Ellipse Fitting for Computer Vision. Implementation and Applications

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Ellipse Fitting for Computer Vision

Implementation and Applications

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Synthesis Lectures on Computer Vision

Editors

Gérard Medioni, University of Southern California Sven Dickinson, University of Toronto

Synthesis Lectures on Computer Visionis edited by Gérard Medioni of the University of Southern California and Sven Dickinson of the University of Toronto. e series publishes 50- to 150 page publications on topics pertaining to computer vision and pattern recognition. e scope will largely follow the purview of premier computer science conferences, such as ICCV, CVPR, and ECCV.

Potential topics include, but not are limited to:

• Applications and Case Studies for Computer Vision

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• Image-Based Modeling

• Image and Video Retrieval

• Medical Image Analysis

• Motion and Tracking

• Object Detection, Recognition, and Categorization

• Segmentation and Grouping

• Sensors

• Shape-from-X

• Stereo and Structure from Motion

• Shape Representation and Matching

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iii

• Statistical Methods and Learning

• Performance Evaluation

• Video Analysis and Event Recognition

Ellipse Fitting for Computer Vision: Implementation and Applications Kenichi Kanatani, Yasuyuki Sugaya, and Yasushi Kanazawa

2016

Background Subtraction: eory and Practice Ahmed Elgammal

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Vision-Based Interaction Matthew Turk and Gang Hua 2013

Camera Networks: e Acquisition and Analysis of Videos over Wide Areas Amit K. Roy-Chowdhury and Bi Song

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Deformable Surface 3D Reconstruction from Monocular Images Mathieu Salzmann and Pascal Fua

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Ellipse Fitting for Computer Vision: Implementation and Applications Kenichi Kanatani, Yasuyuki Sugaya, and Yasushi Kanazawa

www.morganclaypool.com

ISBN: 9781627054584 paperback ISBN: 9781627054980 ebook

DOI 10.2200/S00713ED1V01Y201603COV008

A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON COMPUTER VISION

Lecture #8

Series Editors: Gérard Medioni,University of Southern California Sven Dickinson,University of Toronto

Series ISSN

Print 2153-1056 Electronic 2153-1064

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Ellipse Fitting for Computer Vision

Implementation and Applications

Kenichi Kanatani

Okayama University, Okayama, Japan

Yasuyuki Sugaya

Toyohashi University of Technology, Toyohashi, Aichi, Japan

Yasushi Kanazawa

Toyohashi University of Technology, Toyohashi, Aichi, Japan

SYNTHESIS LECTURES ON COMPUTER VISION #8

M C

&

M or g a n

&

c L ay p o ol p u b l i s h e rs

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ABSTRACT

Because circular objects are projected to ellipses in images, ellipse fitting is a first step for 3- D analysis of circular objects in computer vision applications. For this reason, the study of ellipse fitting began as soon as computers came into use for image analysis in the 1970s, but it is only recently that optimal computation techniques based on the statistical properties of noise were established. ese includerenormalization (1993), which was then improved as FNS (2000) andHEIV (2000). Later, further improvements, called hyperaccurate correction (2006), HyperLS (2009), andhyper-renormalization (2012), were presented. Today, these are regarded as the most accurate fitting methods among all known techniques. is book describes these algorithms as well implementation details and applications to 3-D scene analysis.

We also present general mathematical theories of statistical optimization underlying all ellipse ﬁtting algorithms, including rigorous covariance and bias analyses and the theoretical accuracy limit. e results can be directly applied to other computer vision tasks including computing fundamental matrices and homographies between images.

is book can serve not simply as a reference of ellipse ﬁtting algorithms for researchers, but also as learning material for beginners who want to start computer vision research. e sample program codes are downloadable from the website: https://sites.google.com/a/morganclaypool.com/ellipse-fitting-for- computer-vision-implementation-and-applications/.

