Semi-Differential Invariants for Recognition of Algebraic Curves

Semi-Differential Invariants for Recognition of Algebraic Curves

Yan-Bin Jia and Rinat Ibrayev

Department of Computer Science Iowa State University Ames, IA 50011-1040

July 13, 2004

Object

Model-Based Tactile Recognition

)( 1 n,..,at;axx )( 1 n,..,at;ayy

Tactile data♦ contact (x, y)

Determine

♦ Shape

♦ Location of contact t on the object

Identify curve family

naa ,..,1 Estimate

shape parameters

estimate curvature and derivative w.r.t. arc length s

Models: families ofparametric shapes

Each model:

Related Work

Shape Recognition through TouchGrimson & Lozano-Perez 1984; Fearing 1990; Allen & Michelman 1990; Moll & Erdmann 2002; etc.

Differential & Semi-differenitial Invariants

Padjla & Van Gool 1992; Rivlin & Weiss 1995; Moons et al. 1995; Calabi et al. 1998; Keren et al. 2000; etc.

Vision & Algebraic Invariants Hilbert; Kriegman & Ponce 1990; Forsyth et al. 1991; Mundy & Zisserman 1992; Weiss 1993; Keren 1994; Civi et al. 2003; etc.

Signature Curve

♦ Used in model-based recognition

Requiring global data

♦ Independent of rotation and translation

What if just a few data points?

Plot curvature against its derivative along the curve:

signature curve

-0.75 -0.5 -0.25 0.25 0.5 0.75-0.5

0.5

1

1.5

2

2.5

s

-1 -0.5 0.5 1

-1.5

-1

-0.5

0.5

1

1.5

cubical parabola:

xxy 4.06.0 3

y

x

Eliminate t from and

♦ How to derive?

Differential Invariants

),..,;( 1 naat

),..,(),..,,,..,( 111 nmssm aagI

),..,;( 1 ns aat

♦ Expressions of curvature and derivatives (w.r.t. arc length) Computed from local geometry Small amount of tactile data

invariant

0),..,,,( 1 ns aaf

Independent of position, orientation, and parameterization

Well, ideally so …constant

Independent of point location on the shape

curvature derivative

Parabola

x

y4.0a

8.0a6.0a s

0.5

0.5

pI0.5 1

012

2

012

2

btbtby

atatax

rot, trans, and reparam. aty

atx

2

2

Only 1 parameter instead of 6 Shape remains the same

Invariant:

1

9),(

4

23/2

s

spI 3/2)2(

1

a

evaluated at one point

signature curveshape classification

Semi-Differential Invariants

♦ Differential invariants use one point.

n shape parameters n independent diff. invariants.

up to n+2th derivatives

Numerically unstable!

♦ Semi-differential invariants involve n points.

n curvatures + n 1st derivs

Quadratics: Ellipse

)sin(

)cos(

tby

tax

3/222113/2

23/2

1

3/221

1 )(

1),(),(

)(

abIII spspc

3/4

22

223/2

2113/2

13/22

3/21

2 )(),(),(

1

ab

baIII spspc

♦ Two points involved

♦ Two independent invariants required

1cI

2cI

shape classifiers

Quadratics: Hyperbola

)sinh(

)cosh(

tby

tax

3/21 )(

1

abIc 3/4

22

2 )(ab

baIc

♦ Invariants same as for ellipse

♦ Different value expressions in terms of a, b

♦ distinguishes ellipses (+), hyperbolas (-), parabolas (0) 　　　　　　　　　　　　　　　　　　　1cI

Cubics

x

yt

)(

♦ Eliminate parameter t directly?

High degree resultant polynomial in shape parameters

Computationally very expensive

♦ Reparameterize with slope

Lower the resultant degree

Two slopes related to change of tangential angle (measurable)

Slope depends on rotation

Invariants in terms of ,, s

Invariants for Cubics

ctaty

tx

3

aI scp

6

)1)(3( 222

1

cIs

cp

)3(2

)1(2

22

2

23

2

btaty

tx

aIs

scp

)3(9

)1(82

2/523

1

bIs

scp

2

22

2 3

)1(

cubical parabola semi-cubical parabola

)tan(1

)tan(

1

12

),,,(),,( 22211111 scpscp II

Simulations

Parabola Ellipse Hyperbola Cub. par Semi-cub.

real 0.2198 0.1760 -0.1222 6.9963 6.5107

min 0.2168 0.1711 -0.1369 6.7687 6.3945

max 0.2230 0.1790 -0.1147 7.0289 6.5834

mean 0.2198 0.1756 -0.1225 6.9355 6.5154

♦ Testing invariants (curvature & deriv. est. by finite differences)

♦ Shape recovery

Average error on shape parameter estimation

Parabola Ellipse Hyperbola Cub. Par. Semi-cub.

0.36% 0.40% 1.15% 0.83% 1.23%

Summary over 100 different tests on randomly generated points for each curve

Summary over 100 different shapes for each curve family

Simulations (cont’d)

invariant

data

conic cubical

parabola

semi-cub.

parabola

conic

(ellipse)

-11.97 (min)

15.46 (max)

-0.04 (mean)

2.53 (stdev)

-265.80

5.83

-3.22

26.75

cubical

parabola

-6.38

-0.04

-0.73

1.22

7.80

65.22

29.17

17.19

semi-cub.

parabola

-22.84

28.37

3.37

6.76

8.54

19.03

13.76

3.07

Each cell displays the summary over 100 values

Data from one curve inapplicable for an invariant for a different class.

Recognition Tree

Tactile data

Parabola

pI

2cI1cI

Sign 1cI

Ellipse Hyperbola

a

a, b a, b

yes no

yes no

>0 < 01cpI 2cpI

CubicalParabola

1scpI 2scpI

Semi-CubicalParabola

a, b

a, b

no

no

yes

yes

Cubic

Spline?…

♦ Solve for t after recognition.

Locating Contact

2/32 )1(2

1

ta

322 )1(4

3

ta

ts

♦ Parameter value t determines the contact.

23 st

parabola:

Numerical Curvature Estimation

♦ Noisy tactile data

Curvature – inverse of radius of osculating circle

Derivative of curvature – finite difference

ellipse signature curve

x

y

1

1

(cm)

(cm)

s

(1/cm)

(1/cm )2

♦ A tentative approach

courtesy of Liangchuan Mifor supplying raw data

large errors!

Curvature Estimation – Local Fitting

♦ Curvature estimation

fit a quadratic curve to a few local data points

differentiate the curve fit (1)

♦ Curvature derivative estimation

generate multiple (s, ) pairs in the neighborhood

fit and differentiate again

numerically estimate arc length s using curve fit (1)

Experiments

1cI a b

real 0.373836 1.30145 2.5 1.75

min 0.350559 1.26074 2.38636 1.62636

max 0.404903 1.36736 2.67234 1.83549

mean 0.377728 1.31825 2.51127 1.71959

Summary over 80 different values for the ellipse

2cI

ellipse signature curve

1cI

x

y

(cm)

(cm)

1

1

s

0.03

0.01

(1/cm)

(1/cm )2

Experiments (cont’d)

1cI

x

y

cubic spline signature curve

s

but unstable invariant computation …

Seemingly good curvature & derivative estimates,

Summary & Future Work

♦ Differential invariants for quadratic curves & certain cubic curves

Improvement on robustness to sensor noise

Invariant to point locations on a shape (not just to transformation) Discrimination of families of parametric curves

Unifying shape recognition, recovery, and localization

Numerical estimation of curvature and derivative

Invariant design for more general shape classes (3D)

Computable from local tactile data