Image Processing and Jump Regression Analysis.

By: Qiu, PeihuaMaterial type: TextTextSeries: Wiley Series in Probability and Statistics SerPublisher: Hoboken : John Wiley & Sons, Incorporated, 2005Copyright date: ©2005Edition: 1st edDescription: 1 online resource (340 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9780471733164Subject(s): Image processing | Regression analysisGenre/Form: Electronic books.Additional physical formats: Print version:: Image Processing and Jump Regression AnalysisDDC classification: 006.3/7 LOC classification: TA1637.Q58 2005Online resources: Click to View
Contents:
Image Processing and Jump Regression Analysis -- Contents -- Preface -- 1 Introduction -- 1.1 Images and image representation -- 1.2 Regression curves and sugaces with jumps -- 1.3 Edge detection, image restoration, and jump regression analysis -- 1.4 Statistical process control and some other related topics -- 1.5 Organization of the book -- Problems -- 2 Basic Statistical Concepts and Conventional Smoothing Techniques -- 2.1 Introduction -- 2.2 Some basic statistical concepts and terminologies -- 2.2.1 Populations, samples, and distributions -- 2.2.2 Point estimation of population parameters -- 2.2.3 Confidence intervals and hypothesis testing -- 2.2.4 Maximum likelihood estimation and least squares estimation -- 2.3 Nadaraya- Watson and other kernel smoothing techniques -- 2.3.1 Univariate kernel estimators -- 2.3.2 Some statistical properties of kernel estimators -- 2.3.3 Multivariate kernel estimators -- 2.4 Local polynomial kernel smoothing techniques -- 2.4.1 Univariate local polynomial kernel estimators -- 2.4.2 Some statistical properties -- 2.4.3 Multivariate local polynomial kernel estimators -- 2.4.4 Bandwidth selection -- 2.5 Spline smoothing procedures -- 2.5.1 Univariate smoothing spline estimation -- 2.5.2 Selection of the smoothing parameter -- 2.5.3 Multivariate smoothing spline estimation -- 2.5.4 Regression spline estimation -- 2.6 Wavelet transformation methods -- 2.6.1 Function estimation based on Fourier transformation -- 2.6.2 Univariate wavelet transformations -- 2.6.3 Bivariate wavelet transformations -- Problems -- 3 Estimation of Jump Regression Curves -- 3.1 Introduction -- 3.2 Jump detection when the number of jumps is known -- 3.2.1 Difference kernel estimation procedures -- 3.2.2 Jump detection based on local linear kernel smoothing -- 3.2.3 Estimation of jump regression functions based on semiparametric modeling.
3.2.4 Estimation of jump regression functions by spline smoothing -- 3.2.5 Jump and cusp detection by wavelet transformations -- 3.3 Jump estimation when the number of jumps is unknown -- 3.3.1 Jump detection by comparing three local estimators -- 3.3.2 Estimation of the number of jumps by a sequence of hypothesis tests -- 3.3.3 Jump detection by DAKE -- 3.3.4 Jump detection by local polynomial regression -- 3.4 Jump-preserving curve estimation -- 3.4.1 Jump curve estimation by split linear smoothing -- 3.4.2 Jump-preserving curve fitting based on local piecewise-linear kernel estimation -- 3.4.3 Jump-preserving smoothers based on robust estimation -- 3.5 Some discussions -- Problems -- 4 Estimation of Jump Location Curves of Regression Surfaces -- 4.1 Introduction -- 4.2 Jump detection when the number of jump location curves is known -- 4.2.1 Jump detection by RDKE -- 4.2.2 Minimax edge detection -- 4.2.3 Jump estimation based on a contrast statistic -- 4.2.4 Algorithms for tracking the JLCs -- 4.2.5 Estimation of JLCs by wavelet transformations -- 4.3 Detection of arbitrary jumps by local smoothing -- 4.3.1 Treat JLCs as a pointset in the design space -- 4.3.2 Jump detection by local linear estimation -- 4.3.3 Two modijication procedures -- 4.4 Jump detection in two or more given directions -- 4.4.1 Jump detection in two given directions -- 4.4.2 Measuring the performance of jump detection procedures -- 4.4.3 Connection to the Sobel edge detector -- 4.4.4 Jump detection in more than two given directions -- 4.5 Some discussions -- Problems -- 5 Jump-Preserving Surface Estimation By Local Smoothing -- 5.1 Introduction -- 5.2 A three-stage procedure -- 5.2.1 Jump detection -- 5.2.2 First-order approximation to the JLCs -- 5.2.3 Estimation of jump regression surfaces -- 5.3 Surface reconstruction with thresholding.
