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Least Squares Orthogonal Distance Fitting of Implicit Curves and Surfaces Sung Joon Ahn, Wolfgang Rauh, and Matthias Recknagel Fraunhofer Institute for Manufacturing Engineering and Automation (IPA) Nobelstr. 12, 70569 Stuttgart, Germany {sja, wor, mhr}@ipa.fhg.de http://www.ipa.fhg.de/english/600/Informationstechnik e.php3
Abstract. Curve and surface fitting is a relevant subject in computer vision and coordinate metrology. In this paper, we present a new fitting algorithm for implicit surfaces and plane curves which minimizes the square sum of the orthogonal error distances between the model feature and the given data points. By the new algorithm, the model feature parameters are grouped and simultaneously estimated in terms of form, position, and rotation parameters. The form parameters determine the shape of the model feature, and the position/rotation parameters describe the rigid body motion of the model feature. The proposed algorithm is applicable to any kind of implicit surface and plane curve.
1
Introduction
Fitting of curve or surface to a set of given data points is a very common task carried out with applications of image processing and pattern recognition, e.g. edge detection, information extraction from 2D-image or 3D-range image. In this paper, we are considering least squares fitting algorithms for implicit model features. Algebraic fitting is a procedure whereby model feature is described by implicit equation F (a,X) = 0 with parameters a = (a1 , . . . , aq )T , and the error distances are defined with the deviations of functional values from the expected value (i.e. zero) at each given point. If F (a,Xi ) = 0, the given point Xi does not lie on the model feature (i.e. there is some error-of-fit). Most publications about LS-fitting of implicit features have been concerned with the square sum of algebraic distances or their modifications [5,9,10,12] σ02 =
m i
F 2 (a, Xi )
or
σ02 =
m
[F (a, Xi )/∇F (a, Xi )]2 .
i
In spite of advantages in implementation and computing costs, the algebraic fitting has drawbacks in accuracy, and is not invariant to coordinate transformation. In geometric fitting, also known as best fitting or orthogonal distance fitting, the error distance is defined as the shortest distance (geometric distance) of a given point to the model feature. Sullivan et al. [11] have presented a geometric B. Radig and S. Florczyk (Eds.): DAGM 2001, LNCS 2191, pp. 398–405, 2001. c Springer-Verlag Berlin Heidelberg 2001
Least Squares Orthogonal Distance Fitting of Implicit Curves and Surfaces
399
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I
= ;
2UWKRJRQDO FRQWDFWLQJSRLQW [cL
2). We obtain the initial parameter values set from a sphere fitting and an ellipsoid fitting, successively, with ε1 = ε2 = 1 (Table 2). Superquadric fitting to the 30 points in Table 1 representing a box is terminated after 18 outer iteration cycles for ∆a = 7.9 × 10−7 (Fig. 4).
4
Summary
Dimensional model fitting finds its applications in various fields of science and engineering, and, is a relevant subject in computer vision and coordinate metrology. In this paper, we have presented a new algorithm for orthogonal distance fitting of implicit surfaces and plane curves, by which the estimation parameters are grouped in form/position/rotation parameters, and simultaneously estimated. The new algorithm is universal, and very efficient, from the viewpoint
404
S.J. Ahn, W. Rauh, and M. Recknagel Table 1. Thirty coordinate triples representing a box
X Y Z
−4 3 13
1 16 29
4 −11 10
20 −17 22
−11 1 4
−26 7 −8
−3 −13 1
−7 −26 −15
6 19 9
11 24 19
X Y Z
3 −18 −25
15 9 21
18 −3 22
−21 3 −13
−4 −14 −22
−2 19 11
−14 14 1
20 −17 15
4 −20 −18
6 20 24
X Y Z
22 −8 4
30 −9 12
−8 15 −9
−16 15 −18
8 −13 −15
26 −14 17
−22 12 −13
−2 −22 −20
−3 −3 9
7 1 −5
Table 2. Results of the orthogonal distance fitting to the points set in Table 1 Parameters ˆ a
σ0
a
b
c
ε1
ε2
Sphere Ellipsoid Superquadric σ(ˆ a)
33.8999 14.4338 0.9033 −−
46.5199 46.3303 24.6719 0.1034
−− 25.5975 20.4927 0.1026
−− 9.3304 8.2460 0.0598
−− −− 0.0946 0.0151
−− −− 0.0374 0.0197
Parameters ˆ a
Xo
Yo
Zo
ω
ϕ
κ
Sphere Ellipsoid Superquadric σ(ˆ a)
27.3955 1.6769 1.9096 0.0750
18.2708 −20.8346 −1.2537 0.8719 −1.0234 2.0191 0.0690 0.0774
−− 0.7016 0.6962 0.0046
−− −0.7099 −0.6952 0.0031
−− 0.6925 0.6960 0.0059
of implementation and application to a new model feature. Memory space and computing time costs are proportional to the number of the given points. Our algorithm converges very well, and does not require a necessarily good initial parameter values set, which could also be internally provided (e.g. gravitational center and RMS central distance of the given points set for sphere fitting, sphere parameters for ellipsoid fitting, and ellipsoid parameters for superquadric fitting, etc.). If there is a danger of local minimum estimation, we apply the random walking technique along with the line search [6]. For practical applications, we can individually weight each coordinate of the given data points with the reciprocals of the axis accuracy of the measuring machine (see Eq. (5)). Together with other algorithm for orthogonal distance fitting of parametric features [3], our algorithm is certified by the German authority PTB [7], with a certification grade that the parameter estimation accuracy is higher than 0.1 µm for length unit, and 0.1 µrad for angle unit for all parameters of all tested model features.
