Digital Image Processing (2022) [Burger Burge] [9783031057434]

Preface
Contents
Part I Images and Pixels
1 Digital Images
	1.1 Programming with Images
	1.2 Image Analysis and Computer Vision
	1.3 Types of Digital Images
	1.4 Image Acquisition
		1.4.1 The Pinhole Camera Model
		1.4.2 The “Thin” Lens
		1.4.3 Going Digital
		1.4.4 Image Size and Resolution
		1.4.5 Image Coordinate System
		1.4.6 Pixel Values
	1.5 Image File Formats
		1.5.1 Raster Versus Vector Data
		1.5.2 Tagged Image File Format (TIFF)
		1.5.3 Graphics Interchange Format (GIF)
		1.5.4 Portable Network Graphics (PNG)
		1.5.5 JPEG
		1.5.6 Legacy File Formats
		1.5.7 Bits and Bytes
	1.6 Software for Digital Imaging
	1.7 ImageJ
		1.7.1 Key Features
		1.7.2 Interactive Tools
		1.7.3 Working With ImageJ and Java
	1.8 Exercises
2 Histograms and Image Statistics
	2.1 What is a Histogram?
	2.2 Interpreting Histograms
		2.2.1 Image Acquisition
		2.2.2 Image Defects
	2.3 Calculating Histograms
	2.4 Histograms of Images With More Than 8 Bits
		2.4.1 Binning
		2.4.2 Example
		2.4.3 Implementation
	2.5 Histograms of Color Images
		2.5.1 Intensity Histograms
		2.5.2 Individual Color Channel Histograms
		2.5.3 Combined Color Histograms
	2.6 The Cumulative Histogram
	2.7 Statistical Information from the Histogram
		2.7.1 Mean and Variance
		2.7.2 Median
	2.8 Block Statistics
		2.8.1 Integral Images
		2.8.2 Mean Intensity
		2.8.3 Variance
		2.8.4 Practical Calculation of Integral Images
	2.9 Exercises
3 Point Operations
	3.1 Modifying Image Intensity
		3.1.1 Contrast and Brightness
		3.1.2 Limiting Values by Clamping
		3.1.3 Inverting Images
		3.1.4 Thresholding Operation
	3.2 Point Operations and Histograms
	3.3 Automatic Contrast Adjustment
	3.4 Modified Auto-Contrast Operation
	3.5 Histogram Equalization
	3.6 Histogram Specification
		3.6.1 Frequencies and Probabilities
		3.6.2 Principle of Histogram Specification
		3.6.3 Adjusting to a Piecewise Linear Distribution
		3.6.4 Adjusting to a Given Histogram (Histogram Matching)
		3.6.5 Examples
	3.7 Gamma Correction
		3.7.1 Why “Gamma”?
		3.7.2 Mathematical Definition
		3.7.3 Real Gamma Values
		3.7.4 Applications of Gamma Correction
		3.7.5 Implementation
		3.7.6 Modified Gamma Correction
	3.8 Point Operations in ImageJ
		3.8.1 Point Operations with Lookup Tables
		3.8.2 Arithmetic Operations
		3.8.3 Point Operations Involving Multiple Images
		3.8.4 Methods for Point Operations on Two Images
		3.8.5 ImageJ Plugins Involving Multiple Images
	3.9 Exercises
Part II Filters, Edges and Corners
4 Filters
	4.1 What is a Filter?
	4.2 Linear Filters
		4.2.1 The Filter Kernel
		4.2.2 Applying the Filter
		4.2.3 Implementing Filter Operations
		4.2.4 Filter Plugin Examples
		4.2.5 Integer Coefficients
		4.2.6 Filters of Arbitrary Size
		4.2.7 Types of Linear Filters
	4.3 Formal Properties of Linear Filters
		4.3.1 Linear Convolution
		4.3.2 Formal Properties of Linear Convolution
		4.3.3 Separability of Linear Filters
		4.3.4 Impulse Response of a Filter
	4.4 Nonlinear Filters
		4.4.1 Minimum and Maximum Filters