KEYWORDS

geometric distance minimization, hyperaccurate correction, HyperLS, hyper- renormalization, iterative reweight, KCR lower bound, maximum likelihood, renormalization, robust ﬁtting, Sampson error, statistical error analysis, Taubin method

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vii

Preface

Because circular objects are projected to ellipses in images, ellipse fitting is a first step for 3-D analysis of circular objects in computer vision applications. For this reason, the study of ellipse fitting began as soon as computers came into use for image analysis in the 1970s. e basic principle was to compute the parameters so that the sum of squares of expressions that should ideally be zero is minimized, which is today calledleast squares or algebraic distance minimization. In the 1990s, the notion of optimal computation based on the statistical properties of noise was introduced by researchers including the authors. e first notable example was the authors’renormalization (1993), which was then improved asFNS (2000) and HEIV (2000) by researchers in Australia and the U.S. Later, further improvements, calledhyperaccurate correction (2006), HyperLS (2009), andhyper-renormalization (2012), were presented by the authors. Today, these are regarded as the most accurate fitting methods among all known techniques. is book describes these algorithms as well as underlying theories, implementation details, and applications to 3-D scene analysis.

Most textbooks on computer vision begin with mathematical fundamentals followed by the resulting computational procedures. is book, in contrast,immediately describes computational procedures after a short statement of the purpose and the principle. e theoretical background is brieﬂy explained asComments. us, readers need not worry about mathematical details, which often annoy those who only want to build their vision systems. Rigorous derivations and detailed justiﬁcations are given later in separate sections, but they can be skipped if the interest is not in theories. Sample program codes of the authors are provided via the website¹of the publisher. At the end of each chapter is given a section calledSupplemental Note, describing historical back- grounds, related issues, and reference literature.

Chapters1–4specifically describe ellipse fitting algorithms. Chapter5discusses 3-D analysis of circular objects in the scene extracted by ellipse fitting. In Chapter6, performance com- parison experiments are conducted among the methods described in Chapters1–4. Also, some real image applications of the 3-D analysis of Chapter5are shown. In Chapter7, we point out how procedures of ellipse fitting can straightforwardly be extended to fundamental matrix and homography computation, which play a central role in 3-D analysis by computer vision. Chap- ters8and9give general mathematical theories of statistical optimization underlying all ellipse fitting algorithms. Finally, Chapter10gives a rigorous analysis of the theoretical accuracy limit.

However, beginners and practice-oriented readers can skip these last three chapters.

e authors used the materials in this book as student projects for introductory computer vision research at Okayama University, Japan, and Toyohashi University of Technology, Japan.

By implementing the algorithms themselves, students can learn basic programming know-hows

¹https://sites.google.com/a/morganclaypool.com/ellipse-ﬁtting-for-computer-vision-implementation-and-applications/

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xii PREFACE

and also understand the theoretical background of vision computation as their interest deepens.

We are hoping that this book can serve not simply as a reference of ellipse ﬁtting algorithms for researchers, but also as learning material for beginners who want to start computer vision research.

e theories in this book are the fruit of the authors’ collaborations and interactions with their colleagues and friends for many years. e authors thank Takayuki Okatani of Tohoku University, Japan, Mike Brooks and Wojciech Chojnacki of the University of Adelaide, Aus- tralia, Peter Meer of Rutgers University, U.S., Wolfgang Förstner, of the University of Bonn, Germany, Michael Felsberg of Linköping University, Sweden, Rudolf Mester of the University of Frankfurt, Germany, Prasanna Rangarajan of Southern Methodist University, U.S., Ali Al- Sharadqah of University of East Carolina, U.S., and Alexander Kukush of the University of Kiev, Ukraine. Special thanks are to (late) Professor Nikolai Chernov of the University of Alabama at Birmingham, U.S., without whose inspiration and assistance this work would not have been possible.

Kenichi Kanatani, Yasuyuki Sugaya, and Yasushi Kanazawa March 2016

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1

C H A P T E R 1

Introduction

is chapter describes the basic mathematical formulation to be used in subsequent chapters for ﬁtting an ellipse to observed points. e main focus is on the description of statistical properties of noise in the data in terms of covariance matrices. We point out that two approaches exist for ellipse ﬁtting: “algebraic” and “geometric.” Also, some historical background is mentioned, and related mathematical topics are discussed.

