Geometric interpretation and precision analysis of algebraic ellipse fitting using least squares method

Acta Geod. Geoph. Hung., Vol. 47(4), pp. 430–440 (2012)

DOI: 10.1556/AGeod.47.2012.4.4

GEOMETRIC INTERPRETATION AND PRECISIONANALYSIS OF ALGEBRAIC ELLIPSE FITTING USING

LEAST SQUARES METHOD

O Kurt and O Arslan∗

Engineering Faculty, Geodesy and Photogrammetry Engineering Department, KocaeliUniversity, 41380, Turkey, e-mail: [email protected]

[Manuscript received October 10, 2011; accepted July 2, 2012]

This paper presents a new approach for precision estimation for algebraic ellipsefitting based on combined least squares method. Our approach is based on coordi-nate description of the ellipse geometry to determine the error distances of the fittingmethod. Since it is an effective fitting algorithm the well-known Direct Ellipse Fit-ting method was selected as an algebraic method for precision estimation. Oncean ellipse fitted to the given data points, algebraic distance residuals for each datapoint and fitting accuracy can be computed. Generally, the adopted approach hasrevealed geometrical aspect of precision estimation for algebraic ellipse fitting. Theexperimental results revealed that our approach might be a good choice for precisionestimation of the ellipse fitting method.

Keywords: algebraic ellipse fitting; error distance; fitting; least squares;precision

1. Introduction

The fitting of geometric features is an important problem in several fields ofscience and engineering. Ellipse fitting is one of the classic problems of patternrecognition and a variety of approaches have been used for ellipse fitting. Fit-ting an ellipse is an important task in computer vision, e.g. circle projections maybecome ellipses, and it has many practical applications as face detection, qualitycontrol and analysis of grain. While the problem of model fitting has been success-fully addressed, precision aspect of the fitting method has not been investigatedin a geometrical perspective. On the one hand it is imperative to solve the prob-lem of how ellipse can be fitted to a given data set. But on the other hand it isobvious that precision issue of fitting scheme has a substantial impact on the recog-nition performance. Generally in the past, fitting problems have usually been solvedthrough the least squares (LS) method with respect to elective implementation and

∗Corresponding author

1217-8977/$ 20.00 c©2012 Akademiai Kiado, Budapest

ALGEBRAIC ELLIPSE FITTING 431

acceptable computing costs. LS fitting minimizes the squares sum of error-of-fitin predefined measures. There are two main categories of LS fitting problems forgeometric features, algebraic and geometric fitting, and these are differentiated bytheir respective definition of the error distances involved. By geometric fitting, wemean fitting geometric constraints to observed data and discerning the underlyinggeometric structure from the coefficients of the fitted equations. In the methodthe error distances are defined with the orthogonal, or shortest, distances from thegiven points to the geometric feature to be fitted. By algebraic fitting, a geometricfeature is described by implicit equation F (x; a) = 0 with the parameters vectora = (a1, . . . aq). The error distances are defined with the deviations of the implicitequation from the expected value (i.e. zero) at each given point. The nonequalityof the equation indicates that the given point does not lie on the geometric feature(i.e., there is some error-of-fit) (Safaee-Rad et al. 1991, Rosin 1996, Spath 1997).

As known within the group of LS methods, algebraic and geometric distances re-fer to the parameters being minimized within the second order polynomial equationrepresenting the ellipse. Algebraic least squares solutions are linear and relativelysimple to solve. In spite of the advantages in implementation and computing costs,algebraic fitting has drawbacks in accuracy and in relation to the physical inter-pretation of the fitting parameters and errors. The geometric fitting of ellipse hasattracted much attention and it gives a more meaningful parameter to minimizewith respect to ellipses. Unfortunately, geometric distance minimization methodsare non-linear, requiring iterative solutions. As it is known the LS fitting method isunstable if there are outliers in the data (Yu et al. 2012). Appropriate minimizationcriteria including functions were discussed in the literature. Despite its sensitivityto non-Gaussian noise, LS fitting is probably the most widely used approach forestimating the ellipse’s parameters, due to its computation efficiency as an estima-tor (Gander et al. 1994, Cui et al. 1996). Maximum Likelihood (ML) method hasbeen regarded as the most accurate method for ellipse fitting from time to time. Ahyperaccurate method was proposed based on error analysis of ML. Hyperaccuracycan not be viewed as a dramatic accuracy improvement for fitting practice althoughthe analysis has theoretical significance (Kanatani 2005).