		4.4.2 Median Filter
		4.4.3 Weighted Median Filter
		4.4.4 Other Nonlinear Filters
	4.5 Implementing Filters
		4.5.1 Efficiency of Filter Programs
		4.5.2 Handling Image Borders
		4.5.3 Debugging Filter Programs
	4.6 Filter Operations in ImageJ
		4.6.1 Linear Filters
		4.6.2 Gaussian Filters
		4.6.3 Nonlinear Filters
	4.7 Exercises
5 Edges and Contours
	5.1 What Makes an Edge?
	5.2 Gradient-Based Edge Detection
		5.2.1 Partial Derivatives and the Gradient
		5.2.2 Derivative Filters
	5.3 Simple Edge Operators
		5.3.1 Prewitt and Sobel Edge Operators
		5.3.2 Roberts Edge Operator
		5.3.3 Compass Operators
		5.3.4 Edge Operators in ImageJ
	5.4 Other Edge Operators
		5.4.1 Edge Detection Based on Second Derivatives
		5.4.2 Edges at Different Scales
		5.4.3 From Edges to Contours
	5.5 Canny Edge Operator
		5.5.1 Preprocessing
		5.5.2 Edge Localization
		5.5.3 Edge Tracing and Hysteresis Thresholding
		5.5.4 Additional Information
		5.5.5 Implementation
	5.6 Edge Sharpening
		5.6.1 Edge Sharpening with the Laplacian Filter
		5.6.2 Unsharp Masking
	5.7 Exercises
6 Corner Detection
	6.1 Points of Interest
	6.2 Harris Corner Detector
		6.2.1 The Local Structure Matrix
		6.2.2 Significance of the Local Structure Matrix
		6.2.3 Corner Response Function (CRF)
		6.2.4 Selecting Corner Points
		6.2.5 Examples
	6.3 Alternative Formulations
		6.3.1 Shi-Tomasi Corner Score
		6.3.2 MOPS Corner Score
	6.4 Basic Implementation
		6.4.1 Summary
	6.5 Sub-Pixel Corner Positions
		6.5.1 Position Interpolation by Second-Order Taylor Expansion
		6.5.2 Sub-Pixel Positioning Example
	6.6 Exercises
Part III Binary Images
7 Morphological Filters
	7.1 Shrink and Let Grow
		7.1.1 Pixel Neighborhoods
	7.2 Basic Morphological Operations
		7.2.1 The Structuring Element
		7.2.2 Point Sets
		7.2.3 Dilation
		7.2.4 Erosion
		7.2.5 Formal Properties of Dilation and Erosion
		7.2.6 Designing Morphological Filters
		7.2.7 Application Example: Outline
	7.3 Composite Morphological Operations
		7.3.1 Opening
		7.3.2 Closing
		7.3.3 Properties of Opening and Closing
	7.4 Thinning (Skeletonization)
		7.4.1 Basic Algorithm
		7.4.2 Fast Thinning Algorithm
		7.4.3 Java Implementation
		7.4.4 Built-in Morphological Operations in ImageJ
	7.5 Grayscale Morphology
		7.5.1 Structuring Elements
		7.5.2 Dilation and Erosion
		7.5.3 Grayscale Opening and Closing
	7.6 Exercises
8 Regions in Binary Images
	8.1 Finding Connected Image Regions
		8.1.1 Region Labeling by Flood Filling
		8.1.2 Sequential Region Segmentation
		8.1.3 Region Labeling – Summary
	8.2 Region Contours
		8.2.1 Outer and Inner Contours
		8.2.2 Combining Region Labeling and Contour Detection
	8.3 Representing Image Regions
		8.3.1 Matrix Representation
		8.3.2 Run Length Encoding
		8.3.3 Chain Codes
	8.4 Properties of Binary Regions
		8.4.1 Shape Features
		8.4.2 Geometric Features
	8.5 Statistical Shape Properties
		8.5.1 Centroid
		8.5.2 Moments
		8.5.3 Central Moments
		8.5.4 Normalized Central Moments
		8.5.5 Java Implementation
	8.6 Moment-Based Geometric Properties