1.1 ELLIPSE FITTING

Ellipse fitting means fitting an ellipse equation to points extracted from an image. is is one of the fundamental tasks of computer vision for various reasons. First, we observe many circular objects in man-made scenes indoors and outdoors, and a circle is projected as an ellipse in camera images. If we extract elliptic segments, say, by an edge detection filter, and fit an ellipse equation to them, we can compute the 3-D position of the circular object in the scene (we will discuss such applications in Chapter5). Figure1.1a shows edges extracted from an indoor scene, using an edge detection filter. is scene contains many elliptic arcs, as indicated there. Figure1.1b shows ellipses fitted to them superimposed on the original image. We observe that fitted ellipses are not necessarily exact object shapes, in particular when the observed arc is only a small part of the circumference or when it is continuously connected to a non-elliptic segment (we will discuss this issue in Chapter4).

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Figure 1.1: (a) An edge image and selected elliptic arcs. (b) Ellipses are ﬁtted to the arcs in (a) and superimposed on the original image.

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2 1. INTRODUCTION

Ellipse ﬁtting is also used for detecting not only circular objects in the image but also objects of approximately elliptic shape, e.g., human faces. An important application of ellipse ﬁtting iscamera calibration for determining the position and internal parameters of a camera by taking images of a reference pattern, for which circles are often used for the ease of image processing.

Ellipse ﬁtting is also important as a mathematical prototype of various geometric estimation problems for computer vision. Typical problems include the computation of fundamental matrices and homographies (we will brieﬂy describe these in Chapter7).

1.2 REPRESENTATION OF ELLIPSES

e equation of an ellipse has the form

Ax²C 2Bxy C Cy²C 2f⁰.Dx C Ey/ C f0²F D 0; (1.1) wheref0 is a constant for adjusting the scale. eoretically, it can be 1, but for ﬁnite-length numerical computation it should be chosen so thatx=f0 andy=f0 have approximately the order of 1; this increases the numerical accuracy, avoiding the loss of signiﬁcant digits. In view of this, we take the origin of the imagexy coordinate system at the center of the image, rather than the upper-left corner as is customarily done, and takef0to be the length of the side of a square which we assume to contain the ellipse to be extracted. For example, if we know that an ellipse exists in a_{600 600}pixel region, we letf0 = 600. Since Eq. (1.1) has scale indeterminacy, i.e., the same ellipse is represented ifA,B,C,D,E, andFare simultaneously multiplied by a nonzero constant, we need some kind of normalization. Various types of normalizations have been considered in the past, including

F D 1; (1.2)

A C C D 1; (1.3)

A²C B²C C²C D²C E²C F²D 1; (1.4)

A²C B²C C²C D²C E²D 1; (1.5)

A²C 2B²C C²D 1; (1.6)

AC B² D 1: (1.7)

Among these, Eq. (1.2) is the simplest and most familiar one, but Eq. (1.1) withF = 1 cannot express an ellipse that passes through the origin.0; 0/. Equation (1.3) remedies this. Each of the above normalization equations has its own reasoning, but in this book we adopt Eq. (1.4) (see Supplemental Note (page6) below for the background).

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1.3. LEAST SQUARES APPROACH 3

If we deﬁne the 6-D vectors

ξ_D 0 BB BB BB B@

x² 2xy

y² 2f0x 2f0y f₀²

1 CC CC CC CA

; θ_D

0 BB BB BB B@

A B C D E F

1 CC CC CC CA

; (1.8)

Eq. (1.4) can be written as

.ξ;θ_{/ D 0;} (1.9)

where and hereafter we denote the inner product of vectors a and b by.a;b/. Since the vector θ in Eq. (1.9) has scale indeterminacy, it must be normalized in correspondence with Eqs. (1.2)–(1.7).

Note that the left sides of Eqs. (1.2)–(1.7) can be seen as quadratic forms inA, ...,F; Eqs. (1.2) and (1.3) are linear equations, but we may regard them asF²= 1 and_{.A C C /}² = 1, respectively.

Hence, Eqs. (1.2)–(1.7) are all written in the form

.θ;N θ_{/ D 1;} (1.10)

for some normalization matrix N . e use of Eq. (1.4) corresponds to N = I (the identity), in which case Eq. (1.10) is simply_kθ_k= 1, i.e., normalization to unit norm.

