# Guide to Signals and Patterns in Image Processing: Foundations, Methods and Applications PDF by Apurba Das

### Guide to Signals and Patterns in Image Processing: Foundations, Methods and Applications

By Apurba Das

Contents

1 Introduction to Digital Image …………………………………………………………. 1

1.1 Formation of an Image …………………………………………………………….. 1

1.2 Definition of Signal …………………………………………………………………. 3

1.3 Analog and Digital Image as 2D Signal ……………………………………… 6

1.3.1 Continuous Time Continuous Valued Electrical Signal ……… 7

1.3.2 Continuous Space Continuous Intensity (CSCI) Image ……… 7

1.3.3 Sampling: Discrete Time Continuous Valued (DTCV)

Electrical Signal …………………………………………………………… 8

1.3.4 Concept of Sampling in Images (2D Signal) ……………………. 9

1.3.5 Quantization: Discrete Time Discrete Valued (DTDV)

Electrical Signal …………………………………………………………… 10

1.3.6 Quantization in Images (2D Signal) ……………………………….. 12

1.3.7 Encoding in Images (2D Signal) …………………………………….. 14

1.4 Relationships Between Pixels …………………………………………………… 14

1.4.1 Neighborhood ……………………………………………………………… 14

1.4.3 Distance Measures ……………………………………………………….. 16

1.5 Geometric Transformations ……………………………………………………… 18

1.5.1 Translation …………………………………………………………………… 18

1.5.2 Scaling ………………………………………………………………………… 21

1.5.3 Rotation ………………………………………………………………………. 22

1.5.4 Homogeneous Transformation ……………………………………….. 25

1.5.5 Concatenation of Transformation …………………………………… 27

1.5.6 Affine Transformation …………………………………………………… 30

1.6 Convolution ……………………………………………………………………………. 31

1.6.1 Transformed Domain Simplicity …………………………………….. 33

1.6.2 2D Convolution: Convolution in Image Processing ………….. 35

1.7 Correlation …………………………………………………………………………….. 36

1.7.1 Case Study: Pattern (Shape Feature) Matching Between

Two Objects Using Cross-Correlation …………………………….. 37

1.8 MATLAB Codes …………………………………………………………………….. 40

1.8.1 Sampling of a Sweep Image ………………………………………….. 40

1.8.2 Resolution of Image ……………………………………………………… 40

1.8.3 Quantization of Image …………………………………………………… 41

1.8.4 Correlation Subroutine ………………………………………………….. 42

Reference ………………………………………………………………………………………. 42

2 Image Enhancement in Spatial Domain ………………………………………….. 43

2.1 Intensity Transformations ………………………………………………………… 44

2.1.1 Linear Transformation ………………………………………………….. 44

2.1.2 Contrast Stretching and Thresholding ……………………………… 46

2.1.3 Negative Intensity Transform ………………………………………… 46

2.1.4 Logarithmic Intensity Transformation …………………………….. 46

2.1.5 Power-Law Intensity Transform and Gamma Correction …… 47

2.2 Histogram of an Image …………………………………………………………….. 49

2.2.1 Skewness …………………………………………………………………….. 51

2.2.2 Kurtosis ………………………………………………………………………. 51

2.3 Histogram Equalization and Histogram Specification ………………….. 52

2.4 Image Smoothing ……………………………………………………………………. 58

2.4.1 Mean Filter ………………………………………………………………….. 58

2.4.2 Ordered Statistics Filter ………………………………………………… 61

2.5 Image Sharpening …………………………………………………………………… 63

Derivative ……………………………………………………………………. 65

2.5.2 Image Sharpening by Laplacian mask: Second-Order

Derivative ……………………………………………………………………. 67

2.6 Image Interpolation and Resampling …………………………………………. 68

2.6.1 B-Spline Function ………………………………………………………… 72

2.6.2 Interpolation of 1D Signal by B-Spline …………………………… 74

2.6.3 Interpolation of 2D Image ……………………………………………… 80

2.7 MATLAB codes ……………………………………………………………………… 88

2.7.1 Image Transformation Without Interpolation …………………… 88

2.7.2 Image Rotation with Different Interpolation Techniques …… 89