		8.6.1 Orientation
		8.6.2 Region Eccentricity
		8.6.3 Equivalent Ellipse
		8.6.4 Bounding Box Aligned to the Major Axis
		8.6.5 Invariant Region Moments
	8.7 Projections
	8.8 Topological Region Properties
	8.9 Java Implementation
	8.10 Exercises
9 Automatic Thresholding
	9.1 Global Histogram-Based Thresholding
		9.1.1 Image Statistics from the Histogram
		9.1.2 Simple Threshold Selection
		9.1.3 Iterative Threshold Selection (Isodata Algorithm)
		9.1.4 Otsu’s Method
		9.1.5 Maximum Entropy Thresholding
		9.1.6 Minimum Error Thresholding
	9.2 Local Adaptive Thresholding
		9.2.1 Bernsen’s Method
		9.2.2 Niblack’s Method
	9.3 Java Implementation
		9.3.1 Global Thresholding Methods
		9.3.2 Adaptive Thresholding
	9.4 Summary and Further Reading
	9.5 Exercises
Part IV Geometric Primitives
10 Fitting Straight Lines
	10.1 Straight Line Equations
		10.1.1 Slope-Intercept Form
		10.1.2 Parametric (Point-Vector) Form
		10.1.3 Algebraic Form
		10.1.4 Hessian Normal Form
	10.2 Fitting Lines to Points Sets
		10.2.1 Linear Regression
		10.2.2 Orthogonal Regression
	10.3 Example: Contour Segmentation
	10.4 Java Implementation
	10.5 Exercises
11 Fitting Circles and Ellipses
	11.1 Fitting Circles
		11.1.1 Circle Equations
		11.1.2 Algebraic Circle Fits
		11.1.3 Geometric Circle Fitting
	11.2 Fitting Ellipses
		11.2.1 Algebraic Ellipse Fitting
		11.2.2 Geometric Ellipse Fitting
		11.2.3 Orthogonal Distance Approximations
	11.3 Java Implementation
		11.3.1 Circle Fitting
		11.3.2 Ellipse Fitting
12 Detecting Geometric Primitives
	12.1 Random Sample Consensus (RANSAC)
		12.1.1 How Many Random Draws Are Needed?
		12.1.2 RANSAC Line Detection Algorithm
		12.1.3 Detecting Multiple Lines
		12.1.4 RANSAC Circle Detection
		12.1.5 RANDSAC Ellipse Detection
		12.1.6 RANSAC Extensions and Applications
	12.2 The Hough Transform
		12.2.1 Parameter Space
		12.2.2 Accumulator Map
		12.2.3 A Better Line Representation
		12.2.4 Hough Algorithm
		12.2.5 Hough Transform Extensions
		12.2.6 Hough Transform for Circles and Arcs
		12.2.7 Hough Transform for Ellipses
	12.3 Java Implementation
	12.4 Exercises
Part V Color
13 Color Images
	13.1 RGB Color Images
		13.1.1 Structure of Color Images
		13.1.2 Color Images in ImageJ
	13.2 Color Spaces and Color Conversion
		13.2.1 Conversion to Grayscale
		13.2.2 Desaturating RGB Color Images
		13.2.3 HSV/HSB and HLS Color Spaces
		13.2.4 TV Component Color Spaces: YUV, YIQ, and YCbCr
		13.2.5 Color Spaces for Printing: CMY and CMYK
	13.3 Statistics of Color Images
		13.3.1 How Many Different Colors Are in an Image?
		13.3.2 Color Histograms
	13.4 Color Quantization
		13.4.1 Scalar Color Quantization
		13.4.2 Vector Quantization
		13.4.3 Java Implementation
	13.5 Exercises
14 Colorimetric Color Spaces
	14.1 CIE Color Spaces
		14.1.1 CIE XYZ Color Space
		14.1.2 CIE x, y Chromaticity
		14.1.3 Standard Illuminants
		14.1.4 Gamut
		14.1.5 Variants of the CIE Color Space
	14.2 CIELAB Color Space
		14.2.1 CIEXYZ → CIELAB Conversion
		14.2.2 CIELAB → CIEXYZ Conversion
	14.3 CIELUV Color Space