5HVLGXDOVR
VR DA EA FA HK HK ;R
Abstract. Curve and surface fitting is a relevant subject in computer vision and coordinate metrology. In this paper, we present a new fitting algorithm for implicit surfaces and plane curves which minimizes the square sum of the orthogonal error distances between the model feature and the given data points. By the new algorithm, the model feature parameters are grouped and simultaneously estimated in terms of form, position, and rotation parameters. The form parameters determine the shape of the model feature, and the position/rotation parameters describe the rigid body motion of the model feature. The proposed algorithm is applicable to any kind of implicit surface and plane curve.
1
Introduction
Fitting of curve or surface to a set of given data points is a very common task carried out with applications of image processing and pattern recognition, e.g. edge detection, information extraction from 2D-image or 3D-range image. In this paper, we are considering least squares fitting algorithms for implicit model features. Algebraic fitting is a procedure whereby model feature is described by implicit equation F (a,X) = 0 with parameters a = (a1 , . . . , aq )T , and the error distances are defined with the deviations of functional values from the expected value (i.e. zero) at each given point. If F (a,Xi ) = 0, the given point Xi does not lie on the model feature (i.e. there is some error-of-fit). Most publications about LS-fitting of implicit features have been concerned with the square sum of algebraic distances or their modifications [5,9,10,12] σ02 =
m i
F 2 (a, Xi )
or
σ02 =
m
[F (a, Xi )/∇F (a, Xi )]2 .
i
In spite of advantages in implementation and computing costs, the algebraic fitting has drawbacks in accuracy, and is not invariant to coordinate transformation. In geometric fitting, also known as best fitting or orthogonal distance fitting, the error distance is defined as the shortest distance (geometric distance) of a given point to the model feature. Sullivan et al. [11] have presented a geometric B. Radig and S. Florczyk (Eds.): DAGM 2001, LNCS 2191, pp. 398–405, 2001. c Springer-Verlag Berlin Heidelberg 2001
Least Squares Orthogonal Distance Fitting of Implicit Curves and Surfaces
399
0HDVXUHPHQWSRLQW ;L
I
= ;
2UWKRJRQDO FRQWDFWLQJSRLQW [cL
2). We obtain the initial parameter values set from a sphere fitting and an ellipsoid fitting, successively, with ε1 = ε2 = 1 (Table 2). Superquadric fitting to the 30 points in Table 1 representing a box is terminated after 18 outer iteration cycles for ∆a = 7.9 × 10−7 (Fig. 4).
4
Summary
Dimensional model fitting finds its applications in various fields of science and engineering, and, is a relevant subject in computer vision and coordinate metrology. In this paper, we have presented a new algorithm for orthogonal distance fitting of implicit surfaces and plane curves, by which the estimation parameters are grouped in form/position/rotation parameters, and simultaneously estimated. The new algorithm is universal, and very efficient, from the viewpoint
404
S.J. Ahn, W. Rauh, and M. Recknagel Table 1. Thirty coordinate triples representing a box
X Y Z
−4 3 13
1 16 29
4 −11 10
20 −17 22
−11 1 4
−26 7 −8
−3 −13 1
−7 −26 −15
6 19 9
11 24 19
X Y Z
3 −18 −25
15 9 21
18 −3 22
−21 3 −13
−4 −14 −22
−2 19 11
−14 14 1
20 −17 15
4 −20 −18
6 20 24
X Y Z
22 −8 4
30 −9 12
−8 15 −9
−16 15 −18
8 −13 −15
26 −14 17
−22 12 −13
−2 −22 −20
−3 −3 9
7 1 −5
Table 2. Results of the orthogonal distance fitting to the points set in Table 1 Parameters ˆ a
σ0
a
b
c
ε1
ε2
Sphere Ellipsoid Superquadric σ(ˆ a)
33.8999 14.4338 0.9033 −−
46.5199 46.3303 24.6719 0.1034
−− 25.5975 20.4927 0.1026
−− 9.3304 8.2460 0.0598
−− −− 0.0946 0.0151
−− −− 0.0374 0.0197
Parameters ˆ a
Xo
Yo
Zo
ω
ϕ
κ
Sphere Ellipsoid Superquadric σ(ˆ a)
27.3955 1.6769 1.9096 0.0750
18.2708 −20.8346 −1.2537 0.8719 −1.0234 2.0191 0.0690 0.0774
−− 0.7016 0.6962 0.0046
−− −0.7099 −0.6952 0.0031
−− 0.6925 0.6960 0.0059
of implementation and application to a new model feature. Memory space and computing time costs are proportional to the number of the given points. Our algorithm converges very well, and does not require a necessarily good initial parameter values set, which could also be internally provided (e.g. gravitational center and RMS central distance of the given points set for sphere fitting, sphere parameters for ellipsoid fitting, and ellipsoid parameters for superquadric fitting, etc.). If there is a danger of local minimum estimation, we apply the random walking technique along with the line search [6]. For practical applications, we can individually weight each coordinate of the given data points with the reciprocals of the axis accuracy of the measuring machine (see Eq. (5)). Together with other algorithm for orthogonal distance fitting of parametric features [3], our algorithm is certified by the German authority PTB [7], with a certification grade that the parameter estimation accuracy is higher than 0.1 µm for length unit, and 0.1 µrad for angle unit for all parameters of all tested model features.
5HVLGXDOVR
VR DA EA FA HK HK ;R
