Mathematics for Digital Science 2

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.

• Preface xiChapter 1 Representation of Numbers 11.1 Representation of numbers 21.1.1 Position numbering 21.1.2 The binary system 31.1.3 Octal system and hexadecimal system 61.2 Representation of numbers in a machine 81.2.1 Machine representation of negative numbers 81.2.2 Representation of rational numbers 12Chapter 2 Media Representation 172.1 Character coding 172.2 Image coding 232.2.1 Representation of digital colors 232.2.2 Scanning an image 242.2.3 Image quality 242.3 Sound coding 262.4 Video coding 282.5 Tagging codes 292.5.1 Figures 292.5.2 Bank cards 302.5.3 Barcodes 322.5.4 QR codes 34Chapter 3 Signals and Systems 473.1 Characteristics and varieties 473.1.1 Introduction 473.1.2 Periodicity 493.1.3 Noise 503.2 Fourier analysis 513.2.1 Fourier series expansion 513.2.2 Examples 523.2.3 Special cases 553.2.4 Other development writing 553.2.5 Power 563.3 Dirac distribution 583.4 Convolution 623.4.1 Definition 623.4.2 Dirac distribution and convolution 64Chapter 4 z-transforms Fourier Transforms and Laplace Transforms 674.1 z-transform 684.1.1 Definitions and main results 684.1.2 Application to discrete systems 704.2 Fourier transform 734.2.1 Periodic signals 734.2.2 Non-periodic signals 744.2.3 Main properties of Fourier transforms 794.2.4 Application to analog signals and systems 824.2.5 Transfer function 864.2.6 Autocorrelation and intercorrelation of signals 884.3 Discrete Fourier transform and fast Fourier transform 914.3.1 Discrete Fourier transform 914.3.2 Fast Fourier transform (FFT) 954.4 Laplace transform 1014.4.1 Definition 1014.4.2 Properties 1024.4.3 Differential equation 1034.4.4 Convolution product 104Chapter 5 Digitizing an Analog Signal 1075.1 Introduction 1075.2 Sampling 1085.3 Quantization 1125.4 Coding 116Chapter 6 Modulation 1196.1 Types of modulation 1196.2 Amplitude modulation 1216.2.1 Principle 1216.2.2 Frequency space 1236.2.3 Signal strength 1246.2.4 Overmodulation 1256.2.5 Demodulation 1266.2.6 Single sideband 1276.2.7 Modulation of a binary signal 1276.3 Frequency modulation 1306.3.1 Principle 1306.3.2 Case of a sinusoidal signal 1326.3.3 Spectrum 1336.3.4 Signal power 1366.3.5 FSK modulation 1366.4 Phase modulation 1396.4.1 Principle 1396.4.2 PSK modulation 141Chapter 7 Filtering 1457.1 Definitions and reminders 1457.1.1 Discrete signals 1467.1.2 Analog signals 1477.2 Analog filtering 1487.2.1 General information 1487.2.2 Common filters 1487.2.3 Differential equations and transfer functions 1527.3 Digital filtering 1607.3.1 General information 1607.3.2 Difference equation 1617.3.3 Transfer function 1637.3.4 Filter stability 1667.3.5 Frequency behavior 1687.3.6 FIR filters 1707.3.7 IIR filters 173Chapter 8 The Digital Image 1778.1 Raster and vector images 1778.1.1 Raster images 1778.1.2 Vector images 1798.2 Notions of colorimetry 1798.2.1 Grayscale 1808.2.2 Colors 1848.2.3 True color and indexed color 1888.4 Image display modes 1908.4.1 Matrix coding 1908.4.2 Vector coding 1928.4.3 Fractal curves 1938.5 Compression and compaction 1938.6 Image formats 1958.6.1 Raster image formats 1958.6.2 Vector image formats 196Chapter 9 2D Computer Graphics 1979.1 Basic graphics processing 1979.1.1 Drawing a segment 1979.1.2 Drawing a circle 2019.1.3 Windowing 2029.1.4 Filling and coloring 2049.2 2D geometric transformations 2059.2.1 Homogeneous coordinates 2059.2.2 Translation 2069.2.3 Rotation around the origin 2079.2.4 Dilation 2089.2.5 Symmetries 2099.2.6 Composition of transformations 2109.2.7 Object representation 2119.3 2D parametric curves 2129.3.1 Using cubic curves 2129.3.2 Hermite curves 2139.3.3 Bézier curves 2149.3.4 B-spline curves 216Chapter 10 Concepts in Image Processing and Analysis 21710.1 Image display 21810.1.1 Simple correspondence 21810.1.2 Random threshold display 21810.1.3 Threshold matrix display 22010.2 Basic image analysis tools 22210.2.1 Histogram 22210.2.2 Profiles 22410.2.3 Level search 22410.2.4 Information contained in an image 22410.3 Basic processing 22610.3.1 Histogram transformation 22610.3.2 Changing the shape of the histogram: equalization 23010.3.3 Image subtraction and averaging 23310.4 Filtering 23310.4.1 Filtering in the spatial domain 23310.4.2 Frequency domain filtering 24110.5 Binary images 24510.5.1 Morphological operators 24610.6 Segmentation 24710.6.1 Outline extraction 24810.6.2 Regional segmentation 252Chapter 11 Basics of Image Compression 25711.1 General information 25811.1.1 Coding redundancy 25811.1.2 Interpixel redundancy 25911.1.3 Psychovisual redundancy 26011.1.4 Confidence criteria 26111.1.5 Modeling image compression 26211.2 Lossless compression or compaction 26311.2.1 Variable-length coding 26311.2.2 Bit-plane coding 26611.2.3 Predictive coding 26811.3 Lossy compression 26911.3.1 Predictive coding 26911.3.2 Transform coding 27011.4 An image compression standard: JPEG 272Chapter 12 Elements of Numerical Analysis 27712.1 Numerical solution of a linear system 27812.1.1 Exact solution of a linear Cramerian system 27812.1.2 Principle of iterative methods 28012.1.3 Diagonal iteration and Gauss–Seidel iteration 28212.1.4 Direct methods 28312.2 Numerical solution of fx = 0 28712.2.1 Introduction 28712.2.2 General methods 28812.2.3 Methods applicable to polynomial equations 29412.3 Numerical integration 29712.3.1 Introduction 29712.3.2 Classic methods 29712.3.3 Polynomial interpolation 30212.3.4 Quadrature formulas 30412.3.5 Monte Carlo method 30712.4 Numerical solution of differential equations 31012.4.1 Introduction 31012.4.2 Separate-step algorithms 31112.4.3 Linked-step methods 31812.5 Numerical solution of partial differential equations 32012.5.1 Definitions 32012.5.2 Finite difference method 32112.5.3 Resolution examples 32512.6 Appendices 33112.6.1 Dichotomy method 33112.6.2 Iterative method 33212.6.3 Secant method 33212.6.4 Tangent method 33312.6.5 Monte Carlo method Example 12.7 33412.6.6 Monte Carlo method Example 12.8 336References 339List of Authors 343Index 345