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
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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
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Squared distance minimization
1 21
1
2
n
kk
f e F F
2
1
22
|| '( ) ||
|| ''( ) ||
F P t dt
F P t dt
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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
Initial curve The fitting curve generated by PDM, TDM, SDM in 50 iterations
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Foot point computation
2 1 1 2 1 2( ) /( )kt d t d t d d
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Open curves
,0 ,0 ,0cos (1 cos )outer PD SDe e e
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Initial curves and control points
Specify by user
Compute a quadtree partition ofthe 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 clouds—A tangential flow approach using least squares minimization
Yang Liu, Huaiping Yang, Wenping Wang
Shape Modeling and Applications, 2005 International Conference
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( )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
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Input
Unacceptable point clouds.
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Data Analysis
mUniform cells{B ,m=0,...,M}
Thickness mw
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Initialization and approximation
Random point S
10S il wI
Fitting line LB-spline curve
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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
*2
( ( )) '( )
( ( )) "( ) '( )i i i
i ii i i i
X P t P tt t
X P t P t P t
Sharp corners
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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
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Other cases
Less control pointsEMST with wrong topologyVery 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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,
( , ) ( ) ( )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
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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
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( ) 0
0 1( ) ( ) 0
1 0T
f X
P X f X
( )( , ( )) min || ||
Y V fd P V f P Y
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0
0
f f g
X X X
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Example
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![Page 87: Curve fitting to point clouds Reporter: Lincong Fang Oct 18, 2006](https://reader035.vdocuments.mx/reader035/viewer/2022062518/56649f4c5503460f94c6d06b/html5/thumbnails/87.jpg)
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
![Page 88: Curve fitting to point clouds Reporter: Lincong Fang Oct 18, 2006](https://reader035.vdocuments.mx/reader035/viewer/2022062518/56649f4c5503460f94c6d06b/html5/thumbnails/88.jpg)
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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