5.3.1 Surface reconstruction by local piecewisely linear kernel smoothing -- 5.3.2 Selection of procedure parameters -- 5.4 Surface reconstruction with gradient estimation -- 5.4.1 Gradient estimation and three possible surface estimators -- 5.4.2 Choose one of the three estimators based on the WRMS values -- 5.4.3 Choose one of the three estimators based on their estimated variances -- 5.4.4 A two-step procedure -- 5.5 Surface reconstruction by adaptive weights smoothing -- 5.5.1 Adaptive weights smoothing -- 5.5.2 Selection of procedure parameters -- 5.6 Some discussions -- Problems -- 6 Edge Detection In Image Processing -- 6.1 Introduction -- 6.2 Edge detection based on derivative estimation -- 6.2.1 Edge detection based on first-order derivatives -- 6.2.2 Edge detection based on second-order derivatives -- 6.2.3 Edge detection based on local surface estimation -- 6.3 Canny's edge detection criteria -- 6.3.1 Three criteria for measuring edge detection performance -- 6.3.2 Optimal edge detectors by the three criteria -- 6.3.3 Some modifications -- 6.4 Edge detection by multilevel masks -- 6.4.1 Step edge detection -- 6.4.2 Roof edge detection -- 6.5 Edge detection based on cost minimization -- 6.5.1 A mathematical description of edges -- 6.5.2 Five cost factors and the cost function -- 6.5.3 Minimization using simulated annealing -- 6.6 Edge linking techniques -- 6.6.1 Edge linking by curve estimation -- 6.6.2 Local edge linking based on image gradient estimation -- 6.6.3 Global edge linking by the Hough transform -- 6.7 Some discussions -- Problems -- 7 Edge-Preserving Image Restoration -- 7.1 Introduction -- 7.2 Image restoration by Fourier transformations -- 7.2.1 An image restoration model -- 7.2.2 2 -D Fourier transformations -- 7.2.3 Image restoration by Fourier transformation -- 7.2.4 Image restoration by algebraic approach.
7.3 Image restoration by Markov random field modeling -- 7.3.1 Markov random field modeling and Bayesian estimation -- 7.3.2 Geman and Geman s MAP procedure -- 7.3.3 Besag's ICM procedure and some modifications -- 7.3.4 Image restoration by regularization -- 7.4 Image restoration by local smoothing filters -- 7.4.1 Robust local smoothing filters -- 7.4.2 Adaptive smoothing and bilateral filtering -- 7.4.3 Image restoration by surface estimation -- 7.5 Image restoration by nonlinear difision filtering -- 7.5.1 Diffusion filtering -- 7.5.2 Relationship with adaptive smoothing and bilateral filtering -- 7.5.3 Some generalizations and modifications -- 7.6 Some discussions -- Problems -- References -- Index -- List of Figures -- 1.1 A conventional coordinate system for expressing an image in industry. -- 1.2 A log-transformed C-band, HH-polarization, synthetic aperture radar image of an area near Thetford forest, England. -- 1.3 December sea-level pressures observed by a Bombay weather station in India during 1921-1992. -- 2.1 Probability density curve of the standard normal distribution. -- 2.2 The Nadaraya-Watson (NW) kernel estimator and the local linear kernel (LK) estimator. -- 2.3 Behavior of the Nadaraya-Watson (NW) kernel estimator [plot (a)] and the local linear (LK) kernel estimator [plot (b)] of -- 2.4 Behavior of the Nadaraya- Watson (NW) kernel estimator [plot (a)] and the local linear kernel (LK) estimator [plot (b)] o -- 2.5 Four B-splines when ti, tj+1,tj+2, tj+3, and tj+4 are 0, 0.25, 0.5, 0.75, and 1.0. -- 2.6 The Haar father wavelet, the Haar mother wavelet, the Haar wavelet function y1,0, and the Haar wavelet function y1,1. -- 2.7 When f(x) and y(x) are the Haar father and mother wavelets, the two-dimensional wavelet functions F(x, y), Y(1)(x, y), Y(2)(x, y), and Y(3)(x, y) are displayed.
