semi-differential invariants for recognition of algebraic curves yan-bin jia and rinat ibrayev...
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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