		14.3.1 CIEXYZ → CIELUV Conversion
		14.3.2 CIELUV→ CIEXYZ Conversion
		14.3.3 Measuring Color Differences
	14.4 Standard RGB (sRGB)
		14.4.1 Linear vs. Nonlinear Color Components
		14.4.2 CIEXYZ → sRGB Conversion
		14.4.3 sRGB → CIEXYZ Conversion
		14.4.4 Calculations with Nonlinear sRGB Values
	14.5 Adobe RGB
	14.6 Chromatic Adaptation
		14.6.1 XYZ Scaling
		14.6.2 Bradford Color Adaptation
	14.7 Colorimetric Support in Java
		14.7.1 Profile Connection Space (PCS)
		14.7.2 Color-Related Java Classes
		14.7.3 Implementation of the CIELAB Color Space (Example)
		14.7.4 ICC Profiles
	14.8 Exercises
15 Filters for Color Images
	15.1 Linear Filters
		15.1.1 Monochromatic Application of Linear Filters
		15.1.2 Color Space Matters
		15.1.3 Linear Filtering with Circular Values
	15.2 Nonlinear Color Filters
		15.2.1 Scalar Median Filter
		15.2.2 Vector Median Filter
		15.2.3 Sharpening Vector Median Filter
	15.3 Java Implementation
	15.4 Further Reading
	15.5 Exercises
16 Edge Detection in Color Images
	16.1 Monochromatic Techniques
	16.2 Edges in Vector-Valued Images
		16.2.1 Multi-Dimensional Gradients
		16.2.2 The Jacobian Matrix
		16.2.3 Squared Local Contrast
		16.2.4 Color Edge Magnitude
		16.2.5 Color Edge Orientation
		16.2.6 Grayscale Gradients Revisited
	16.3 Canny Edge Detector for Color Images
	16.4 Other Color Edge Operators
	16.5 Java Implementation
	16.6 Exercises
17 Edge-Preserving Smoothing Filters
	17.1 Kuwahara-Type Filters
		17.1.1 Application to Color Images
	17.2 Bilateral Filter
		17.2.1 Domain Filter
		17.2.2 Range Filter
		17.2.3 Bilateral Filter: General Idea
		17.2.4 Bilateral Filter with Gaussian Kernels
		17.2.5 Application to Color Images
		17.2.6 Efficient Implementation by x/y Separation
		17.2.7 Further Reading
	17.3 Anisotropic Diffusion Filters
		17.3.1 Homogeneous Diffusion and the Heat Equation
		17.3.2 The Perona-Malik Filter
		17.3.3 Perona-Malik Filter for Color Images
		17.3.4 Geometry Preserving Anisotropic Diffusion
		17.3.5 Tschumperlé-Deriche Algorithm
	17.4 Java Implementation
	17.5 Exercises
Part VI Spectral Techniques
18 Introduction to Spectral Methods
	18.1 The Fourier Transform
		18.1.1 Sine and Cosine Functions
		18.1.2 Fourier Series Representation of Periodic Functions
		18.1.3 Fourier Integral
		18.1.4 Fourier Spectrum and Transformation
		18.1.5 Fourier Transform Pairs
		18.1.6 Important Properties of the Fourier Transform
	18.2 Working with Discrete Signals
		18.2.1 Sampling
		Assume
	18.3 The Discrete Fourier Transform (DFT)
		18.3.1 Definition of the DFT
		18.3.2 Discrete Basis Functions
		18.3.3 Aliasing Again!
		18.3.4 Units in Signal and Frequency Space
		18.3.5 Power Spectrum
	18.4 Implementing the DFT
		18.4.1 Direct Implementation
		18.4.2 Fast Fourier Transform (FFT)
	18.5 Exercises
19 The Discrete Fourier Transform in 2D
	19.1 Definition of the 2D DFT
		19.1.1 2D Basis Functions
		19.1.2 Implementing the 2D DFT
	19.2 Visualizing the 2D Fourier Transform
		19.2.1 Range of Spectral Values
		19.2.2 Centered Representation of the DFT Spectrum