3.1 The true regression function f and the jump detection criterion MDKE dejined by expression (3.2) when c = 0,n = 100, and hn = 0.1. -- 3.2 The jump detection criterion MDKE and the jump detection criterion MDLK. -- 3.3 True regression function f, fc, fr, fl, and |fr-fl|. -- 3.4 If f(x) = 5x2 + I(x E [0.5, 1]), n = 100, and k = 7, then B1(i) ~ B1(xi), 4 <- i <- 97. The quantities {B1(xi), 4 <- i <- 97} include information about both the continuity and thejump parts off. With the use of the diference operator dejined in equation (3.20) -- 3.5 Slope estimators {B1(i)} from the Bombay sea-level pressure data and values of the jump detection criterion {D1(i)}. -- 3.6 Sea-level pressures observed by a Bombay weather station in India during 1921-1992, estimated regression function with a detected jump accommodated, and conventional local linear kernel estimator of the regression function. -- 3.7 I f f'(x) = 5x2 + I(x E (0.5, 1]), n = 100, and k is chosen to be 11, then B2(i) ~ B2(xi) for 6 <- i <- 95. After using the difference operatol -- which is similar to that in equation (3.20), we get {J2(xi)}. -- 3.8 The true regression function and al,0, ar,0, and f in the case when no noise is involved in the data. -- 4.1 Upper- and lower-sided supports of the two kernel functions K1* and K2* and two one-sided supports of the rotational kernel functions K1(q,.,.) and K2(q,.,.). -- 4.2 Two possible pointset estimators of the true JLC. -- 4.3 Three types of singular points of the JLCs. -- 4.4 At the design point (xi, yj), the jump detection criterion dij, is defined as the minimum length of the vectors vij - vN1 and vij - vN2, where vij, vN1, and vN2 are the gradient vectors of the fitted LS planes obtained in N(xi, yi), N(xi, yj), -- 4.5 True jump regression surface and jump location curve.
4.6 The gradient vector vij of the fitted LS plane at each design point.
Summary: The first text to bridge the gap between image processing and jump regression analysis Recent statistical tools developed to estimate jump curves and surfaces have broad applications, specifically in the area of image processing. Often, significant differences in technical terminologies make communication between the disciplines of image processing and jump regression analysis difficult. In easy-to-understand language, Image Processing and Jump Regression Analysis builds a bridge between the worlds of computer graphics and statistics by addressing both the connections and the differences between these two disciplines. The author provides a systematic analysis of the methodology behind nonparametric jump regression analysis by outlining procedures that are easy to use, simple to compute, and have proven statistical theory behind them. Key topics include: Conventional smoothing procedures Estimation of jump regression curves Estimation of jump location curves of regression surfaces Jump-preserving surface reconstruction based on local smoothing Edge detection in image processing Edge-preserving image restoration With mathematical proofs kept to a minimum, this book is uniquely accessible to a broad readership. It may be used as a primary text in nonparametric regression analysis and image processing as well as a reference guide for academicians and industry professionals focused on image processing or curve/surface estimation.
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Image Processing and Jump Regression Analysis -- Contents -- Preface -- 1 Introduction -- 1.1 Images and image representation -- 1.2 Regression curves and sugaces with jumps -- 1.3 Edge detection, image restoration, and jump regression analysis -- 1.4 Statistical process control and some other related topics -- 1.5 Organization of the book -- Problems -- 2 Basic Statistical Concepts and Conventional Smoothing Techniques -- 2.1 Introduction -- 2.2 Some basic statistical concepts and terminologies -- 2.2.1 Populations, samples, and distributions -- 2.2.2 Point estimation of population parameters -- 2.2.3 Confidence intervals and hypothesis testing -- 2.2.4 Maximum likelihood estimation and least squares estimation -- 2.3 Nadaraya- Watson and other kernel smoothing techniques -- 2.3.1 Univariate kernel estimators -- 2.3.2 Some statistical properties of kernel estimators -- 2.3.3 Multivariate kernel estimators -- 2.4 Local polynomial kernel smoothing techniques -- 2.4.1 Univariate local polynomial kernel estimators -- 2.4.2 Some statistical properties -- 2.4.3 Multivariate local polynomial kernel estimators -- 2.4.4 Bandwidth selection -- 2.5 Spline smoothing procedures -- 2.5.1 Univariate smoothing spline estimation -- 2.5.2 Selection of the smoothing parameter -- 2.5.3 Multivariate smoothing spline estimation -- 2.5.4 Regression spline estimation -- 2.6 Wavelet transformation methods -- 2.6.1 Function estimation based on Fourier transformation -- 2.6.2 Univariate wavelet transformations -- 2.6.3 Bivariate wavelet transformations -- Problems -- 3 Estimation of Jump Regression Curves -- 3.1 Introduction -- 3.2 Jump detection when the number of jumps is known -- 3.2.1 Difference kernel estimation procedures -- 3.2.2 Jump detection based on local linear kernel smoothing -- 3.2.3 Estimation of jump regression functions based on semiparametric modeling.