2.7.3 Mean and Median Filter Response on Noisy Image ………….. 90

2.7.4 Image Sharpening by Laplacian Mask …………………………….. 91

Reference ………………………………………………………………………………………. 91

3 Interpretation and Processing of Image in Frequency Domain …………. 93

3.1 Concept of Frequency in Image ………………………………………………… 94

3.1.1 Fourier Series ………………………………………………………………. 94

3.1.2 Interpretation and Direction of Frequency in Image ………….. 99

3.2 Phase Congruency and Edge Detection in an Image ……………………. 100

3.3 Fourier Transform for Continuous and Discrete Time Signals ………. 106

3.3.1 Discrete Time Fourier Transform ……………………………………. 107

3.3.2 DFT and FFT ………………………………………………………………. 109

3.4 DFT of Digital Image ………………………………………………………………. 110

3.5 Translation and Scaling Properties of 2D Fourier Transform ………… 112

3.5.1 Translation: Dragging the LF (DC) Component at the

Center of the 2D Spectra ……………………………………………….. 112

3.5.2 Scaling: Space-Frequency Relationship in Image …………….. 114

3.6 Concept of Image Filtering in Frequency Domain ………………………. 120

3.7 Smoothing Filter …………………………………………………………………….. 124

3.7.1 Ideal LPF …………………………………………………………………….. 125

3.7.2 Butterworth LPF ………………………………………………………….. 126

3.7.3 Gaussian LPF ………………………………………………………………. 129

3.8 Sharpening Filter …………………………………………………………………….. 130

3.8.1 Ideal HPF ……………………………………………………………………. 132

3.8.2 Butterworth HPF ………………………………………………………….. 133

3.8.3 Gaussian HPF ………………………………………………………………. 135

3.9 Case Studies …………………………………………………………………………… 136

3.9.1 Importance of Phase over Amplitude in DFT Spectrum …….. 136

3.9.2 DFT over DFT …………………………………………………………….. 138

3.10 Matlab Codes ……………………………………………………………………….. 143

3.10.1 Ideal 2D Filters in Frequency Domain …………………………… 143

3.10.2 Subroutine of 2D Butterworth Filter ……………………………… 145

3.10.3 Subroutine of 2D Gaussian Filter …………………………………. 145

3.10.4 Importance of Phase over Amplitude in Image Spectrum … 146

References ……………………………………………………………………………………… 147

4 Color Science and Color Technology ……………………………………………….. 149

4.1 Light and Primary Colors …………………………………………………………. 149

4.1.1 Device-Dependent Primary Colors: Additive Color Model … 151

4.1.2 Device-Dependent Primary Colors: Subtractive Color

Model …………………………………………………………………………. 152

4.1.3 Reflectance and Its Spectra ……………………………………………. 154

4.2 Psycho-Visual Color: Human Vision System ……………………………… 158

4.2.1 Photoreceptors: Rods and Cones ……………………………………. 158

4.3 Color Description Systems ………………………………………………………. 161

4.3.1 Munsell System ……………………………………………………………. 162

4.3.2 Pantone System ……………………………………………………………. 163

4.4 Colorimetry: CIE Standards ……………………………………………………… 166

4.4.1 CIE Standard Illuminant ……………………………………………….. 167

4.4.2 CIE Standard Observer …………………………………………………. 169

4.5 CIE Color Spaces ……………………………………………………………………. 170

4.5.1 Non-uniform Perceptual Color Spaces ……………………………. 171

4.5.2 Uniform Perceptual Color Spaces …………………………………… 174

4.5.3 Xerox/YES Color Space ……………………………………………….. 176

4.6 Halftone Screening ………………………………………………………………….. 177

4.6.1 Moire Pattern and Screen Angle …………………………………….. 178

4.6.2 Growth Sequence of Halftone Dot ………………………………….. 179

4.7 Color Management ………………………………………………………………….. 181

4.7.1 Profile Connection Space (PCS) …………………………………….. 184

4.7.2 Gamut Mapping …………………………………………………………… 184

4.7.3 Rendering Intents …………………………………………………………. 185

4.8 Matlab Codes …………………………………………………………………………. 189

4.8.1 Halftone Screening by Error Diffusion ……………………………. 189

4.8.2 Error Diffusion Subroutine ……………………………………………. 190

Reference ………………………………………………………………………………………. 190