	19.3 Frequencies and Orientation in 2D
		19.3.1 Effective Frequency
		19.3.2 Frequency Limits and Aliasing in 2D
		19.3.3 Orientation
		19.3.4 Normalizing the Geometry of the 2D Spectrum
		19.3.5 Effects of Periodicity
		19.3.6 Windowing
		19.3.7 Common Windowing Functions
	19.4 2D Fourier Transform Examples
	19.5 Applications of the DFT
		19.5.1 Linear Filter Operations in Frequency Space
		19.5.2 Linear Convolution and Correlation
		19.5.3 Inverse Filters
	19.6 Exercises
20 The Discrete Cosine Transform (DCT)
	20.1 One-Dimensional DCT
		20.1.1 DCT Basis Functions
		20.1.2 Implementing the 1D DCT
	20.2 Two-Dimensional DCT
		20.2.1 Examples
		20.2.2 Separability
	20.3 Java Implementation
	20.4 Other Spectral Transforms
	20.5 Exercises
Part VII Image Transformations
21 Geometric Operations
	21.1 Coordinate Transformations in 2D
		21.1.1 Linear Coordinate Transformations
		21.1.2 Homogeneous Coordinates
		21.1.3 Affine (Three-Point) Mapping
		21.1.4 Projective (Four-Point) Mapping
		21.1.5 Bilinear Mapping
		21.1.6 Log-Polar Mapping
		21.1.7 Other Nonlinear Transformations
		21.1.8 Piecewise Image Transformations
	21.2 Resampling the Image
		21.2.1 Source-to-Target Mapping
		21.2.2 Target-to-Source Mapping
	21.3 Java Implementation
		21.3.1 Geometric Transformations
		21.3.2 Image Transformations
		21.3.3 Examples
	21.4 Exercises
22 Pixel Interpolation
	22.1 Interpolation in 1D: Simple Methods
		22.1.1 Nearest-Neighbor Interpolation
		22.1.2 Linear Interpolation
		22.1.3 Ideal Low-Pass Filter
	22.2 Interpolation by Convolution
	22.3 Cubic Interpolation
	22.4 Spline Interpolation
		22.4.1 Catmull-Rom Interpolation
		22.4.2 Cubic B-spline Approximation
		22.4.3 Mitchell-Netravali Approximation
		22.4.4 Lanczos Interpolation
	22.5 Interpolation in 2D
		22.5.1 Nearest-Neighbor Interpolation in 2D
		22.5.2 Bilinear Interpolation
		22.5.3 Bicubic and Spline Interpolation in 2D
		22.5.4 Lanczos Interpolation in 2D
		22.5.5 Examples and Discussion
	22.6 Aliasing
		22.6.1 Sampling the Interpolated Image
		22.6.2 Space-Variant Low-Pass Filtering
	22.7 Java Implementation
	22.8 Exercises
Part VIII
Image Matching
23 Image Matching and Registration
	23.1 Template Matching in Intensity Images
		23.1.1 Distance between Image Patterns
		23.1.2 Matching Under Rotation and Scaling
		23.1.3 Java Implementation
	23.2 Matching Binary Images
		23.2.1 Direct Comparison of Binary Images
		23.2.2 The Distance Transform
		23.2.3 Chamfer Matching
		23.2.4 Java Implementation
	23.3 Exercises
24 Non-Rigid Image Matching
	24.1 The Lucas-Kanade Technique
		24.1.1 Registration in 1D
		24.1.2 Extension to Multi-Dimensional Functions
	24.2 The Lucas-Kanade Algorithm
		24.2.1 Summary of the Algorithm
	24.3 Inverse Compositional Algorithm
	24.4 Linear Transformation Parameters
		24.4.1 Pure Translation
		24.4.2 Affine Transformation
		24.4.3 Projective Transformation
		24.4.4 Concatenating Linear Transformations
		24.4.5 Coordinate Frames
	24.5 Example
	24.6 Java Implementation
	24.7 Exercises
Part IX Local Features