1.3 LEAST SQUARES APPROACH

Fitting an ellipse in the form of Eq. (1.1) to a sequence of points.x1; y1/, ...,.xN; yN/in the presence of noise (Fig.1.2) is to ﬁndA,B,C,D,E, andF such that

Ax²_˛C 2Bx^˛y˛C Cy˛²C 2f⁰.Dx˛C Ey^˛/ C f0²F 0; ˛ D 1; :::; N: (1.11) If we write ξ˛for the value obtained by replacingxandyin the 6-D vector ξ of Eq. (1.8) byx˛

andy˛, respectively, Eq. (1.11) can be equivalently written as

.ξ˛;θ_{/ 0;} ˛ D 1; :::; N: (1.12)

(x , y )_α _α

Figure 1.2: Fitting an ellipse to a noisy point sequence.

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Our task is to compute such a unit vector θ. e simplest and the most naive method is the followingleast squares.

Procedure 1.1 (Least squares) 1. Compute the_{6 6}matrix

M _D ¹ N

XN

˛D1

ξ˛ξ_˛^>: (1.13)

2. Solve the eigenvalue problem

M θ_Dθ; (1.14)

and return the unit eigenvector θ for the smallest eigenvalue.

Comments.is is a straightforward generalization of line ﬁtting to a point sequence (_,!Prob- lem1.1); we minimize the sum of the squares

J D 1 N

XN

˛D1

.ξ˛;θ/²D 1 N

XN

˛D1

θ^>ξ˛ξ^>_˛θ_{D .}θ; 1 N

XN

˛D1

ξ˛ξ^>_˛θ_{/ D .}θ;M θ/; (1.15)

subject to_kθ_k= 1. As is well known in linear algebra, the minimum of this quadratic form in θis given by the unit eigenvector θ of M for the smallest eigenvalue. Equation (1.15) is often called thealgebraic distance, and Procedure1.1is also known asalgebraic distance minimization. It is sometimes calledDLT (direct linear transformation). Since the computation is very easy and the solution is immediately obtained, this method has been widely used. However, when the input point sequence covers only a small part of the ellipse circumference, it often produces a small and flat ellipse very different from the true shape (we will see such examples in Chapter6). Still, this is a prototype of all existing ellipse fitting algorithms. How we can improve this method is the main theme of this book.

1.4 NOISE AND COVARIANCE MATRICES

e reason for the poor accuracy of Procedure1.1 is that the properties of image noise are not considered; for accurate ﬁtting, we need to take the statistical properties of noise into consider- ation. Suppose the datax˛ andy˛ are disturbed from their true values _Nx˛ and _Ny˛ byx˛ and

y˛:

x˛ D Nx^˛C x^˛; y˛ D Ny^˛C y^˛: (1.16) Substituting this into ξ˛, we can write

ξ˛D Nξ˛C ¹ξ˛C ²ξ˛; (1.17)

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1.4. NOISE AND COVARIANCE MATRICES 5

whereξN˛ is the value of ξ˛ obtained by replacingx˛ andy˛ by their true values _Nx˛ and _Ny˛, respectively, while1ξ˛and2ξ˛are, respectively, the ﬁrst-order noise term (the linear expression inx˛ andy˛) and the second-order noise term (the quadratic expression inx˛ andy˛).

From Eq. (1.8), we obtain the following expressions:

1ξ˛D 0 BB BB BB B@

2 Nx^˛x˛

2x˛Ny^˛C 2 Nx^˛y˛

2 Ny^˛y˛

2f0x˛

2f0y˛

0

1 CC CC CC CA

; 2ξ˛ D 0 BB BB BB B@

x_˛² 2x˛y˛

y_˛² 0 0 0

1 CC CC CC CA

: (1.18)

We regard the noise termsx˛andy˛as random variables and deﬁne the covariance matrix of ξ˛by

