arc-length based curvature estimator thomas lewiner, joão d. gomes jr., hélio lopes, marcos...
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Arc-Length Based CurvatureEstimator
Thomas Lewiner, João D. Gomes Jr. , Hélio Lopes, Marcos Craizer
{ tomlew , jgomes , lopes , craizer }@ mat.puc-rio.br
Scope
Digital Curves
Gaussian convolution : [Worring & Smeulders, 1993]
FFT :[Estrozi, Campos, Rios, Cesar & Costa, 1999]
Sampled Curve
3-Points Methods
• Angle Among Three Points[Coeurjoly et al.,2001]
• External Angle [Gumhold, 2004]
3-Points Methods
• Circumscribed Circle [Coeurjolly & Svensson,2003]
• Derivatives Estimations Among Three Points [Belyaev, 2004]
ba
ArBr
a
ArOr
b
OrBrr
)()()()()()(
Least Square Methods
• Rigid Parabola Fitting [Pouget & Cazals,2003]
• Circle Fitting [Pratt,1987]
Rigid Parabola Fitting
Rotated Parabola
Circle Fitting
• Circle fit in low curvature
• A = 1
0)( 22 DCyBxyxA
Objectives
Robust computation of:
• Tangent Vector
• Normal Vector
• Curvature
with a least-square approach
Parametric Parabola Fitting
• We shall fit our data to parabolas of the form:
2
2
2
1)(
2
1)(
sysyysy
sxsxxsx
jjj
jjj
Model
2
2
)(2
1
)(2
1
jij
jijji
jij
jijji
sysyyy
sxsxxx where sji approximates
the arc-length between pi and pj
Estimation of sji
jilli
jkk
ji
when ,1
• The arc-length estimator from pj to pi is defined as
1
when ,j
ikk
ji jill
Weighted Least Squares Approach
qj
qji
jij
jijjiijjx lxlxxxwxxE 22 ))(
2
1(),(
qj
qji
jij
jijjiijjy lylyyywyyE 22 ))(
2
1(),(
Solution
qj
qjiji
jii
qj
qjiji
jii
qj
qjiji
jii
qj
qjiji
jii
qj
qji
jii
qj
qji
jii
qj
qji
jii
j
j
j
j
yylwh
yylwg
xxlwf
xxlwe
lwc
lwb
lwa
bac
bgahy
bac
bhcgy
bac
beafx
bac
bfcex
)()(2
1
)(
)()(2
1
)(
)(4
1
)(2
1
)(
: where,
22
2
22
2
42
32
22
2
2
2
2
Methods
• Independent Coordinates– Use xj’, xj’’, yj’, yj’’ as above
• Dependent Coordinates (if y’j > x’j)
))(1()( 2jjj xysigny
j
jjj y
xxy
Curvature
22 )()(ˆ
jj
jjjj
yx
yxyxk
Example Eight Curve
Comparison with Rigid Parabola Fitting
Parametric Parabola FittingRigid Parabola Fitting
Comparison withCircle Fitting
Circle fitting Parametric Parabola Fitting
Numerical Errors
Rigid Parabola Fitting Dependent
Ill-conditioned matrixes
Improvements
ji
ji
xy
yx
ji
ji
yy
xx
TT
TT
yy
xx
Dependent Rotated
Calibration
Uniformly Sampled Not Uniformly Sampled
Calibration: Noisy case
Uniformly Sampled Not Uniformly Sampled
q = 1 q = 1
q = 5 q = 5
Example of curves
Future Works
• Cubic fitting
• Curves in the space
• Surfaces
Thanks!!!!!