One of the widely known algebraic fitting method is Direct Ellipse Fittingmethod which is specific to ellipses and not iterative at the same time. Here thefitting problem is linear and can be solved in a deterministic manner with a pro-gram that solves the generalized eigenvectors problem. The standard LS methodtries to minimize a distance measure from the data points to the ellipse. However,the minimization of the Euclidean distances from the data points to the ellipse hasbeen computationally impractical, because there is no closed form expression forthe real geometric distance from a point to a general algebraic ellipse and itera-tive methods are required to compute it. Thus, the real geometric distance hasbeen approximated (Ahn et al. 2001, Kanatani and Sugaya 2007). To overcomethis problem a new approach has been developed in the study. After the fittingprocedure the geometrical error distances can be computed and accuracy (precisionin fact) analysis of the fitting be made in a practical manner. To accomplish thisaim, an explicit mathematical method of precision that reveals geometrical proper-

Acta Geod. Geoph. Hung. 47, 2012

432 O KURT and O ARSLAN

ties of the fitting method has been developed within the framework of least squaresconcept. In this paper we propose a precision analysis concept for algebraic fittingbased on ‘combined least squares’ method for ellipse fitting. The well-known DirectEllipse Fitting method was selected for precision analysis among from the otheralgebraic methods as it is an effective fitting algorithm in this work. Our approachis based on coordinate description of the ellipse geometry to determine the precisiondegree (via estimating unknown parameters, computing standard deviation and theresidual values etc.) of the fitting method.

2. Proposed scheme

Our approach is based on the coordinate description of the ellipse geometry andwe begin our study of the problem by discussing its geometric aspect. The diagramshown in Fig. 1 illustrates our discussion. An ellipse is to be defined with twocoordinate system. First coordinate system (x/y set of axes) has its origin at thecentre of ellipse with semi-major axis (a) and semi-minor axis (b). Consider thissystem as ellipse coordinate system. Second coordinate system (X/Y) specifies thelocation of the ellipse with respect to other objects. Ellipse is rotated anti-clockwisethrough an angle α. Any point on the ellipse is able to directly be determined withrespect to measured data point (Xi, Yi) if the origin of the ellipse (XC , YC), theellipse axes and the rotation of any axis (i.e. α) are known. Unknown parametersare (a, b,XC , YC , α). If the error distances are explicitly be defined for a fittingmethod precision analysis can be implemented. By algebraic fitting an ellipse isdescribed by implicit equation with the parameters vector and error distances aredefined with the deviations of the implicit equation from the expected value at eachgiven point. This distance is denoted as di in the figure.

The corresponding points on the ellipse for the given data point for algebraicfitting are (X i, Y i). The orthogonal components of the error distances (vXi, vY i)are also shown in the graph. Once the ellipse is fitted to data points performingalgebraic fitting method, the ellipse parameters are determined and the residual val-ues (algebraic) can be computed for the fitting method. The next sections describehow to compute the precision values of the parameters for algebraic fitting methodin detail.

Precision analysis of algebraic fitting method

A widely-known analysis of the fitting approaches was done by Fitzgibbon(Fitzgibbon et al. 1996, 1999). In this section we provide an overview of thiswell-known approach: An ellipse is a special case of a general conic which canbe described by an implicit second order polynomial

f(X,Y ) = A ·X2 + B ·X · Y + C · Y 2 +D ·X + E · Y + F = 0 (1)

with an ellipse-specific constraint

B2 − 4 ·A · C < 0 , (2)



Fig. 1. Geometrical relationship between ellipse coordinate system and any other reference system

where A,B,C,D,E, F are the coefficients of the ellipse and (X,Y ) are coordinatesof points lying on it. The polynomial F (X,Y ) is called the algebraic distance ofthe point (X,Y ) to the given conic. By introducing the vectors below

a = [A B C D E F ]T X =[X2 X · Y Y 2 X Y 1

](3)

it can be rewritten to the vector form

fa(X) = X · a = 0 . (4)

