Over the past century, advancements in computer science have consistently resulted from extensive mathematical work. Even today, innovations in the digital domain continue to be grounded in a strong mathematical foundation. To succeed in this profession, both today’s students and tomorrow’s computer engineers need a solid mathematical background.
The goal of this book series is to offer a solid foundation of the knowledge essential to working in the digital sector. Across three volumes, it explores fundamental principles, digital information, data analysis, and optimization. Whether the reader is pursuing initial training or looking to deepen their expertise, the Mathematics for Digital Science series revisits familiar concepts, helping them refresh and expand their knowledge while also introducing equally essential, newer topics.
Preface xi
Chapter 1 Representation of Numbers 1
1.1 Representation of numbers 2
1.1.1 Position numbering 2
1.1.2 The binary system 3
1.1.3 Octal system and hexadecimal system 6
1.2 Representation of numbers in a machine 8
1.2.1 Machine representation of negative numbers 8
1.2.2 Representation of rational numbers 12
Chapter 2 Media Representation 17
2.1 Character coding 17
2.2 Image coding 23
2.2.1 Representation of digital colors 23
2.2.2 Scanning an image 24
2.2.3 Image quality 24
2.3 Sound coding 26
2.4 Video coding 28
2.5 Tagging codes 29
2.5.1 Figures 29
2.5.2 Bank cards 30
2.5.3 Barcodes 32
2.5.4 QR codes 34
Chapter 3 Signals and Systems 47
3.1 Characteristics and varieties 47
3.1.1 Introduction 47
3.1.2 Periodicity 49
3.1.3 Noise 50
3.2 Fourier analysis 51
3.2.1 Fourier series expansion 51
3.2.2 Examples 52
3.2.3 Special cases 55
3.2.4 Other development writing 55
3.2.5 Power 56
3.3 Dirac distribution 58
3.4 Convolution 62
3.4.1 Definition 62
3.4.2 Dirac distribution and convolution 64
Chapter 4 z-transforms Fourier Transforms and Laplace Transforms 67
4.1 z-transform 68
4.1.1 Definitions and main results 68
4.1.2 Application to discrete systems 70
4.2 Fourier transform 73
4.2.1 Periodic signals 73
4.2.2 Non-periodic signals 74
4.2.3 Main properties of Fourier transforms 79
4.2.4 Application to analog signals and systems 82
4.2.5 Transfer function 86
4.2.6 Autocorrelation and intercorrelation of signals 88
4.3 Discrete Fourier transform and fast Fourier transform 91
4.3.1 Discrete Fourier transform 91
4.3.2 Fast Fourier transform (FFT) 95
4.4 Laplace transform 101
4.4.1 Definition 101
4.4.2 Properties 102
4.4.3 Differential equation 103
4.4.4 Convolution product 104
Chapter 5 Digitizing an Analog Signal 107
5.1 Introduction 107
5.2 Sampling 108
5.3 Quantization 112
5.4 Coding 116
Chapter 6 Modulation 119
6.1 Types of modulation 119
6.2 Amplitude modulation 121
6.2.1 Principle 121
6.2.2 Frequency space 123
6.2.3 Signal strength 124
6.2.4 Overmodulation 125
6.2.5 Demodulation 126
6.2.6 Single sideband 127
6.2.7 Modulation of a binary signal 127
6.3 Frequency modulation 130
6.3.1 Principle 130
6.3.2 Case of a sinusoidal signal 132
6.3.3 Spectrum 133
6.3.4 Signal power 136
6.3.5 FSK modulation 136
6.4 Phase modulation 139
6.4.1 Principle 139
6.4.2 PSK modulation 141
Chapter 7 Filtering 145
7.1 Definitions and reminders 145
7.1.1 Discrete signals 146
7.1.2 Analog signals 147
7.2 Analog filtering 148
7.2.1 General information 148
7.2.2 Common filters 148
7.2.3 Differential equations and transfer functions 152
7.3 Digital filtering 160
7.3.1 General information 160
7.3.2 Difference equation 161
7.3.3 Transfer function 163
7.3.4 Filter stability 166
7.3.5 Frequency behavior 168
7.3.6 FIR filters 170
7.3.7 IIR filters 173
Chapter 8 The Digital Image 177
8.1 Raster and vector images 177
8.1.1 Raster images 177
8.1.2 Vector images 179
8.2 Notions of colorimetry 179
8.2.1 Grayscale 180
8.2.2 Colors 184
8.2.3 True color and indexed color 188
