Preface 15
Acknowledgments 19
The Book Web Site 20
About the Authors 21

Chapter 1 Introduction 23
1.1 What Is Digital Image Processing? 23
1.2 The Origins of Digital Image Processing 25
1.3 Examples of Fields that Use Digital Image Processing 29
1.3.1 Gamma-Ray Imaging 30
1.3.2 X-Ray Imaging 31
1.3.3 Imaging in the Ultraviolet Band 33
1.3.4 Imaging in the Visible and Infrared Bands 34
1.3.5 Imaging in the Microwave Band 40
1.3.6 Imaging in the Radio Band 42
1.3.7 Examples in which Other Imaging Modalities Are Used 42
1.4 Fundamental Steps in Digital Image Processing 47
1.5 Components of an Image Processing System 50
Summary 53
References and Further Reading 53

Chapter 2 Digital Image Fundamentals 57
2.1 Elements of Visual Perception 58
2.1.1 Structure of the Human Eye 58
2.1.2 Image Formation in the Eye 60
2.1.3 Brightness Adaptation and Discrimination 61
2.2 Light and the Electromagnetic Spectrum 65
2.3 Image Sensing and Acquisition 68
2.3.1 Image Acquisition Using a Single Sensor 70
2.3.2 Image Acquisition Using Sensor Strips 70
2.3.3 Image Acquisition Using Sensor Arrays 72
2.3.4 A Simple Image Formation Model 72
2.4 Image Sampling and Quantization 74
2.4.1 Basic Concepts in Sampling and Quantization 74
2.4.2 Representing Digital Images 77
2.4.3 Spatial and Intensity Resolution 81
2.4.4 Image Interpolation 87
2.5 Some Basic Relationships between Pixels 90
2.5.1 Neighbors of a Pixel 90
2.5.2 Adjacency, Connectivity, Regions, and Boundaries 90
2.5.3 Distance Measures 93
2.6 An Introduction to the Mathematical Tools Used in Digital Image Processing 94
2.6.1 Array versus Matrix Operations 94
2.6.2 Linear versus Nonlinear Operations 95
2.6.3 Arithmetic Operations 96
2.6.4 Set and Logical Operations 102
2.6.5 Spatial Operations 107
2.6.6 Vector and Matrix Operations 114
2.6.7 Image Transforms 115
2.6.8 Probabilistic Methods 118
Summary 120
References and Further Reading 120
Problems 121

Chapter 3 Intensity Transformations and Spatial Filtering 126
3.1 Background 127
3.1.1 The Basics of Intensity Transformations and Spatial Filtering 127
3.1.2 About the Examples in This Chapter 129
3.2 Some Basic Intensity Transformation Functions 129
3.2.1 Image Negatives 130
3.2.2 Log Transformations 131
3.2.3 Power-Law (Gamma) Transformations 132
3.2.4 Piecewise-Linear Transformation Functions 137
3.3 Histogram Processing 142
3.3.1 Histogram Equalization 144
3.3.2 Histogram Matching (Specification) 150
3.3.3 Local Histogram Processing 161
3.3.4 Using Histogram Statistics for Image Enhancement 161
3.4 Fundamentals of Spatial Filtering 166
3.4.1 The Mechanics of Spatial Filtering 167
3.4.2 Spatial Correlation and Convolution 168
3.4.3 Vector Representation of Linear Filtering 172
3.4.4 Generating Spatial Filter Masks 173
3.5 Smoothing Spatial Filters 174
3.5.1 Smoothing Linear Filters 174
3.5.2 Order-Statistic (Nonlinear) Filters 178
3.6 Sharpening Spatial Filters 179
3.6.1 Foundation 180
3.6.2 Using the Second Derivative for Image Sharpening-The Laplacian 182
3.6.3 Unsharp Masking and Highboost Filtering 184
3.6.4 Using First-Order Derivatives for (Nonlinear) Image Sharpening—The Gradient 187
3.7 Combining Spatial Enhancement Methods 191
3.8 Using Fuzzy Techniques for Intensity Transformations and Spatial Filtering 195
3.8.1 Introduction 195
3.8.2 Principles of Fuzzy Set Theory 196
3.8.3 Using Fuzzy Sets 200
3.8.4 Using Fuzzy Sets for Intensity Transformations 208
3.8.5 Using Fuzzy Sets for Spatial Filtering 211
Summary 214
References and Further Reading 214
Problems 215

