Quantum Image Processing in Practice E-Book

Artyom M. Grigoryan

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 460

Veröffentlichungsjahr: 2025

Beschreibung

Comprehensive resource addressing the need for a quantum image processing machine learning model that can outperform classical neural networks

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.

Leseprobe

Quantum Image Processing in Practice

A Mathematical Toolbox

Artyom M. Grigoryan

Sos S. Agaian

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To AnoushandTo Sarkis and Gayane

Preface

The modern world has witnessed remarkable applications in the dynamic field of image processing, where operations transform an image to enhance it or extract vital information. It is a vibrant and diverse field encompassing various applications, such as facial recognition, image segmentation and compression, noise reduction, and more. These applications require sophisticated techniques to transform, enhance, and extract image information. However, these techniques also demand substantial computational resources for image storage and processing, which pose significant challenges for scalability and efficiency. Therefore, there is a critical need for more advanced and innovative methods to handle visual information. On the other hand, quantum computing defines a probabilistic approach to represent classical information using methods from quantum theory. Quantum computing offers a probabilistic and parallel approach to computation, which differs fundamentally from the deterministic and sequential approach of classical computing. The basic unit of quantum information, the qubit, can exist in a superposition of two states until measured, which enables quantum parallelism and entanglement. These quantum phenomena can provide exponential speedups and enhanced security for specific computational tasks, such as factoring large numbers, searching unsorted databases, simulating quantum systems, and solving linear systems of equations.

Quantum image processing (QIP) is a research branch of quantum information and computing that aims to exploit the advantages of quantum computing for image processing. QIP studies how to encode and process images using various quantum image representations and operations in a quantum computer. QIP has the potential to outperform classical image processing in terms of computing speed, security, and minimum storage requirements. However, QIP also faces many challenges and open questions, such as quantum superiority, reading the classical data, measurement, noise and error correction, scalability and compatibility, and the practical implementation of QIP algorithms and circuits.

In this book, we provide a comprehensive introduction to QIP, covering the theoretical foundations, methodological developments, quaternion color imaging, and practical QIP applications. We describe the existing quantum image representations and their operations, such as geometric transformations, color transformations, filtering, and enhancement. We also explore the emerging topics and applications of QIP, such as quantum image filtration in the frequency domain, convolution, and fast unitary transforms. We discuss the current state of QIP research, addressing the controversies and opportunities, as well as the challenges and future directions of QIP. We illustrate the QIP algorithms and circuits with detailed examples, diagrams, and code snippets using the Qiskit framework. We also provide exercises and references for further learning and research.

Organization of the Book

This book is organized into 14 chapters, as follows:

Chapter 1: IntroductionThis chapter provides an overview of the main concepts and motivations of quantum computing and image processing. It outlines the structure and objectives of the book.

Chapter 2: Basic Concepts of QubitsThis chapter delves into the core concepts and principles, such as computational qubit states, superposition, operations on qubits, permutations, elementary gates, and qubit measurement. It presents the operations and gates in matrix and graphical notations and illustrates them with examples; 3‐D model of qubits is presented together with the known Bloch sphere.

Chapter 3: Understanding 2‐Qubit SystemsThis chapter focuses on 2‐qubit systems, which are the building blocks of multi‐qubit systems. It discusses the mathematical tools and techniques for manipulating 2‐qubit systems, such as projection operators, Kronecker product and sum, qubit entanglement, orthogonality, and unitary transformations. It also describes the elementary operations and main gates for 2‐qubit systems, such as CNOT, SWAP, local and controlled gates, and explores their properties and applications with practical examples.

Chapter 4: Multi‐Qubit Superpositions and OperationsThis chapter extends the concepts and methods of 2‐qubit systems to multi‐qubit systems, which are essential for quantum image processing. It examines multi‐qubit superpositions of different types. Many 3‐qubit gates with 1 and 2 control bits are described with matrices and circuit elements. It also highlights the key 3‐qubit gates, such as Toffoli and Fredkin, bit SWAP, and Hadamard gates, and shows how they can be used to implement classical logic functions and reversible circuits.

Chapter 5: Fast Transforms in Quantum ComputationThis chapter introduces the quantum analogs of the classical fast transforms, such as the discrete paired, Fourier, and Hadamard transforms which are widely used in image processing. It provides detailed descriptions of the algorithms and implementations of these quantum‐fast transforms, supported by examples and circuit designs. It also compares the advantages and disadvantages of these quantum fast transforms concerning their classical counterparts. Examples and circuits of these transforms and their inverses on 2‐, 3‐, and 4‐qubits are presented. The paired transform is the core of the Fourier and Hadamard transforms. Therefore, the quantum paired transform is described in detail. The 1‐D quantum circular convolution for phase filters with circuits is also presented with examples.

Chapter 6: Quantum Signal‐Induced Heap TransformThis chapter presents a novel concept of quantum fast transform, which refers to the so‐called discrete signal‐induced heap transform (DsiHT), which can generate a unique unitary and fast transform for any given signal. It explains the theory and algorithm of quantum signal‐induced heap transform (QsiHT) and demonstrates its applications in quantum cosine and Hartley transforms with quantum circuits. It also shows how DsiHT can be used to factorize and decompose any transform in a set of rotated gates and permutation, as well to initiate any quantum superposition from the basis state 0.

Chapter 7: Quantum Image Representation with ExamplesThis chapter describes the various quantum image representations proposed in the literature and compares their features and limitations. It covers the models for both grayscale and color images, such as QLM, NEQR, FRQI, RKL, GQIP, QIRHSI, and MQFTR. It also presents the 2 × 2 model of color image respresentation as a single grayscale image in quantum computation. It explores the quantum representation of different color models, such as RGB, CMY, XYZ, HSV, and HSI, and discusses the challenges and opportunities of quantum color image processing.

