Fundamentals of Digital Image Processing

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1 \L\.6 Gw.i Fundamentals of Digital Image Processing A Practical Approach with Examples in Matlab Chris Solomon School of Physical Sciences, University of Kent, Canterbury, UK Toby Breckon School of Engineering, Cranfield University, Bedfordshire, UK ~WILEY-BLACKWELL A John Wiley & Sons, Ltd., Publication

2 Contents Preface Using the book website xi XV 1 Representation What is an image? Image Layout Image colour Resolution and quantization Bit-plane splicing Image formats Image data types Image compression Colour spaces RGB RGB to grey-scale image conversion Perceptual colour space Images in Matlab Reading, writing and querying images Basic display of images Accessing pixel values Converting im age types 17 Exercises 18 2 Formation How is an image formed? The mathematics of image formation Introduction Linear imaging systems Linear superposition integral The Dirac delta or impulse function The point-spread function 28

3 vi CONTENTS Linear shift-invariant systems and the convolution integral Convolution: its importance and meaning Multiple convolution: N imaging elements in a linear shift-invariant system Digital convolution The engineering of image formation The camera The digitization process Quantization Digitization hardware Resolution versus performance Noise 44 Exercises 46 3 Pixels What is a pixel? Operations upon pixels Arithmetic operations on images Image addition and subtraction Multiplication and division Logical operations on images Thresholding Point-based operations on images Logarithmic transform Exponential transform Power-law (gamma) transform Application: gamma correction Pixel distributions: histograms Histograms for threshold selection Adaptive thresholding Contrast stretching Histogram equalization Histogram equalization theory Histogram equalization theory: discrete case Histogram equalization in practice Histogram matching Histogram-matching theory Histogram-matching theory: discrete case Histogram matching in practice Adaptive histogram equalization Histogram operations on colour images 79 Exercises 81

4 CONTENTS vii 4 Enhancement Why perform enhancement? Enhancement via image filtering Pixel neighbourhoods Filter kernels and the mechanics of linear filtering Nonlinear spatial filtering Filtering for noise removal Mean filtering Median filtering Rank filtering Gaussian filtering Filtering for edge detection Derivative filters for discontinuities First-order edge detection Linearly separable filtering Second-order edge detection Laplacian edge detection Laplacian of Gaussian Zero-crossing detector Edge enhancement Laplacian edge sharpening The unsharp mask filter 107 Exercises Fourier transforms and frequency-domain processing Frequency space: a friendly introduction Frequency space: the fundamental idea The Fourier series Calculation of the Fourier spectrum Complex Fourier series The 1-D Fourier transform The inverse Fourier transform and reciprocity The 2-D Fourier transform Understanding the Fourier transform: frequency-space filtering Linear systems and Fourier transforms The convolution theorem The optical transfer function Digital Fourier transforms: the discrete fast Fourier transform Sampled data: the discrete Fourier transform The centred discrete Fourier transform Image restoration Imaging models Nature of the point-spread function and noise 142

5 viii CONTENTS 6.3 Restoration by the inverse Fourier filter The Wiener- Helstrom Filter Origin of the Wiener-Helstrom filter Acceptable solutions to the imaging equation Constrained deconvolution Estimating an unknown point-spread function or optical transfer function Blind deconvolution Iterative deconvolution and the Lucy-Richardson algorithm Matrix formulation of image restoration The standard Least-squares solution Constrained Least-squares restoration Stochastic input distributions and Bayesian estimators The generalized Gauss- Markov estimator Geometry The descriptio n of shape Shape-preserving transformations Shape transformation and homogeneous coordinates The general 2-D affine transformation Affine transformation in homogeneous coordinates The Procrustes transformation Procrustes alignment The projective transform Nonlinear transformations Warping: the spatial transformation of an image Overdetermined spatial transformations The piecewise warp The piecewise affine warp Warping: forward and reverse mapping Morphological processing Introduction Binary images: foreground, background and connectedness Structuring elements and neighbourhoods Dilation and erosion Dilation, erosion and structuring elements within Matlab Structuring element decomposition and Matlab Effects and uses of erosion and dilation Application of erosion to particle sizing Morphological opening and closing The rolling-ball analogy Boundary extraction Extracting connected components 213

6 CONTENTS ix 8.11 Region filling The hit-or-miss transformation Generalization of hit-or-miss Relaxing constraints in hit-or-miss: 'don't care' pixels Morphological thinning Skeletonization Opening by reconstruction Grey-scale erosion and dilation Grey-scale structuring elements: general case Grey-scale erosion and dilation with flat structuring elements Grey-scale opening and closing The top-hat transformation Summary 231 Exercises Features Landmarks and shape vectors Single-parameter shape descriptors Signatures and the radial Fourier expansion Statistical moments as region descriptors Texture features based on statistical measures Principal component analysis Principal component analysis: an illustrative example Theory of principal component analysis: version Theory of principal component analysis: version Principal axes and principal components Summary of properties of principal component analysis Dimensionality reduction: the purpose of principal com ponent analysis Principal components analysis on an ensemble of digital images Representation of out-of-sample exam ples using principal component analysis Key example: eigenfaces and the human face Image Segmentation Image segmentation Use of image properties and features in segmentation Intensity thresholding Problems with global thresholding Region growing and region splitting Split-and-merge algorithm The challenge of edge detection The Laplacian of Gaussian and difference of Gaussians filters The Canny edge detector 271

7 ,. X CONTENTS d 10.9 Interest operators Watershed segmentation Segmentation functions Image segmentation with Markov random fields Parameter estimation Neighbourhood weighting parameter On Minimizing U(x ly): the iterated conditional modes algorithm Classification The purpose of automated classification 11.2 Supervised and unsupervised classification 11.3 Classification: a simple example 11.4 Design of classification systems 11.5 Simple classifiers: prototypes and minimum distance criteria 11.6 Linear discriminant functions 11.7 Linear discriminant functions in N dimensions 11.8 Extension of the minimum distance classifier and the Mahalanobis distance 11.9 Bayesian classification: definitions The Bayes decision rule The multivariate normal density Bayesian classifiers for multivariate normal distributions The Fisher Linear discriminant Risk and cost functions Ensemble classifiers Combining weak classifiers: the AdaBoost method Unsupervised Learning: k-means clustering 313 Further reading 317 Index 319

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