Quantum Image Processing in Practice explores the transformative potential of quantum color image processing across various domains, including biomedicine, entertainment, economics, and industry. The rapid growth of image data, especially in facial recognition and autonomous vehicles, demands more efficient processing techniques. Quantum computing promises to accelerate digital image processing (DIP) to meet this demand.
This book covers the role of quantum image processing (QIP) in quantum information processing, including mathematical foundations, quantum operations, image processing using quantum filters, quantum image representation, and quantum neural networks. It aims to inspire practical applications and foster innovation in this promising field.
Topics include: - Qubits and Quantum Logic Gates: Introduces qubits, the fundamental data unit in quantum computing, and their manipulation using quantum logic gates like Pauli matrices, rotations, the CNOT gate, and Hadamard matrices. The concept of entanglement, where qubits become interconnected, is also explored, highlighting its importance for applications like quantum teleportation and cryptography. - Two and Multiple Qubit Systems: Demonstrates the importance of using two qubits to process color images, enabling image enhancement, noise reduction, edge detection, and feature extraction. Covers the tensor product, Kronecker sum, SWAP gate, and local and controlled gates. Extends to multi-qubit superpositions, exploring local and control gates for three qubits, such as the Toffoli and Fredkin gates, and describes the measurement of superpositions using projection operators. - Transforms and Quantum Image Representations: Covers the Hadamard, Fourier, and Heap transforms and their circuits in quantum computation, highlighting their applications in signal and image processing. Introduces the quantum signal-induced heap transform for image enhancement, classification, compression, and filtration. Explores quantum representations and operations for images using the RGB, XYZ, CMY, HSI, and HSV color models, providing numerous examples. - Fourier Transform Qubit Representation: Introduces a new model of quantum image representation, the Fourier transform qubit representation. Describes the algorithm and circuit for calculating the 2-D quantum Fourier transform, enabling advancements in quantum imaging techniques. - New Operations and Hypercomplex Algebra: Presents new operations on qubits and quantum representations, including multiplication, division, and inverse operations. Explores hypercomplex algebra, specifically quaternion algebra, for its potential in color image processing. - Quantum Neural Networks (QNNs): Discusses QNNs and their circuit implementation as advancements in machine learning driven by quantum mechanics. Summarizes various applications of QNNs and current trends and future developments in this rapidly evolving field.
The book also addresses challenges and opportunities in QIP research, aiming to inspire practical applications and innovation. It is a valuable resource for researchers, students, and professionals interested in the intersection of quantum computing and color image processing applications, as well as those in visual communications, multimedia systems, computer vision, entertainment, and biomedical applications.
Table of Contents
Preface xiii
Acknowledgments xvii
About the Companion Website xix
Part I Mathematical Foundation of Quantum Computation 1
1 Introduction 3
References 4
2 Basic Concepts of Qubits 5
2.1 Measurement of the Qubit 7
2.1.1 Operations on Qubits 10
2.1.2 Elementary Gates 10
References 14
3 Understanding of Two Qubit Systems 15
3.1 Measurement of 2-Qubits 16
3.1.1 Projection Operators 17
3.2 Operation of Kronecker Product 20
3.2.1 Tensor Product of Single Qubits 21
3.3 Operation of Kronecker Sum 22
3.3.1 Properties on Matrices 23
3.3.2 Orthogonality of Matrices 23
3.4 Permutations 24
3.4.1 Elementary Operations on 2-Qubits 25
References 36
4 Multi-qubit Superpositions and Operations 37
4.1 Elementary Operations on Multi-qubits 38
4.2 3-Qubit Operations with Local Gates 38
4.3 3-Qubit Operations with Control Bits 41
4.4 3-Qubit Operations with 2 Control Bits 43
4.5 Known 3-Qubit Gates 49
4.6 Projection Operators 51
References 52
5 Fast Transforms in Quantum Computation 53
5.1 Fast Discrete Paired Transform 53
5.2 The Quantum Circuits for the Paired Transform 57
5.3 The Inverse DPT 58
5.3.1 The First Circuit for the Inverse QPT 59
5.4 Fast Discrete Hadamard Transform 60
5.5 Quantum Fourier Transform 65
5.5.1 The Paired DFT 65
5.5.2 Algorithm of the 4-Qubit QFT 75
5.5.3 The Known Algorithm of the QFT 77
5.6 Method of 1D Quantum Convolution for Phase Filters 81