Chapter 4 Filtering in the Frequency Domain 221
4.1 Background 222
4.1.1 A Brief History of the Fourier Series and Transform 222
4.1.2 About the Examples in this Chapter 223
4.2 Preliminary Concepts 224
4.2.1 Complex Numbers 224
4.2.2 Fourier Series 225
4.2.3 Impulses and Their Sifting Property 225
4.2.4 The Fourier Transform of Functions of One Continuous Variable 227
4.2.5 Convolution 231
4.3 Sampling and the Fourier Transform of Sampled Functions 233
4.3.1 Sampling 233
4.3.2 The Fourier Transform of Sampled Functions 234
4.3.3 The Sampling Theorem 235
4.3.4 Aliasing 239
4.3.5 Function Reconstruction (Recovery) from Sampled Data 241
4.4 The Discrete Fourier Transform (DFT) of One Variable 242
4.4.1 Obtaining the DFT from the Continuous Transform of a Sampled Function 243
4.4.2 Relationship Between the Sampling and Frequency Intervals 245
4.5 Extension to Functions of Two Variables 247
4.5.1 The 2-D Impulse and Its Sifting Property 247
4.5.2 The 2-D Continuous Fourier Transform Pair 248
4.5.3 Two-Dimensional Sampling and the 2-D Sampling Theorem 249
4.5.4 Aliasing in Images 250
4.5.5 The 2-D Discrete Fourier Transform and Its Inverse 257
4.6 Some Properties of the 2-D Discrete Fourier Transform 258
4.6.1 Relationships Between Spatial and Frequency Intervals 258
4.6.2 Translation and Rotation 258
4.6.3 Periodicity 259
4.6.4 Symmetry Properties 261
4.6.5 Fourier Spectrum and Phase Angle 267
4.6.6 The 2-D Convolution Theorem 271
4.6.7 Summary of 2-D Discrete Fourier Transform Properties 275
4.7 The Basics of Filtering in the Frequency Domain 277
4.7.1 Additional Characteristics of the Frequency Domain 277
4.7.2 Frequency Domain Filtering Fundamentals 279
4.7.3 Summary of Steps for Filtering in the Frequency Domain 285
4.7.4 Correspondence Between Filtering in the Spatial and Frequency Domains 285
4.8 Image Smoothing Using Frequency Domain Filters 291
4.8.1 Ideal Lowpass Filters 291
4.8.2 Butterworth Lowpass Filters 295
4.8.3 Gaussian Lowpass Filters 298
4.8.4 Additional Examples of Lowpass Filtering 299
4.9 Image Sharpening Using Frequency Domain Filters 302
4.9.1 Ideal Highpass Filters 303
4.9.2 Butterworth Highpass Filters 306
4.9.3 Gaussian Highpass Filters 307
4.9.4 The Laplacian in the Frequency Domain 308
4.9.5 Unsharp Masking, Highboost Filtering, and High-Frequency-Emphasis Filtering 310
4.9.6 Homomorphic Filtering 311
4.10 Selective Filtering 316
4.10.1 Bandreject and Bandpass Filters 316
4.10.2 Notch Filters 316
4.11 Implementation 320
4.11.1 Separability of the 2-D DFT 320
4.11.2 Computing the IDFT Using a DFT Algorithm 321
4.11.3 The Fast Fourier Transform (FFT) 321
4.11.4 Some Comments on Filter Design 325
Summary 325
References and Further Reading 326
Problems 326

Chapter 5 Image Restoration and Reconstruction 333
5.1 A Model of the Image Degradation/Restoration Process 334
5.2 Noise Models 335
5.2.1 Spatial and Frequency Properties of Noise 335
5.2.2 Some Important Noise Probability Density Functions 336
5.2.3 Periodic Noise 340
5.2.4 Estimation of Noise Parameters 341
5.3 Restoration in the Presence of Noise Only—Spatial Filtering 344
5.3.1 Mean Filters 344
5.3.2 Order-Statistic Filters 347
5.3.3 Adaptive Filters 352
5.4 Periodic Noise Reduction by Frequency Domain Filtering 357
5.4.1 Bandreject Filters 357
5.4.2 Bandpass Filters 358
5.4.3 Notch Filters 359
5.4.4 Optimum Notch Filtering 360
5.5 Linear, Position-Invariant Degradations 365
5.6 Estimating the Degradation Function 368
5.6.1 Estimation by Image Observation 368
5.6.2 Estimation by Experimentation 369
5.6.3 Estimation by Modeling 369
5.7 Inverse Filtering 373
5.8 Minimum Mean Square Error (Wiener) Filtering 374
5.9 Constrained Least Squares Filtering 379
5.10 Geometric Mean Filter 383
5.11 Image Reconstruction from Projections 384
5.11.1 Introduction 384
5.11.2 Principles of Computed Tomography (CT) 387
5.11.3 Projections and the Radon Transform 390
5.11.4 The Fourier-Slice Theorem