The fitting of a general conic to a set of points (Xi, Yi), i ∈ {1, 2, . . . , n} may beapproached by minimizing the sum of squared algebraic distances of the points tothe conic which is represented by coefficients a:

mina

n∑i=1

f (Xi, Yi)2 = min

a

n∑i=1

(fa(Xi))2 = min

a

n∑i=1

(Xia)2 . (5)

The problem in Eq. (5) can be solved directly by the standard LS approach, butthe result of such fitting is a general conic and it needs not to be an ellipse. To ensurean ellipse-specificity of the solution, the appropriate constraint Eq. (2) has to beconsidered. There were many attempts to make the fitting process computationallyeffective. Finally, Fitzgibbon et al. (1996) proposed a direct least squares basedellipse-specific method. They introduce a new method for fitting ellipses, ratherthan general conics, to segmented data.

In the following section a mathematical model is formulated that links the pa-rameters of the ellipse to measured point coordinate values to estimate the precision



of fitting method. Regarding the principle of LS, functional model consisting un-known parameters and coordinates of measured point i, is written as:

fi

(X i, Y i, a, b,XC , YC , α

)= fi

(λi,β

)= 0, i ∈ {1, 2, . . . , n} , (6)

where n: the number of points measured.

β ∈ {a, b,XC, YC , α} , (7)

λi = λi + vλi λ ∈ {X,Y } (8)

ellipse parameters given (Eq. 7) and LS estimate of observations for point i (Eq. 8).Functional model is established in two phases via Fig. 1. At first stage ith point

is represented asb2x2

i + a2y2i − a2b2 = 0 (9)

using standard ellipse equation through ellipse parameters (a, b) at the coordinatesystem (x/y). At the second stage the relationship between the x/y system and theX/Y system is given by

xi = cosα(Xi −XC

)+ sinα

(Y i − YC

)yi = − sinα

(Xi −XC

)+ cosα

(Y i − YC

).

(10)

Substituting Eqs (10) into Eq. (9) and expanding the terms yields the functionalmodel as,

fi

(X i, Y i,β

)= b2{cosα

(Xi −XC

)+ sinα

(Y i − YC

)}2+

+ a2{− sinα

(Xi −XC

)+ cosα

(Y i − YC

)}2 − a2b2 = 0 .(11)

The solution of Eq. (11) involves linearising the mathematical model and solvingfor the parameters using combined least squares technique (Leick 1995). The processof estimating the ellipse parameters must be initialized by specifying an initial guessfor the parameter values. If Eq. (11) is expanded in Taylor series form and secondand higher order term is ignored, mathematical model for a point is obtained invector form as

A · v + B · δ + w = 0 (12a)

⎡⎢⎢⎢⎣

A1 0 . . . 00 A2 . . . 0...

.... . .

...0 0 . . . An

⎤⎥⎥⎥⎦ ·

⎡⎢⎢⎢⎣

v1

v2

...vn

⎤⎥⎥⎥⎦+

⎡⎢⎢⎢⎣

B1

B2

...Bn

⎤⎥⎥⎥⎦ ·

⎡⎢⎢⎢⎢⎢⎢⎢⎣

δa

δb

δXC

δYC

δα

⎤⎥⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎣

w1

w2

...wn

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣

01

02

...0n

⎤⎥⎥⎥⎦ ,

(12b)where n is the number of data points. This combined least squares method issometimes called as “conditional adjustment with unknowns” in LS literature; where



A is a matrix of partial differentials of the model equations with respect to theobservations, v is a vector of observation residuals, B is a coefficient matrix (partialdifferentials of the model equations with respect to the ellipse parameters), δ is avector of corrections (i.e. unknowns) and w is the closure vector. The LS estimateof δ is based on the minimization of the function vTv. A solution is obtained byintroducing a vector of Lagrange multipliers, k, and minimizing the function (Ω) asdemonstrated in the following.

