Circular and Linear Regression: Fitting Circles and Lines by Least Squares, 1st Edition (Hardback) book cover

Circular and Linear Regression

Fitting Circles and Lines by Least Squares, 1st Edition

By Nikolai Chernov

CRC Press

286 pages | 65 B/W Illus.

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Hardback: 9781439835906
pub: 2010-06-22
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Find the right algorithm for your image processing application

Exploring the recent achievements that have occurred since the mid-1990s, Circular and Linear Regression: Fitting Circles and Lines by Least Squares explains how to use modern algorithms to fit geometric contours (circles and circular arcs) to observed data in image processing and computer vision. The author covers all facets—geometric, statistical, and computational—of the methods. He looks at how the numerical algorithms relate to one another through underlying ideas, compares the strengths and weaknesses of each algorithm, and illustrates how to combine the algorithms to achieve the best performance.

After introducing errors-in-variables (EIV) regression analysis and its history, the book summarizes the solution of the linear EIV problem and highlights its main geometric and statistical properties. It next describes the theory of fitting circles by least squares, before focusing on practical geometric and algebraic circle fitting methods. The text then covers the statistical analysis of curve and circle fitting methods. The last chapter presents a sample of "exotic" circle fits, including some mathematically sophisticated procedures that use complex numbers and conformal mappings of the complex plane.

Essential for understanding the advantages and limitations of the practical schemes, this book thoroughly addresses the theoretical aspects of the fitting problem. It also identifies obscure issues that may be relevant in future research.


Overall, the book provides an excellent reference for those interested in the geometric, algebraic, and numerical aspects of fitting lines, curves, and circles to data. It gives an insightful discussion of the geometric and algebraic aspects of measurement error models and provides a thorough discussion of the various algorithms for fitting such models to data, by offering advice and comparisons concerning their strengths and weaknesses under different circumstances.

Journal of the American Statistical Association, 2012

… the book contains useful insights and will be of benefit to those interested in the theoretical aspects of fitting lines and circles to data. Some mathematical background is assumed by the author, but the book is written excellently and the exposition is clearly structured. The book is unique in the way that it has brought together a variety of topics into one volume of work. I would recommend that statisticians who are interested in measurement error models, and those who are interested in fitting circles to data (such as those working in computer vision) at the very least investigate this book.

Journal of the Royal Statistical Society: Series A, July 2011

Table of Contents

Introduction and Historic Overview

Classical regression

Errors-in-variables (EIV) model

Geometric fit

Solving a general EIV problem

Nonlinear nature of the "linear" EIV

Statistical properties of the orthogonal fit

Relation to total least squares (TLS)

Nonlinear models: general overview

Nonlinear models: EIV versus orthogonal fit

Fitting Lines


Existence and uniqueness

Matrix solution

Error analysis: exact results

Asymptotic models: large n versus small σ

Asymptotic properties of estimators

Approximative analysis

Finite-size efficiency

Asymptotic efficiency

Fitting Circles: Theory




Multivariate interpretation of circle fit


Local minima

Plateaus and valleys

Proof of two valley theorem

Singular case

Geometric Circle Fits

Classical minimization schemes

Gauss–Newton method

Levenberg–Marquardt correction

Trust region

Levenberg–Marquardt for circles: full version

Levenberg–Marquardt for circles: reduced version

A modification of Levenberg–Marquardt circle fit

Späth algorithm for circles

Landau algorithm for circles

Divergence and how to avoid it

Invariance under translations and rotations

The case of known angular differences

Algebraic Circle Fits

Simple algebraic fit (Kåsa method)

Advantages of the Kåsa method

Drawbacks of the Kåsa method

Chernov–Ososkov modification

Pratt circle fit

Implementation of the Pratt fit

Advantages of the Pratt algorithm

Experimental test

Taubin circle fit

Implementation of the Taubin fit

General algebraic circle fits

A real data example

Initialization of iterative schemes

Statistical Analysis of Curve Fits

Statistical models

Comparative analysis of statistical models

Maximum likelihood estimators (MLEs)

Distribution and moments of the MLE

General algebraic fits

Error analysis: a general scheme

Small noise and "moderate sample size"

Variance and essential bias of the MLE

Kanatani–Cramer–Rao lower bound

Bias and inconsistency in the large sample limit

Consistent fit and adjusted least squares

Statistical Analysis of Circle Fits

Error analysis of geometric circle fit

Cramer–Rao lower bound for the circle fit

Error analysis of algebraic circle fits

Variance and bias of algebraic circle fits

Comparison of algebraic circle fits

Algebraic circle fits in natural parameters

Inconsistency of circular fits

Bias reduction and consistent fits via Huber

Asymptotically unbiased and consistent circle fits

Kukush–Markovsky–van Huffel method

Renormalization method of Kanatani: 1st order

Renormalization method of Kanatani: 2nd order

Various "Exotic" Circle Fits

Riemann sphere

Simple Riemann fits

Riemann fit: the SWFL version

Properties of the Riemann fit

Inversion-based fits

The RTKD inversion-based fit

The iterative RTKD fit

Karimäki fit

Analysis of Karimäki fit

Numerical tests and conclusions



About the Author

Nikolai Chernov is a professor of mathematics at the University of Alabama at Birmingham.

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Programming / Algorithms
MATHEMATICS / Probability & Statistics / General