25 Scale-Invariant Feature Transform (SIFT)
	25.1 Interest Points at Multiple Scales
		25.1.1 The LoG Filter
		25.1.2 Gaussian Scale Space
		25.1.3 LoG/DoG Scale Space
		25.1.4 Hierarchical Scale Space
		25.1.5 Scale Space Structure in SIFT
	25.2 Key Point Selection and Refinement
		25.2.1 Local Extrema Detection
		25.2.2 Position Refinement
		25.2.3 Suppressing Responses to Edge-Like Structures
	25.3 Creating Local Descriptors
		25.3.1 Finding Dominant Orientations
		25.3.2 SIFT Descriptor Construction
	25.4 SIFT Algorithm Summary
	25.5 Matching SIFT Features
		25.5.1 Feature Distance and Match Quality
		25.5.2 Examples
	25.6 Efficient Feature Matching
	25.7 Java Implementation
		25.7.1 SIFT Feature Extraction
		25.7.2 SIFT Feature Matching
	25.8 Exercises
26 Maximally Stable Extremal Regions (MSER)
	26.1 Threshold Sets and Extremal Regions
	26.2 Building the Component Tree
		26.2.1 Component Tree Algorithms
		26.2.2 Component Tree Algorithm 1: Global Immersion
		26.2.3 Component Tree Algorithm 2: Local Flooding
		26.2.4 Component Tree Examples
	26.3 Extracting MSERs from the Component Tree
		26.3.1 Component Size Variation (Growth Rate)
		26.3.2 Maximally Stable Components
		26.3.3 Constraints on Component Size and Diversity
		26.3.4 MSER Feature Statistics and Equivalent Ellipse
		26.3.5 Additional Constraints
		26.3.6 Detecting Dark And Bright Blobs
		26.3.7 MSER Examples
	26.4 Matching MSERs
	26.5 Local Affine Frames
	26.6 Summary
Appendix
Appendix A Mathematical Symbols and Notation
	A.1 Symbols
	A.2 Sets
		A.2.1 Basic Set Symbols and Operators
		A.2.2 Destructive Set Operators
		A.2.3 Relations, Mappings and Functions
	A.3 Sequences
		A.3.1 Adding and Removing Elements
		A.3.2 “Stack”-Type Sequences
		A.3.3 “Queue”-Type Sequences
		A.3.4 Sorting Sequences
	A.4 Tuples and Objects
		A.4.1 Type Definition and Instantiation
		A.4.2 Accessing Object Components
		A.4.3 Duplication
	A.5 Complex Numbers
Appendix B Linear Algebra
	B.1 Vectors and Matrices
		B.1.1 Column and Row Vectors
		B.1.2 Extracting Submatrices and Vectors
		B.1.3 Length (Norm) of a Vector
	B.2 Matrix Multiplication
		B.2.1 Scalar Multiplication
		B.2.2 Product of Two Matrices
		B.2.3 Matrix-Vector Products
	B.3 Vector Products
		B.3.1 Dot (Scalar) Product
		B.3.2 Outer Product
		B.3.3 Cross Product
	B.4 Trace and Determinant of a Square Matrix
	B.5 Eigenvalues and Eigenvectors
		B.5.1 Calculating Eigenvalues
		B.5.2 Generalized Symmetric Eigenproblems
	B.6 Homogeneous Coordinates
	B.7 Basic Matrix-Vector Operations with the Apache Commons Math Library
		B.7.1 Vectors and Matrices
		B.7.2 Matrix-Vector Multiplication
		B.7.3 Vector Products
		B.7.4 Inverse of a Square Matrix
		B.7.5 Eigenvalues and Eigenvectors
	B.8 Solving Systems of Linear Equations
		B.8.1 Exact Solutions
		B.8.2 Over-Determined System (Least-Squares Solutions)
		B.8.3 Solving Homogeneous Linear Systems
Appendix C Nonlinear Least Squares
	C.1 Nonlinear Least-Squares Fitting
	C.2 Solution Methods
		C.2.1 Implementation With Apache Commons Math