Ω = vT · v − 2 · kT · (A · v + B · δ + w) → min . (13)

A group of equations of combined LS method is given below. LS estimate ofunknown corrections can be computed as

δ = −Qβ ·(

n∑i=1

BTi ·wi/Ai ·AT

i

), (14)

where

Ai =[(

∂fi

∂Xi

)(∂fi

∂Y i

)], Bi =

[(∂fi

∂a

)(∂fi

∂b

)(∂fi

∂XC

)(∂fi

∂Y C

)(∂fi

∂α

)](15)

(see appendix for details). Since the ellipse parameters are determined by usingthe Direct Ellipse fitting method, there is no need to re-compute these parametersfrom the Eq. (14). The parameters are also used for the calculation of A and Bmatrices. Using the matrices (A and B) above, cofactor matrix of unknowns canbe computed as

Qβ =

(n∑

i=1

BTi · Bi/Ai · AT

i

)−1

. (16)

LS residuals vector can be calculated as in the equation below

vi =[vXi

vYi

]= AT

i · ki . (17)

Lagrange multipliers are computed from the Eq. (18):

ki = −(Bi · δ + wi)/Ai · ATi . (18)

It should be noted that by algebraic fitting, the error distances are not orthogonal(shortest) distances from the given points to the ellipse. By using the residualsvector, standard deviation values are calculated as

σ = ±

√√√√√n∑

i=1

vTi vi

n− u, (19)



where u is the number of unknown parameter (u = 5). Variance-covariance matrixof unknowns are computed as:

Kβ = σ2 ·Qβ . (20)

LS estimate of observations can be computed as

λi = λi + vi =

[Xi

Y i

]=

[Xi

Yi

]+

[vXi

vYi

]i ∈ {1, 2, . . . , n} . (21)

Here the solution is provided in the iterative way considering the difference ofstandard deviation values of consecutive iteration steps. It should be noted thatsince we will concentrate on the precision estimation for the algebraic (i.e. directfitting) method, we will use fitted ellipse parameters (i.e. δ), estimated with directellipse fitting method as previously explained. Substituting this ellipse parametersvector in Eq. (12) as unknown parameters, one can compute the residual vectorsemploying the solution scheme above. Hence, standard deviation criterion can becomputed using the residuals for the algebraic method from the Eq. (19). It is worthto note that the adopted approach can also be applied to another ellipse fittingmethod in a similar way. There has not been any strategy to estimate the precisionof fitting method in this way at the literature. The approach makes it easier tointerpret the precision estimation of fitting scheme in a geometrical perspective.The precision outcome of fitting procedure is explicit visually and computationallypractical. So the proposed approach can be viewed as a novelty for the precisionestimation of algebraic ellipse fitting method.

3. Experimental results

This section describes two experiments that illustrate the features of the pro-posed algorithm for direct ellipse fitting method considering the precision analysis.A new Matlab application code has been written for the experiments regarding ourapproach based on the combined LS theory. To evaluate the performance of theproposed algorithm it was seemed to be useful to compute the ellipse parametersvia algebraic method for the experiments using a well-known Matlab (MatWorks)implementation of the method. In order for other researchers to quickly assessthe method, the Matlab implementation of the Direct LS fitting method can easilybe found in the literature (Pilu et al. 1996). For the purpose of both fitting andprecision analysis our Matlab implementation is evaluated for the experiments.

For the first experiment we have used the point data set, consisted of 8 coordi-nate pairs, shown in Fig. 2. In the figure algebraically fitted ellipse, it’s computederror distances (or residual values) for each data point and error bars for each pointare shown. Algebraic error distances are denoted as dashed lines and error bars aredenoted as vertical solid lines. The distances of error bars are shown as doubled forclarity and comparison in the figure. Data points are denoted as small circle pointsand corresponding points on fitted ellipse denoted as square symbols in the fig-ure. Data point coordinates, computed algebraic residual values, fitting accuracies,ellipse parameters for the final fit and their accuracies are given in Table I.



Fig. 2. Fitted ellipse for algebraic distance fitting method, algebraic error distances and error bars(double sized) for the first experiment

Fig. 3. Fitted ellipse for algebraic distance fitting method, algebraic error distances and error bars(double sized) for the second experiment

We have taken 6 coordinate pairs shown in Fig. 3 as another precision estimationexample. The distances of error bars are shown as doubled for clarity in the figure.The algebraic distance residuals for the fitted ellipse are shown in the figure. In thissecond experiment the computed algebraic distance residual values for the directellipse fitting method are given in Table II. Ellipse parameters for the final fit andtheir accuracies are also given.