5 Wavelets: Multiresolution Image Processing ……………………………………. 191

5.1 Introduction ……………………………………………………………………………. 191

5.2 Short-Time Fourier Transform ………………………………………………….. 191

5.2.1 Continuous-time STFT …………………………………………………. 194

5.2.2 Discrete-time STFT ………………………………………………………. 194

5.2.3 Spectrogram ………………………………………………………………… 194

5.2.4 Limitation ……………………………………………………………………. 195

5.3 Wavelet Function and Scaling Function …………………………………….. 196

5.4 Wavelet Series ………………………………………………………………………… 202

5.5 Discrete Wavelet Transform and Multiresolution analysis ……………. 203

5.5.1 Analysis Filter Bank ……………………………………………………… 205

5.5.2 Synthesis Filter Bank ……………………………………………………. 206

5.6 Image Decomposition Using DWT ……………………………………………. 207

5.6.1 Concept of 2D Signal Decomposition Using Analysis

Filter …………………………………………………………………………… 207

5.6.2 DWT on Images (Fig. 5.16) …………………………………………… 208

5.7 Image Compression Using DWT: EZW Encoding ………………………. 210

5.7.1 Relationship Between Decomposed Sub-bands ………………… 211

5.7.2 Successive Approximation Quantization in EZW …………….. 212

5.7.3 EZW Encoding Algorithm …………………………………………….. 213

5.7.4 Image Compression using EZW: An Example …………………. 214

5.7.5 Experimental Results of Image Compression Using EZW …. 216

5.8 MATLAB Programs ………………………………………………………………… 218

5.8.1 Haar Scaling and Wavelet Function ………………………………… 218

5.8.2 Wavelet Series Expansion ……………………………………………… 219

5.8.3 Wavelet Decomposition of Image (4 level) ……………………… 220

5.8.4 Image Compression by EZW Encoding …………………………… 221

References ……………………………………………………………………………………… 221

6 Compression and Encoding of Image: Image Formats …………………….. 223

6.1 Redundancy: Fundamentals of Compression ………………………………. 224

6.2 Entropy: The Measure of Information ……………………………………….. 226

6.3 Entropy Coding ………………………………………………………………………. 227

6.3.1 Shannon–Fano Coding ………………………………………………….. 228

6.3.2 Huffman Coding ………………………………………………………….. 229

6.4 Lossy Compression …………………………………………………………………. 230

6.4.1 Block Truncation Compression (BTC) ……………………………. 230

6.4.2 Vector Quantization Compression (VQC) ……………………….. 233

6.5 Lossless Compression ……………………………………………………………… 236

6.5.1 Run Length Coding (RLC) ……………………………………………. 236

6.5.2 Block Coding ………………………………………………………………. 237

6.6 QPAC: Quality Preserving Adaptive Compression ……………………… 238

6.7 Some Common Image Formats ………………………………………………… 239

6.7.1 C++ Code for Reading BMP Image ……………………………….. 240

6.7.2 JPEG ………………………………………………………………………….. 245

6.7.3 GIF …………………………………………………………………………….. 254

6.8 Matlab Codes and Pseudocodes ………………………………………………… 258

6.8.1 Block Truncation Compression (BTC) ……………………………. 258

6.8.2 JPEG Compression ………………………………………………………. 261

6.8.3 GIF: LZW Compression ……………………………………………….. 266

6.8.4 GIF: LZW Decompression ……………………………………………. 267

References ……………………………………………………………………………………… 267

7 Morphology-Based Image Processing ……………………………………………… 269

7.1 Basics of Set Theory ……………………………………………………………….. 269

7.2 Logic Operations on Binary Images ………………………………………….. 271

7.3 Dilation and Erosion ……………………………………………………………….. 273

7.3.1 Dilation ………………………………………………………………………. 273

7.3.2 Erosion ……………………………………………………………………….. 276

7.4 Opening and Closing ……………………………………………………………….. 279

7.5 Hit–Miss Transform ………………………………………………………………… 280

7.6 Morphological Algorithms for Feature Extraction ………………………. 280

7.6.1 Boundary Extraction …………………………………………………….. 282

7.6.2 Region Filling ……………………………………………………………… 283

7.6.3 Pixel Connectivity ………………………………………………………… 284