V Œξ˛ D EŒ¹ξ˛1ξ^>_˛; (1.19) where_EŒdenotes expectation over the noise distribution. If we assume thatx˛andy˛are sampled from independent Gaussian distributions of mean 0 and standard deviation, we obtain EŒx˛ D EŒy^˛ D 0; EŒx_˛² D EŒy˛² D ²; EŒx˛y˛ D 0: (1.20) Substituting Eq. (1.18) and using this relationship, we obtain the covariance matrix in Eq. (1.19) in the following form:

V Œξ˛ D ²V0Œξ˛; V0Œξ˛ D 4 0 BB BB BB B@

Nx˛² Nx^˛Ny^˛ 0 f0Nx^˛ 0 0 Nx^˛Ny^˛ Nx²˛C Ny˛² Nx^˛Ny^˛ f0Ny^˛ f0Nx^˛ 0 0 Nx^˛Ny^˛ Ny˛² 0 f0Ny^˛ 0 f0Nx^˛ f0Ny^˛ 0 f₀² 0 0 0 f0Nx^˛ f0Ny^˛ 0 f₀² 0

0 0 0 0 0 0

1 CC CC CC CA

: (1.21)

Since all the elements ofV Œξ˛have the multiple², we factor it out and callV0Œξ˛thenormalized covariance matrix. We also call the standard deviation thenoise level. e diagonal elements of the covariance matrixV Œξ˛indicate the noise susceptibility of each component of ξ˛, and the oﬀ-diagonal elements measure their pair-wise correlation.

e covariance matrix of Eq. (1.19) is deﬁned in terms of the ﬁrst-order noise term1ξ˛

alone. It is known that incorporation of the second-order term 2ξ˛ has little inﬂuence over ﬁnal results. is is because2ξ˛is very small as compared with1ξ˛. Note that the elements ofV0Œξ˛in Eq. (1.21) contain true values _Nx˛ and _Ny˛. ey are replaced by observed valuesx˛

andy˛ in actual computation. It is known that this replacement has practically no eﬀect in the ﬁnal results.

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1.5 ELLIPSE FITTING APPROACHES

In the following chapters, we describe typical ellipse fitting methods that incorporate the above noise properties. We will see that all the methods we considerdo not require knowledge of the noise level, which is very difficult to estimate in real problems. e qualitative properties of noise are all encoded in the normalized covariance matrixV0Œξ˛, which gives sufficient information for designing high accuracy fitting schemes. In general terms, there exist two approaches for ellipse fittings: algebraic and geometric.

Algebraic methods: We solves some algebraic equation for computing θ. e resulting solution may or may not minimize some cost function. In other words, the equation need not have the form of_rθJ = 0 for some cost functionJ. Rather, we can modify the equation in any way so that the resulting solution θ is as close to its true valueθ^Nas possible. us, our task is to ﬁnda good equation to solve. To this end, we need detailed statistical error analysis.

Geometric methods: We minimize some cost functionJ. Hence, the solution is uniquely deter- mined once the costJ is deﬁned. us, our task is to ﬁnda good cost to minimize. For this, we need to consider the geometry of the ellipse and the data points. We also need to devise a convenient minimization algorithm, since minimization of a given cost is not always easy.

e meaning of these two approaches will be better understood by seeing the actual procedures described in the subsequent chapters. ere are, however, a lot of overlaps between the two approaches.

1.6 SUPPLEMENTAL NOTE

e study of ellipse fitting began as soon as computers came into use for image analysis in the 1970s. Since then, numerous fitting techniques have been proposed, and even today new methods appear one after another. Since we are unable cite all the literature, we mention only some of the earlest work:Albano[1974],Bookstein[1979],Cooper and Yalabik[1979],Gnanadesikan [1977],Nakagawa and Rosenfeld[1979],Paton[1970]. Beside fitting an ellipse to data points, a voting scheme calledHough transform for accumulating evidences in the parameter space was also studied as a means of ellipse fitting [Davis,1989].

