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!!!!!