7.6.4 Convex Hull ………………………………………………………………… 285

7.6.5 Thinning ……………………………………………………………………… 286

7.6.6 Thickening ………………………………………………………………….. 288

7.6.7 Object Skeletons ………………………………………………………….. 289

7.6.8 Pruning ……………………………………………………………………….. 290

7.7 Case Studies …………………………………………………………………………… 293

7.7.1 Boundary Detection ……………………………………………………… 293

7.7.2 Region Filling ……………………………………………………………… 295

7.7.3 Binary Skeleton ……………………………………………………………. 295

7.8 MATLAB Codes …………………………………………………………………….. 296

7.8.1 Dilation ………………………………………………………………………. 296

7.8.2 Erosion ……………………………………………………………………….. 297

7.8.3 Boundary Detection ……………………………………………………… 297

Reference ………………………………………………………………………………………. 298

8 Patterns in Images and Their Applications ……………………………………… 299

8.1 Introduction to Pattern …………………………………………………………….. 299

8.2 Features …………………………………………………………………………………. 300

8.2.1 Feature Selection and Extraction ……………………………………. 301

8.3 Principal Component Analysis ………………………………………………….. 302

8.3.1 Algorithm of PCA ………………………………………………………… 303

8.3.2 Application of PCA in Face Recognition …………………………. 305

8.3.3 Limitations of PCA-Based Face Recognition …………………… 307

8.4 Face Detection Based on Haar-Like Features ……………………………… 308

8.5 Elastic Branch Graph Matching and Face Manifold …………………….. 311

8.6 Decision Tree and Feature Hierarchy ………………………………………… 314

8.6.1 Information Gain ………………………………………………………….. 314

8.6.2 Information Gain Ratio …………………………………………………. 315

8.6.3 Selection of Optimized Set of Features …………………………… 316

8.6.4 Feature Hierarchy for Gabor Features in Face

Recognition …………………………………………………………………. 317

8.7 Scale Invariant Feature Transform …………………………………………….. 319

8.7.1 Scale–Space Concept: Multiscale Singularity Tree …………… 320

8.7.2 SIFT: Representation of Image in Scale–Space ………………… 322

8.7.3 SIFT: Detection of Local Scale–Space Extrema ……………….. 324

8.7.4 SIFT: Accurate Keypoint Localization ……………………………. 325

8.7.5 SIFT: Orientation Assignment ……………………………………….. 326

8.7.6 SIFT: Keypoint Descriptor …………………………………………….. 327

8.7.7 SIFT: Results ……………………………………………………………….. 328

8.8 Histogram of Oriented Gradient ……………………………………………….. 332

8.8.1 HOG: Dividing Image into Blocks …………………………………. 333

8.8.2 HOG: Quantization of Gradient Histogram ……………………… 333

8.8.3 HOG: Feature Vector Synthesis ……………………………………… 335

8.8.4 HOG: Design of Classifier by Training …………………………… 335

8.9 Matlab Codes …………………………………………………………………………. 336

8.9.1 PCA of a 2D data set …………………………………………………….. 336

8.9.2 Scale–Space: Multiscale Singularity Tree ……………………….. 337

Reference ………………………………………………………………………………………. 338

9 Psycho-visual pattern recognition: Computer Vision ………………………. 341

9.1 Introduction ……………………………………………………………………………. 341

9.2 Receptive Field ………………………………………………………………………. 342

9.2.1 On-Center Off-Surround ……………………………………………….. 343

9.2.2 Off-Center On-Surround ……………………………………………….. 344

9.2.3 Edge Detection in Retinal Receptive Field ………………………. 345

9.3 Modeling of Retinal Receptive Field from Optical Illusions …………. 347

9.3.1 Optical Illusions: A study ………………………………………………. 347

9.3.2 Illustration of the Illusions in Terms of DoG Model of

Retinal Receptive Field …………………………………………………. 348

9.4 Three Levels of Psycho-Visual System for Pattern Recognition ……. 352

9.5 Neuro-Visually Inspired Figure-Ground Segregation …………………… 353

9.5.1 The Detailed Algorithm for NFGS ………………………………….. 355

9.6 “Where” and “What” Visual Pathways: Modeling in Computer

Vision ……………………………………………………………………………………. 358

References ……………………………………………………………………………………… 362