It should be emphasized that the proposed method is fast and efficient com-putationally. The computation time of the proposed method is as fast as thefitting method. On the other hand, needles to say that the proposed approach



Table I.

a) Computed residual values for 8data points for the first experiment

Points Residuals

x y VXiVYi

1 7 0.6394 –0.35012 6 –0.5428 0.11225 8 –0.0304 –0.16967 7 0.0095 0.01429 5 –0.3635 –0.10413 7 –0.3463 0.58666 2 0.0016 0.23918 4 0.6161 –0.1992

δ ±1.3971

b) Result of the algebraic ellipse fitting to the coordinate pairs shown in Fig. 2

Parameters a b XC YC α (degree)

3.7757 2.6423 5.0639 5.0698 157.8770Accuracy ±0.4258 ±0.5994 ±0.4530 ±0.4336 ±0.3755

can efficiently be implemented on other ellipse fitting methods to determine fittingaccuracies.

4. Conclusion

Many different ellipse fitting techniques depend on a suitable error term. Eval-uating the true Euclidean distance from the data points to the ellipse is a complexissue at fitting and it is usual practice to approximate it by some measure that issimpler to calculate. To overcome this problem a new approach has been proposedin the study. After the algebraic fitting procedure, the geometrical error distancescan be computed and accuracy analysis of the fitting method be made in a practi-cal manner. The problem of precision estimation for ellipse fitting is evaluated asa geometric concept by employing combined LS method. Although the presentedmethod was applied to the direct ellipse fitting as an algebraic method in the paper,it can also be used for another fitting method in order to estimate precision. Whenan ellipse fitted to the given data points, algebraic distance residuals for each datapoint can be determined. Thus the presented solution is considered to be explana-tory for precision analysis of fitting in a geometrical perspective. On the other handthe algorithm is computationally fast and efficient. This approach may also be usedfor the purpose of comparing the different geometric fitting algorithms.



Table II.

a) Computed residual values for 6 datapoints for the second experiment

Data points Residuals

x y VXiVYi

1986.22 985.92 0.1643 –0.49701989.76 982.64 –0.0188 0.42661990.33 984.99 –1.8776 1.08031994.04 984.01 0.1049 –0.45771996.22 986.55 0.0217 0.17262000.24 986.17 –1.4350 –0.5228

σ ±2.7807

b) Result of the algebraic ellipse fitting to the coordinate pairs shown in Fig. 3

Parameters a b XC YC α (degree)

7.0200 1.7062 1991.8269 984.9256 6.4286Accuracy ±7.7228 ±1.4257 ±7.2308 ±1.7007 ±0.4194

Appendix

Expansion of partial derivatives of Eq. (15):

Ai =[(

∂fi

∂Xi

)(∂fi

∂Y i

)](∂fi

∂Xi

)= 2(b2xi cosα− a2yi sinα)

(∂fi

∂Y i

)= 2(b2xi sinα+ a2yi cosα)

Bi =[(

∂fi

∂a

)(∂fi

∂b

)(∂fi

∂XC

)(∂fi

∂Y C

)(∂fi

∂α

)](∂fi

∂a

)= 2a(y2

i − b2)

(∂fi

∂b

)= 2b(x2

i − a2)

(∂fi

∂XC

)= −(∂fi

∂Xi

)


440 O KURT and O ARSLAN(∂fi

∂Y C

)= −(∂fi

∂Y i

)(∂fi

∂α

)= 2xiyi(b2 − a2)

wi = b2x2i + a2y2

i − a2b2 i ∈ {1, 2, . . . , n} .

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Volume, 103–151.MatWorks. Matlab Package. http://www.mathworks.comPilu M, Fitzgibbon A W, Fisher R B 1996: In: Proc. IEEE ICIP, Lausanne, SwitzerlandRosin P L 1996: Graphical Models Image Process, 58, 494–502.Safaee-Rad R, Tchoukanov I, Benhabib B, Smith K C 1991: Image Understanding, 54,

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Geometric interpretation and precision analysis of algebraic ellipse fitting using least squares method