In the 1990s, a paradigm shift occurred. It was first thought that the purpose of ellipse fitting was to find an ellipse thatapproximately passes near observed points. However, some researchers, including the authors, turned their attention to finding an ellipse thatexactly passes through the true points that would be observed in the absence of noise. us, the problem turned to astatistical problem for estimating the true points subject to the constraint that they are on some ellipse. It follows that the goodness of the fitted ellipse is measured not by how close it is to the observed points but by how close it is to thetrue shape. is type of paradigm shift has also occurred in other problems including fundamental matrix and homography computation for 3-D

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1.6. SUPPLEMENTAL NOTE 7

analysis. Today, statistical analysis is one of the main tools for accurate geometric computation for computer vision.

e ellipse ﬁtting techniques discussed in this book are naturally extended to general curves and surfaces in a general space in the form

11C ²2C C ⁿnD 0; (1.22) where1, ...,n are functions of coordinatesx1,x2, ..., and1, ...,n are unknowns to be de- termined. is equation is written in the form of.ξ;θ/= 0 in terms of the vector ξ = .i/of observations and the vector θ =.i/of unknowns. en, all techniques and analysis for ellipse fitting can apply. Evidently, Eq. (1.22) includes all polynomial curves in 2-D and all polynomial surfaces in 3-D, but all algebraic functions also can be expressed in the form of Eq. (1.22) after canceling denominators. For general nonlinear surfaces, too, we can usually write the equation in the form of Eq. (1.22) after an appropriate reparameterization, as long as the problem is to estimate the “coefficients” of linear/nonlinear terms. Terms that are not multiplied by unknown coefficients are regarded as being multiplied by 1, which is also regarded as an unknown. e resulting set of coefficients can be viewed as a vector of unknown magnitude, or a “homogeneous vector,” and we can write the equation in the form.ξ;θ/= 0. us, the theory in this book has wide applicability beyond ellipse fitting.

In Eq. (1.1), we introduce the scaling constantf0 to makex=f0 andy=f0have the order of 1, and this also make the vector ξ in Eq. (1.8) have magnitude O.f₀²/so that ξ=f₀² is approximately a unit vector. e necessities and eﬀects of such scaling for numerical computation is discussed byHartley[1997] in relation to fundamental matrix computation (we will discuss this in Chapter7), which is also a fundamental problem of computer vision and has the same mathematical structure as ellipse ﬁtting. In this book, we introduce the scaling constantf0 and take the coordinate origin at the center of the image based on the same reasoning.

e normalization using Eq. (1.2) was adopted by Albano[1974], Cooper and Yalabik [1979], andRosin[1993]. Many authors used Eq. (1.4), but some authors preferred Eq. (1.5) [Gnanadesikan,1977]. e use of Eq. (1.6) was proposed byBookstein[1979], who argued that it leads to “coordinate invariance” in the sense that the ellipse fitted by least squares after the coordinate system is translated and rotated is the same as the originally fitted ellipse translated and rotated accordingly. In this respect, Eqs. (1.3) and (1.7) also have that invariance. e normalization using Eq. (1.7) was proposed byFitzgibbon et al.[1999] so that the resulting least-squares fit is guaranteed to be an ellipse, while other equations can theoretically produce a parabola or a hyperbola (we will discuss this in Chapter4).

Today, we need not worry about the coordinate invariance, which is a concern of the past.

As long as we regard ellipse fitting as statistical estimation and use Eq. (1.10) for normalization, all statistically meaningful methods are automatically invariant to the choice of the coordinate system. is is because the normalization matrix N in Eq. (1.10) is defined as a function of the covariance matrixV Œξ˛of Eq. (1.19). If we change the coordinate system, e.g., adding transla- tion, rotation, and other arbitrary mapping, the covariance matrixV Œξ˛defined by Eq. (1.19)

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also changes, and the ﬁtted ellipse after the coordinate change using the transformed covariance matrix is the same as the ellipse ﬁtted in the original coordinate system using the original covariance matrix and transformed afterwards.

e fact that the all the normalization equations of Eqs. (1.2)–(1.7) are written in the form of Eq. (1.10) poses an interesting question: What N is the “best,” if we are to minimize the algebraic distance of Eq. (1.15) subject to.θ;N θ/= 1? is problem was studied by the authors’

group [Kanatani and Rangarajan,2011,Kanatani et al.,2011,Rangarajan and Kanatani,2009], and the matrix N that gives rise to the highest accuracy was found after a detailed error analysis.

e method was namedHyperLS (this will be described in the next chapter).