3.2.4 Estimation of jump regression functions by spline smoothing -- 3.2.5 Jump and cusp detection by wavelet transformations -- 3.3 Jump estimation when the number of jumps is unknown -- 3.3.1 Jump detection by comparing three local estimators -- 3.3.2 Estimation of the number of jumps by a sequence of hypothesis tests -- 3.3.3 Jump detection by DAKE -- 3.3.4 Jump detection by local polynomial regression -- 3.4 Jump-preserving curve estimation -- 3.4.1 Jump curve estimation by split linear smoothing -- 3.4.2 Jump-preserving curve fitting based on local piecewise-linear kernel estimation -- 3.4.3 Jump-preserving smoothers based on robust estimation -- 3.5 Some discussions -- Problems -- 4 Estimation of Jump Location Curves of Regression Surfaces -- 4.1 Introduction -- 4.2 Jump detection when the number of jump location curves is known -- 4.2.1 Jump detection by RDKE -- 4.2.2 Minimax edge detection -- 4.2.3 Jump estimation based on a contrast statistic -- 4.2.4 Algorithms for tracking the JLCs -- 4.2.5 Estimation of JLCs by wavelet transformations -- 4.3 Detection of arbitrary jumps by local smoothing -- 4.3.1 Treat JLCs as a pointset in the design space -- 4.3.2 Jump detection by local linear estimation -- 4.3.3 Two modijication procedures -- 4.4 Jump detection in two or more given directions -- 4.4.1 Jump detection in two given directions -- 4.4.2 Measuring the performance of jump detection procedures -- 4.4.3 Connection to the Sobel edge detector -- 4.4.4 Jump detection in more than two given directions -- 4.5 Some discussions -- Problems -- 5 Jump-Preserving Surface Estimation By Local Smoothing -- 5.1 Introduction -- 5.2 A three-stage procedure -- 5.2.1 Jump detection -- 5.2.2 First-order approximation to the JLCs -- 5.2.3 Estimation of jump regression surfaces -- 5.3 Surface reconstruction with thresholding.

5.3.1 Surface reconstruction by local piecewisely linear kernel smoothing -- 5.3.2 Selection of procedure parameters -- 5.4 Surface reconstruction with gradient estimation -- 5.4.1 Gradient estimation and three possible surface estimators -- 5.4.2 Choose one of the three estimators based on the WRMS values -- 5.4.3 Choose one of the three estimators based on their estimated variances -- 5.4.4 A two-step procedure -- 5.5 Surface reconstruction by adaptive weights smoothing -- 5.5.1 Adaptive weights smoothing -- 5.5.2 Selection of procedure parameters -- 5.6 Some discussions -- Problems -- 6 Edge Detection In Image Processing -- 6.1 Introduction -- 6.2 Edge detection based on derivative estimation -- 6.2.1 Edge detection based on first-order derivatives -- 6.2.2 Edge detection based on second-order derivatives -- 6.2.3 Edge detection based on local surface estimation -- 6.3 Canny's edge detection criteria -- 6.3.1 Three criteria for measuring edge detection performance -- 6.3.2 Optimal edge detectors by the three criteria -- 6.3.3 Some modifications -- 6.4 Edge detection by multilevel masks -- 6.4.1 Step edge detection -- 6.4.2 Roof edge detection -- 6.5 Edge detection based on cost minimization -- 6.5.1 A mathematical description of edges -- 6.5.2 Five cost factors and the cost function -- 6.5.3 Minimization using simulated annealing -- 6.6 Edge linking techniques -- 6.6.1 Edge linking by curve estimation -- 6.6.2 Local edge linking based on image gradient estimation -- 6.6.3 Global edge linking by the Hough transform -- 6.7 Some discussions -- Problems -- 7 Edge-Preserving Image Restoration -- 7.1 Introduction -- 7.2 Image restoration by Fourier transformations -- 7.2.1 An image restoration model -- 7.2.2 2 -D Fourier transformations -- 7.2.3 Image restoration by Fourier transformation -- 7.2.4 Image restoration by algebraic approach.

