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IEEE Signal Processing Magazine

Quaternion-Based Arithmetic in Quantum Information Processing: A promising approach for efficient color quantum imaging

Artyom M. Grigoryan, Sos С. Agaian

Published Mar 14, 2024• 8 citations
Publisher

Abstract

Classical color image processing, image recognition, and machine learning introduce nonlinearity, causing the collapse of the quantum state into classical probability perceptrons after measurements, due to the inherent linearity of quantum computing. To address this challenge, quaternion-based arithmetic offers a promising approach. By treating the primary color components as a single unit using quaternion algebra, nonlinear relationships can be implemented, effectively manipulating higher-dimensional color data. This article aims to achieve efficient and accurate color quantum image processing (QIP) by introducing new quaternion quantum-based color imaging tools based on multiplicative arithmetic on two-qubits and quantum superpositions. The approach includes the concept of a quaternion Fourier transform (QFT) in two-qubit-based color image representation. To end, we discuss possible applications of the proposed methods in color quantum imaging.

DOI
10.1109/msp.2023.3327627

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Classical color image processing, image recognition, and machine learning introduce nonlinearity, causing the collapse of the quantum state into classical probability perceptrons after measurements, due to the inherent linearity of quantum computing. To address this challenge, quaternion-based arithmetic offers a promising approach. By treating the primary color components as a single unit using quaternion algebra, nonlinear relationships can be implemented, effectively manipulating higher-dimensional color data. This article aims to achieve efficient and accurate color quantum image processing (QIP) by introducing new quaternion quantum-based color imaging tools based on multiplicative arithmetic on two-qubits and quantum superpositions. The approach includes the concept of a quaternion Fourier transform (QFT) in two-qubit-based color image representation. To end, we discuss possible applications of the proposed methods in color quantum imaging.
In quaternion algebra the color images are processed not by separating their color components, but these components are presented and processed together as one unit. In framework of the quaternion image representation, many effective methods of processing color images can be developed, including image enhancement in spatial and frequency domains. In this paper, we unite two approaches for processing color images, by proposing a quaternion representation in quantum imaging, which includes the color images in the RGB model together with the grayscale component or brightness. The concept of quaternion two-qubit is considered and applied for image representation in each quantum pixel. The colors at each pixel are processed as one unit in quaternion representation. Other new models for quaternion image representation are also described. It is shown that a quaternion image or four component image of 𝑁 × 𝑀 pixels, can be represented by (𝑟 + 𝑠 + 2) qubits, when 𝑁 = 2𝑟 and 𝑀 = 2𝑠, 𝑟, 𝑠 > 1. The number of qubits for representing the image can be reduced to (𝑟 + 𝑠), when using the quaternion 2-qubit concept.
This work focuses into the emerging field of quantum image representation, which is a key for processing grayscale and color images on quantum computing platforms. We explore foundational methods like the qubit lattice model, the flexible representation for quantum images, the novel enhanced quantum representation, and the generalized quantum image representation, and categorize them into state-based, amplitude-based, and quaternion-based representations - the latter offering holistic color processing. A novel discrete signal representation mapping data on the unit circle will be introduced to simplify the Fourier transform concepts in quantum systems. We will examine crucial quantum operations, including quantum gradients, multiplication, square roots, Fourier transform, and filtering. Challenges in implementing the circular convolution on quantum hardware will be addressed with practical examples such as quantum low-pass and high-pass filters and 3-qubit convolution circuits. Additionally, we will discuss ongoing research into quantum algorithms inspired by classical transform processing, including the Fourier transforms, that leverage quantum states’ unique properties to achieve efficient encoding, transformation, and decoding of images.
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.
The Fourier transform is a popular tool in many areas of practical mathematics, including data compression and signal processing. The underlying principle of the Fourier transform is that functions with the right characteristics may be expressed as a linear combination of trigonometric functions. Decomposing a signal using trigonometric functions is like taking a time-domain signal and separating its frequency-domain components; this opens up new and more efficient avenues for signal analysis and manipulation. An increasing number of people are looking at quantum image processing (QIP) as a way to enhance the efficiency of conventional methods and their applications by taking use of quantum computing's characteristics. Filtering and rectifying the frequency domain to get important picture information is the goal of both image design and interactive processing in this procedure. We demonstrate the operation of quantum operators and states in the context of quaternions and demonstrate how it may be used to expand the quantum complex Fourier transform (QCFT) to the quantum quaternion Fourier transform (QQFT). We demonstrate why convolution cannot be used to quantum image processing, and we create space-domain and quaternion-Fourier spectrum filters, including an edge detector and a quantum median filter. An enhanced method is used to provide the groundwork for the realization of picture design and interactive technology, drawing on the knowledge of FT and inverse transform.
Quantum Image Processing (QIP)is an exciting new field showing a lot of promise as a powerful addition to the arsenal of Image Processing techniques. Representing image pixel by pixel using classical information requires an enormous amount of computational resources. Hence, exploring methods to represent images in a different paradigm of information is important. In this work, we study the representation of images in Quantum Information. The main motivation for this pursuit is the ability of storing N bits of classical information in only log(2N) quantum bits (qubits). The promising first step was the exponentially efficient implementation of the Fourier transform in quantum computers as compared to Fast Fourier Transform in classical computers. In addition, images encoded in quantum information could obey unique quantum properties like superposition or entanglement.
There is a growing interest in quantum image processing (QIP) that rose from the desire to exploit the properties of quantum computing to improve the performance of classical techniques and their applications. Since the introduction of a quaternion by Hamilton in 1843, quaternions had been used in a lot of applications. We show how the quantum states and quantum operators work in the field of quaternions and how we can use it to extend the quantum complex Fourier transform (QCFT) to the quantum quaternion Fourier transform (QQFT). We propose for the first time the novel algorithm for the QQFT. We explain the impossibility of the use of convolution in quantum image processing; we design filters in the quaternion Fourier spectrum and the space domain an edge detector and a quantum median filter.