Chapter 8: Image Representation on the Unit Circle and MQFTRThis chapter focuses on a specific type of quantum image representation, called the multi‐qubit Fourier transform representation (MQFTR), which encodes the image information on the unit circle using the Kronecker product of qubits. It explains the advantages and disadvantages of MQFTR. It presents some extensions and variations of MQFTR, such as MQFTR with phase shift and MQFTR with amplitude modulation. It also describes the quantum schemes for 2‐D quantum Fourier transform with examples for 4 × 4 and 8 × 8 images.

Chapter 9: New Arithmetic on QubitsThis chapter introduces some novel concepts and methods of arithmetic operations on qubits, such as multiplication, division, conjugate, and inverse. It extends these operations to multi‐qubit superpositions and discusses their properties and applications. It also shows how these operations can be implemented using quantum circuits and algorithms in image summation, linear convolution and filtration with examples.

Chapter 10: Quaternion‐Based Arithmetic in Quantum Image ProcessingThis chapter explores the non‐commutative quaternion arithmetic in quantum color image processing, which can offer some advantages over conventional complex arithmetic. It differentiates between the traditional and commutative quaternion algebras and discusses their properties and applications. It presents a new concept of the multiplicative group of 2‐qubits and describes the main properties of the multiplication of 2‐qubits and 2‐qubit‐based superpositions. It also describes the graphical representation of 2‐qubits.

Chapter 11: Quantum Schemes for Multiplication of 2‐QubitsThis chapter presents detailed quantum 2‐qubit multiplication circuits that underlie many quantum arithmetic operations. It showcases a few circuits composed using the QsiHT concept and compares their performance and complexity. It shows how to design the quantum 4 × 4‐gates of multiplication with 4, 5, and 6 rotations. It also describes 12 Hadamard matrices as multiplication gates.

Chapter 12: Quaternion Qubit Image Representation (QQIR)This chapter introduces a new quantum image representation, called the quaternion qubit image representation (QQIR), which combines the features of 4‐D quaternion arithmetic and MQFTR. It covers some basic operations in QQIR, such as square root, power, and exponentiation, and shows how they can be used for image processing. It explores some advanced operations in QQIR, such as the convolution and gradient calculation, and demonstrates their applications in image filtering, edge detection, and feature extraction. It also describes the concept of the quantum quaternion Fourier transform (QQFT) and ideal filtration by this transform.

Chapter 13: Quantum Neural Networks (QNN)This chapter bridges quantum computing and machine learning and discusses the development and applications of quantum neural networks inspired by classical neural networks. It highlights the differences, synergies, and challenges of quantum and classical neural networks and reviews some existing models and architectures of quantum neural networks. It also explores some potential applications of quantum neural networks in image processing, such as image classification, recognition, segmentation, restoration, and reconstruction.

Chapter 14: Conclusion and Opportunities and Challenges of Quantum Image ProcessingThis chapter summarizes the main contributions and findings of the book and reflects on the current state and future directions of quantum image processing. It discusses the opportunities and challenges of quantum image processing, such as quantum superiority, noise, and error correction, scalability and compatibility, and practical implementation. It also provides some suggestions and recommendations for further research and development in quantum image processing.

Designed for a diverse audience, from students to professionals, this book is accessible to those with some background in linear algebra, quantum mechanics, and image processing. Many examples and references throughout the text encourage further exploration and deeper understanding. We are grateful to everyone who read this book. We hope the reader will enjoy learning about quantum imaging and its applications and gain some ideas and inspiration from the methodologies and concepts discussed in this book. This book explores the emerging interdisciplinary field of quantum imaging, introducing fundamental concepts, state‐of‐the‐art techniques, and applications.

An Invitation to Innovation: We also hope that this book will arouse readers' interest in QIP and inspire them to contribute to its development. Join us on a journey where the classical and quantum worlds intertwine, unlocking a future brimming with unprecedented potential for image processing innovation.

We appreciate all who assisted in the preparation of this book. We are grateful to Meruzhan Grigoryan, Alexis Gomez, and the reviewers for many suggestions and recommendations.

May 2024

Artyom Grigoryan and Sos S. Agaian

Acknowledgments

Artyom Grigoryan: I would like to express my deepest gratitude to my daughter, Anoush Grigoryan, for her unwavering support. My heartfelt thanks go to my brother, Meruzhan Grigoryan, for his insightful comments and suggestions, which significantly improved the material in Chapter 6. I am also thankful to our student, Alexis Gomez at UTSA, for his valuable assistance in developing Python codes for the examples in this book.

Sos Agaian: I want to express my sincere appreciation and dedication to my wife, Gayane Abrahamian, and my son, Sarkis Agaian, for their steadfast support throughout the writing process. Their constant encouragement, motivation, and love were the foundation of this work. This book would not have come to fruition without their presence and assistance.

Additionally, we extend our gratitude to Kavipriya for her diligent support, the reviewers for their invaluable feedback, senior commissioning editor Sandra Grayson for her expert guidance, senior editorial assistant Becky Cowan for her meticulous attention to detail, and cover designer Jose Bacede for his creative contributions.

Finally, we are grateful for the opportunity to present “Quantum Image Processing: A Mathematical Toolbox,” a work that has been a labor of love and dedication.

About the Companion Website

This book is accompanied by a companion website

www.wiley.com/go/grigoryan/quantumimageprocessing 

This website includes:

QuAlgorithms

Part IMathematical Foundation of Quantum Computation