7.3 Image restoration by Markov random field modeling -- 7.3.1 Markov random field modeling and Bayesian estimation -- 7.3.2 Geman and Geman s MAP procedure -- 7.3.3 Besag's ICM procedure and some modifications -- 7.3.4 Image restoration by regularization -- 7.4 Image restoration by local smoothing filters -- 7.4.1 Robust local smoothing filters -- 7.4.2 Adaptive smoothing and bilateral filtering -- 7.4.3 Image restoration by surface estimation -- 7.5 Image restoration by nonlinear difision filtering -- 7.5.1 Diffusion filtering -- 7.5.2 Relationship with adaptive smoothing and bilateral filtering -- 7.5.3 Some generalizations and modifications -- 7.6 Some discussions -- Problems -- References -- Index -- List of Figures -- 1.1 A conventional coordinate system for expressing an image in industry. -- 1.2 A log-transformed C-band, HH-polarization, synthetic aperture radar image of an area near Thetford forest, England. -- 1.3 December sea-level pressures observed by a Bombay weather station in India during 1921-1992. -- 2.1 Probability density curve of the standard normal distribution. -- 2.2 The Nadaraya-Watson (NW) kernel estimator and the local linear kernel (LK) estimator. -- 2.3 Behavior of the Nadaraya-Watson (NW) kernel estimator [plot (a)] and the local linear (LK) kernel estimator [plot (b)] of -- 2.4 Behavior of the Nadaraya- Watson (NW) kernel estimator [plot (a)] and the local linear kernel (LK) estimator [plot (b)] o -- 2.5 Four B-splines when ti, tj+1,tj+2, tj+3, and tj+4 are 0, 0.25, 0.5, 0.75, and 1.0. -- 2.6 The Haar father wavelet, the Haar mother wavelet, the Haar wavelet function y1,0, and the Haar wavelet function y1,1. -- 2.7 When f(x) and y(x) are the Haar father and mother wavelets, the two-dimensional wavelet functions F(x, y), Y(1)(x, y), Y(2)(x, y), and Y(3)(x, y) are displayed.

3.1 The true regression function f and the jump detection criterion MDKE dejined by expression (3.2) when c = 0,n = 100, and hn = 0.1. -- 3.2 The jump detection criterion MDKE and the jump detection criterion MDLK. -- 3.3 True regression function f, fc, fr, fl, and |fr-fl|. -- 3.4 If f(x) = 5x2 + I(x E [0.5, 1]), n = 100, and k = 7, then B1(i) ~ B1(xi), 4 <- i <- 97. The quantities {B1(xi), 4 <- i <- 97} include information about both the continuity and thejump parts off. With the use of the diference operator dejined in equation (3.20) -- 3.5 Slope estimators {B1(i)} from the Bombay sea-level pressure data and values of the jump detection criterion {D1(i)}. -- 3.6 Sea-level pressures observed by a Bombay weather station in India during 1921-1992, estimated regression function with a detected jump accommodated, and conventional local linear kernel estimator of the regression function. -- 3.7 I f f'(x) = 5x2 + I(x E (0.5, 1]), n = 100, and k is chosen to be 11, then B2(i) ~ B2(xi) for 6 <- i <- 95. After using the difference operatol -- which is similar to that in equation (3.20), we get {J2(xi)}. -- 3.8 The true regression function and al,0, ar,0, and f in the case when no noise is involved in the data. -- 4.1 Upper- and lower-sided supports of the two kernel functions K1* and K2* and two one-sided supports of the rotational kernel functions K1(q,.,.) and K2(q,.,.). -- 4.2 Two possible pointset estimators of the true JLC. -- 4.3 Three types of singular points of the JLCs. -- 4.4 At the design point (xi, yj), the jump detection criterion dij, is defined as the minimum length of the vectors vij - vN1 and vij - vN2, where vij, vN1, and vN2 are the gradient vectors of the fitted LS planes obtained in N(xi, yi), N(xi, yj), -- 4.5 True jump regression surface and jump location curve.

4.6 The gradient vector vij of the fitted LS plane at each design point.

The first text to bridge the gap between image processing and jump regression analysis Recent statistical tools developed to estimate jump curves and surfaces have broad applications, specifically in the area of image processing. Often, significant differences in technical terminologies make communication between the disciplines of image processing and jump regression analysis difficult. In easy-to-understand language, Image Processing and Jump Regression Analysis builds a bridge between the worlds of computer graphics and statistics by addressing both the connections and the differences between these two disciplines. The author provides a systematic analysis of the methodology behind nonparametric jump regression analysis by outlining procedures that are easy to use, simple to compute, and have proven statistical theory behind them. Key topics include: Conventional smoothing procedures Estimation of jump regression curves Estimation of jump location curves of regression surfaces Jump-preserving surface reconstruction based on local smoothing Edge detection in image processing Edge-preserving image restoration With mathematical proofs kept to a minimum, this book is uniquely accessible to a broad readership. It may be used as a primary text in nonparametric regression analysis and image processing as well as a reference guide for academicians and industry professionals focused on image processing or curve/surface estimation.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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