References 85
6 Quantum Signal-Induced Heap Transform 87
6.1 Definition 87
6.1.1 The Algorithm of the Strong DsiHT 89
6.1.2 Initialization of the Quantum State by the DsiHT 94
6.2 DsiHT-Based Factorization of Real Matrices 97
6.2.1 Quantum Circuits for DCT-II 98
6.2.2 Quantum Circuits for the DCT-IV 105
6.2.3 Quantum Circuits for the Discrete Hartley Transform 107
6.3 Complex DsiHT 110
References 111
Part II Applications in Image Processing 113
7 Quantum Image Representation with Examples 115
7.1 Models of Representation of Grayscale Images 116
7.1.1 Quantum Pixel Model (QPM) 116
7.1.2 Qubit Lattice Model (QLM) 122
7.1.3 Flexible Representation for Quantum Images 123
7.1.4 Representation of Amplitudes 125
7.1.5 Gradient and Sum Operators 128
7.1.6 Real Ket Model 130
7.1.7 General and Novel Enhanced Quantum Representations (GQIR and NEQR) 131
7.2 Color Image Quantum Representations 135
7.2.1 Quantum Color Pixel in the RGB Model 135
7.2.1.1 3-Color Quantum Qubit Model 136
7.2.2 NASS Representation 137
7.2.3 NASSTC Model 137
7.2.4 Novel Quantum Representation of Color Images (NCQI) 137
7.2.5 Multi-channel Representation of Images (MCRI) 139
7.2.6 Quantum Image Representation in HSI Model (QIRHSI) 141
7.2.7 Transformation 2 × 2 Model for Color Images 142
References 145
8 Image Representation on the Unit Circle and MQFTR 147
8.1.1 Preparation for FTQR 147
8.1.2 Constant Signal and Global Phase 148
8.1.3 Inverse Transform 149
8.1.4 Property of Phase 150
8.2 Operations with Kronecker Product 150
8.3 FTQR Model for Grayscale Image 151
8.4 Color Image FTQR Models 151
8.5 The 2D Quantum Fourier Transform 153
8.5.1 Algorithm of the 2D QFT 153
8.5.2 Examples in Qiskit 157
References 159
9 New Operations of Qubits 161
9.1 Multiplication 161
9.1.1 Conjugate Qubit 162
9.1.2 Inverse Qubit 162
9.1.3 Division of Qubits 163
9.1.4 Operations on Qubits with Relative Phases 163
9.1.5 Quadratic Qubit Equations 164
9.1.6 Multiplication of n-Qubit Superpositions 165
9.1.7 Conjugate Superposition 167
9.1.8 Division of Multi-qubit Superpositions 167
9.1.9 Operations on Left-Sided Superpositions 167
9.1.10 Quantum Sum of Signals 168
9.2 Quantum Fourier Transform Representation 169
9.3 Linear Filter (Low-Pass Filtration) 170
9.3.1 General Method of Filtration by Ideal Filters 173
9.3.2 Application: Linear Convolution of Signals 174
References 176
10 Quaternion-Based Arithmetic in Quantum Image Processing 177
10.1 Noncommutative Quaternion Arithmetic 178
10.2 Commutative Quaternion Arithmetic 180
10.3 Geometry of the Quaternions 182
10.4 Multiplicative Group on 2-Qubits 184
10.4.1 2-Qubit to the Power 188
10.4.2 Second Model of Quaternion and 2-Qubits 190
References 193
11 Quantum Schemes for Multiplication of 2-Qubits 195
11.1 Schemes for the 4×4 Gate A q 1 196
11.2 The 4×4 Gate with 4 Rotations 202
11.3 Examples of 12 Hadamard Matrices 205
11.4 The General Case: 4×4 Gate with 5 Rotations 210
11.5 Division of 2-Qubits 213
11.6 Multiplication Circuit by 2nd 2-Qubit (Aq2) 214
References 218
12 Quaternion Qubit Image Representation (QQIR) 219
12.1 Model 2 for Quaternion Images 220
12.1.1 Comments: Abstract Models with Quaternion Exponential Function 221
12.1.2 Multiplication of Colors 222
12.1.3 2-Qubit Superposition of Quaternion Images 222
12.2 Examples in Color Image Processing 224
12.2.1 Grayscale-2-Quaternion Image Model 224
12.3 Quantum Quaternion Fourier Transform 227
12.4 Ideal Filters on QQIR 228
12.4.1 Algorithm of Filtration G p = Y p F p by Ideal Filters 229
12.5 Cyclic Convolution of 2-Qubit Superpositions 230
12.6 Windowed Convolution 230
12.6.1 Edges and Contours of Images 235
12.6.2 Gradients and Thresholding 235
12.7 Convolution Quantum Representation 238
12.7.1 Gradient Operators and Numerical Simulations 241
12.8 Other Gradient Operators 244
12.9 Gradient and Smooth Operators by Multiplication 246
12.9.1 Challenges 248
References 248
13 Quantum Neural Networks: Harnessing Quantum Mechanics for Machine Learning 251
13.1 Introduction in Quantum Neural Networks: A New Frontier in Machine Learning 251
13.2 McCulloch-Pitts Processing Element 254
13.3 Building Blocks: Layers and Architectures 258
13.4 Artificial Neural Network Architectures: From Simple to Complex 259
13.5 Key Properties and Operations of Artificial Neural Networks 261
13.5.1 Reinforcement Learning: Learning Through Trial and Reward 262
13.6 Quantum Neural Networks: A Computational Model Inspired by Quantum Mechanics 263
13.7 The Main Difference Between QNNs and CNNs 271
13.8 Applications of QNN in Image Processing 276
13.9 The Current and Future Trends and Developments in Quantum Neural Networks 281
References 282
14 Conclusion and Opportunities and Challenges of Quantum Image Processing 285
References 288
Index 291