Minimizing the sum of squares in the form of Eq. (1.15) is a natural idea, but readers may wonder why we minimize^P^N_˛D1.ξ˛;θ/². Why not minimize, say, the absolute sum PN

˛D1j.ξ˛;θ_/jor the maximum max_˛D1^N _j.ξ˛;θ_/j? is is the issue of the choice of thenorm. A class of criteria, calledLp-norms, exist for measuring the magnitude of ann-D vector x =.xi/:

kx_kp

Xⁿ

i D1

jxⁱj^p

1=p

: (1.23)

eL2-norm, or thesquare norm,

kx_k2 vu ut

Xn

i D1

jxⁱj²; (1.24)

is widely used for linear algebra. If we let_{p ! 1}in Eq. (1.23), it approaches kx_k₁ maxⁿ

i D1 jxⁱj; (1.25)

called theL₁-norm, or the maximum norm, where components with large_jxijhave a dominant eﬀect and those with small_jxijare ignored. Conversely, if let_{p !}0 in Eq. (1.23), those components with large_jxijare ignored. eL1-norm

kx_k1 Xn

i D1

jxⁱj (1.26)

is often called theaverage norm, because it eﬀectively measures the average.1=N /Pn

i D1jxⁱj. In the limit of_{p !}0, we obtain

kx_k0 jfij jxⁱj ¤ 0gj; (1.27) where the right side means the number of nonzero components. is is called theL0-norm, or theHamming distance.

Least squares for minimizing theL2-norm is the most widely used approach for statistical optimization for two reasons. One is the computational simplicity: diﬀerentiation of a sum

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1.6. SUPPLEMENTAL NOTE 9

of squares leads to linear expressions, so the solution is immediately obtained by solving linear equations or an eigenvalue problem. e other reason is that it corresponds tomaximum likelihood estimation (we will discuss this in Chapter9), provided the discrepancies to be minimized arise from independent and identical Gaussian noise. Indeed, the scheme of least squares was invented by Gauss, who introduced the “Gaussian distribution,” which he himself called the “normal distribution,” as the standard noise model (physicists usually use the former term, while statisticians prefer the latter). In spite of added computational complexity, however, minimization of theLp- norm forp <2, typicallyp= 1, has its merit, because the eﬀect of those terms with large absolute values are suppressed. is suggests that terms irrelevant for estimation, calledoutliers, are automatically ignored. Estimation methods that are not very susceptible to outliers are said to be robust (we will discuss robust ellipse ﬁtting in Chapter4). For this reason,L1-minimization is frequently used in some computer vision applications.

PROBLEMS

1.1. Fitting a line in the formn1x C n²y C n³f0 = 0 to points.x˛; y˛/,˛= 1, ...,N, can be viewed as the problem for computing n =.ni/, which can be normalized to a unit vector, such that.ξ˛;n_/0,˛= 1, ..,N, with

ξ˛ D 0

@ x˛

y˛

f0

1

A ; n_D

0

@ n1

n2

n3

1

A : (1.28)

(1) Write down the least-squares procedure for this computation.

(2) If the noise termsx˛andy˛are sampled from independent Gaussian distributions of mean 0 and variance², how is their covariance matrixV Œξ˛deﬁned?

Ellipse Fitting for Computer Vision. Implementation and Applications

Ellipse Fitting for Computer Vision

Implementation and Applications

Synthesis Lectures on Computer Vision

Ellipse Fitting for Computer Vision

Implementation and Applications

M C

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ABSTRACT

KEYWORDS

Contents

1

2

3

4

5

6

7

8

9

10

Preface

C H A P T E R 1

Introduction

1.1 ELLIPSE FITTING

1.2 REPRESENTATION OF ELLIPSES

1.3 LEAST SQUARES APPROACH

1.4 NOISE AND COVARIANCE MATRICES

1.5 ELLIPSE FITTING APPROACHES

1.6 SUPPLEMENTAL NOTE

PROBLEMS