		C.2.2 Example 1: One-Dimensional Curve Fitting
	C.3 Multi-Dimensional NLS Problems
		C.3.1 Example 2: Geometric Circle Fitting
		C.3.2 Numerical Estimation of Partial Derivatives
Appendix D Elements from Calculus
	D.1 Scalar and Vector Fields
		D.1.1 The Jacobian Matrix
		D.1.2 Gradients
		D.1.3 Maximum Gradient Direction
		D.1.4 Divergence of a Vector Field
		D.1.5 The Laplacian Operator
		D.1.6 The Hessian Matrix
	D.2 Taylor Series Expansion
		D.2.1 Single-Variable Functions
		D.2.2 Multi-Variable Functions
		D.2.3 Finding Function Extrema by 2nd-Order Taylor Expansion
	D.3 Estimating Derivatives of Discrete Functions
		D.3.1 First-order Derivatives
		D.3.2 Second-Order Derivatives
		D.3.3 Alternative Formulations
Appendix E Sub-Pixel Maximum Finding
	E.1 Second-Order Interpolation in 1D
	E.2 Subpixel Interpolation in 2D
		E.2.1 Quadratic Functions in 2D
		E.2.2 Method A: Second-Order Taylor Interpolation
		E.2.3 Method B: Least-Squares Quadratic Interpolation
		E.2.4 Quartic Interpolation
Appendix F Geometry
	F.1 Straight Lines
		F.1.1 Conversions Between Different Line Equations
		F.1.2 Intersections of Algebraic Lines
		F.1.3 Intersections of Lines in Hessian Normal Form
		F.1.4 Numeric Line Fitting Examples
	F.2 Circles
		F.2.1 Circle Equations and Conversions
		F.2.2 Circle From 3 Points
	F.3 Ellipses
		F.3.1 Ellipse Equations
		F.3.2 Converting Between Algebraic and Geometric Parameters
		F.3.3 Ellipse From 5 Points
Appendix G Statistical Prerequisites
	G.1 Mean, Variance, and Covariance
		G.1.1 Mean
		G.1.2 Variance and Covariance
		G.1.3 Biased vs. Unbiased Variance
	G.2 The Covariance Matrix
		G.2.1 Example
		G.2.2 Practical Calculation
	G.3 Mahalanobis Distance
		G.3.1 Definition
		G.3.2 Relation to the Euclidean Distance
		G.3.3 Numerical Considerations
		G.3.4 Pre-Mapping Data For Efficient Mahalanobis Matching
	G.4 The Gaussian Distribution
		G.4.1 Maximum Likelihood Estimation
		G.4.2 Gaussian Mixtures
		G.4.3 Creating Gaussian Noise
Appendix H Gaussian Filters
	H.1 Cascading Gaussian Filters
	H.2 Gaussian Filters and Scale Space
	H.3 Effects of Gaussian Filtering in the Frequency Domain
	H.4 LoG-Approximation by Difference of Gaussians (DoG)
Appendix I Writing ImageJ Plugins
	I.1 ImageJ Plugins
		I.1.1 Program Structure
		I.1.2 A First Example: Inverting an Image
		I.1.3 Plugin My_Inverter_A (PlugInFilter)
		I.1.4 Plugin My_Inverter_B (PlugIn)
		I.1.5 When To Use PlugIn Or PlugInFilter?
		I.1.6 Executing ImageJ “Commands”
		I.1.7 ImageJ’s Command Recorder
Appendix J Java Notes
	J.1 Arithmetic
		J.1.1 Integer Division
		J.1.2 Modulus Operator
		J.1.3 Mathematical Functions in Class Math
		J.1.4 Numerical Rounding
		J.1.5 Inverse Tangent Function
		J.1.6 Unsigned Byte Data
		J.1.7 Classes Float and Double
		J.1.8 Testing Floating-Point Values Against Zero
	J.2 Arrays
		J.2.1 Creating Arrays
		J.2.2 Array Size
		J.2.3 Accessing Array Elements
		J.2.4 2D Arrays
		J.2.5 Arrays of Objects
		J.2.6 Searching for Minimum and Maximum Values
		J.2.7 Sorting Arrays
References
Index
About the Authors