Chernov Circular and Linear Regression

Fitting Circles and Lines by Least Squares
1. Auflage 2010
ISBN: 978-1-4398-3591-3
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 286 Seiten

Reihe: Chapman & Hall/CRC Monographs on Statistics & Applied Probability

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.

Zielgruppe

Statisticians and practitioners using circle fitting; researchers in nuclear physics, computer vision, image processing, and applied (industrial) math.

Autoren/Hrsg.

Chernov, Nikolai

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

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
Parametrization
Existence and uniqueness
Matrix solution
Error analysis: exact results
Asymptotic models: large n versus small s
Asymptotic properties of estimators
Approximative analysis
Finite-size efficiency
Asymptotic efficiency

Fitting Circles: Theory
Introduction
Parametrization
(Non)existence
Multivariate interpretation of circle fit
(Non)uniqueness
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
Bibliography
Index

Über Autor(innen)

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