8.4 Image display modes 190
8.4.1 Matrix coding 190
8.4.2 Vector coding 192
8.4.3 Fractal curves 193
8.5 Compression and compaction 193
8.6 Image formats 195
8.6.1 Raster image formats 195
8.6.2 Vector image formats 196
Chapter 9 2D Computer Graphics 197
9.1 Basic graphics processing 197
9.1.1 Drawing a segment 197
9.1.2 Drawing a circle 201
9.1.3 Windowing 202
9.1.4 Filling and coloring 204
9.2 2D geometric transformations 205
9.2.1 Homogeneous coordinates 205
9.2.2 Translation 206
9.2.3 Rotation around the origin 207
9.2.4 Dilation 208
9.2.5 Symmetries 209
9.2.6 Composition of transformations 210
9.2.7 Object representation 211
9.3 2D parametric curves 212
9.3.1 Using cubic curves 212
9.3.2 Hermite curves 213
9.3.3 Bézier curves 214
9.3.4 B-spline curves 216
Chapter 10 Concepts in Image Processing and Analysis 217
10.1 Image display 218
10.1.1 Simple correspondence 218
10.1.2 Random threshold display 218
10.1.3 Threshold matrix display 220
10.2 Basic image analysis tools 222
10.2.1 Histogram 222
10.2.2 Profiles 224
10.2.3 Level search 224
10.2.4 Information contained in an image 224
10.3 Basic processing 226
10.3.1 Histogram transformation 226
10.3.2 Changing the shape of the histogram: equalization 230
10.3.3 Image subtraction and averaging 233
10.4 Filtering 233
10.4.1 Filtering in the spatial domain 233
10.4.2 Frequency domain filtering 241
10.5 Binary images 245
10.5.1 Morphological operators 246
10.6 Segmentation 247
10.6.1 Outline extraction 248
10.6.2 Regional segmentation 252
Chapter 11 Basics of Image Compression 257
11.1 General information 258
11.1.1 Coding redundancy 258
11.1.2 Interpixel redundancy 259
11.1.3 Psychovisual redundancy 260
11.1.4 Confidence criteria 261
11.1.5 Modeling image compression 262
11.2 Lossless compression or compaction 263
11.2.1 Variable-length coding 263
11.2.2 Bit-plane coding 266
11.2.3 Predictive coding 268
11.3 Lossy compression 269
11.3.1 Predictive coding 269
11.3.2 Transform coding 270
11.4 An image compression standard: JPEG 272
Chapter 12 Elements of Numerical Analysis 277
12.1 Numerical solution of a linear system 278
12.1.1 Exact solution of a linear Cramerian system 278
12.1.2 Principle of iterative methods 280
12.1.3 Diagonal iteration and GaussSeidel iteration 282
12.1.4 Direct methods 283
12.2 Numerical solution of fx = 0 287
12.2.1 Introduction 287
12.2.2 General methods 288
12.2.3 Methods applicable to polynomial equations 294
12.3 Numerical integration 297
12.3.1 Introduction 297
12.3.2 Classic methods 297
12.3.3 Polynomial interpolation 302
12.3.4 Quadrature formulas 304
12.3.5 Monte Carlo method 307
12.4 Numerical solution of differential equations 310
12.4.1 Introduction 310
12.4.2 Separate-step algorithms 311
12.4.3 Linked-step methods 318
12.5 Numerical solution of partial differential equations 320
12.5.1 Definitions 320
12.5.2 Finite difference method 321
12.5.3 Resolution examples 325
12.6 Appendices 331
12.6.1 Dichotomy method 331
12.6.2 Iterative method 332
12.6.3 Secant method 332
12.6.4 Tangent method 333
12.6.5 Monte Carlo method Example 12.7 334
12.6.6 Monte Carlo method Example 12.8 336
References 339
List of Authors 343
Index 345
Gérard-Michel Cochard is Professor Emeritus at Université de Picardie Jules Verne, France, where he has held various senior positions. He has also served at the French Ministry of Education and the CNAM (Conservatoire National des Arts et Métiers). His research is conducted at the Eco-PRocédés, Optimisation et Aide à la Décision (EPROAD) laboratory, France.
Mhand Hifi is Professor of Computer Science at Université de Picardie Jules Verne, France, where he heads the EPROAD UR 4669 laboratory and manages the ROD team. As an expert in operations research and NP-hard problem-solving, he actively contributes to numerous international conferences and journals in the field.