10 Appendix A: Digital Differentiation and Edge Detection ………………….. 365

10.1 Edge in an Image ……………………………………………………………………. 365

10.2 Digital Differentiation ……………………………………………………………… 366

10.2.1 Digital Differentiation of One-Dimensional (1D) Signal …… 367

10.3 Digital Differentiation for Edge Detection …………………………………. 369

10.4 Convolution and Correlation for Edge Detection ………………………… 371

10.5 Prewitt and Sobel Mask for Edge Detection of Digital Image ………. 374

10.6 Canny Edge Detector ………………………………………………………………. 375

10.6.1 Noise Reduction …………………………………………………………… 375

10.6.2 Non-Maxima Suppression …………………………………………….. 376

10.6.3 Hysteresis Thresholding ………………………………………………… 376

10.7 MATLAB Codes …………………………………………………………………….. 378

10.7.1 Digital Differentiation of 1D Signal ……………………………….. 378

10.7.2 Detection of Edges in Orthogonal Directions by Convolu-tion Interpretation of Digital Differentiation ……………………. 379

References ……………………………………………………………………………………… 381

11 Appendix B: Elementary Probability Theory …………………………………… 383

11.1 Concept of Probability …………………………………………………………….. 385

11.1.1 Random Experiments and Sample Space …………………………. 385

11.1.2 Events …………………………………………………………………………. 385

11.1.3 Probability: Understanding Approaches ………………………….. 386

11.2 Random Variable …………………………………………………………………….. 386

11.3 Mean, Variance, Skewness, and Kurtosis ……………………………………. 387

11.4 Cumulative Distribution Function …………………………………………….. 389

11.5 Probability Density Function ……………………………………………………. 392

11.5.1 Uniform PDF ……………………………………………………………….. 393

11.6 Frequently Used Probability Distribution …………………………………… 393

References ……………………………………………………………………………………… 397

12 Appendix C: Frequently Used MATLAB Functions ………………………… 399

12.1 plot() ……………………………………………………………………………………… 399

12.1.1 Syntax ………………………………………………………………………… 399

12.1.2 Description ………………………………………………………………….. 399

12.2 imshow() ……………………………………………………………………………….. 400

12.2.1 Syntax ………………………………………………………………………… 400

12.2.2 Description ………………………………………………………………….. 400

12.3 drawnow() ……………………………………………………………………………… 401

12.3.1 Syntax ………………………………………………………………………… 401

12.3.2 Description ………………………………………………………………….. 401

12.4 stairs() …………………………………………………………………………………… 401

12.5 int2str() ………………………………………………………………………………….. 402

12.5.1 Syntax ………………………………………………………………………… 402

12.5.2 Description ………………………………………………………………….. 402

12.6 conv() ……………………………………………………………………………………. 402

12.7 conv2() ………………………………………………………………………………….. 403

12.8 Two-Dimensional Convolution …………………………………………………. 403

12.9 ginput() ………………………………………………………………………………….. 404

12.10 bitget() …………………………………………………………………………………. 405

12.11 bitset() …………………………………………………………………………………. 405

12.12 dec2bin() ……………………………………………………………………………… 406

12.12.1 Syntax ………………………………………………………………………. 406

12.12.2 Description ………………………………………………………………… 406

12.13 fft2() ……………………………………………………………………………………. 407

12.13.1 Syntax ………………………………………………………………………. 407

12.13.2 Description ………………………………………………………………… 407

12.14 fftshift() ……………………………………………………………………………….. 407

12.14.1 Syntax ………………………………………………………………………. 407

12.14.2 Description ………………………………………………………………… 408

12.15 wavefun() …………………………………………………………………………….. 409

12.15.1 Syntax ………………………………………………………………………. 409

12.15.2 Description ………………………………………………………………… 410

12.16 Fourier Synthesizer GUI ………………………………………………………… 412

Reference ………………………………………………………………………………………. 412

Index …………………………………………………………………………………………………… 413

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