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.
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.
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.
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.
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.
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.
Quantum information processing and its subfield, quantum image processing, are rapidly growing fields as a result of advancements in the practicality of quantum mechanics. In this paper, we propose a quantum algorithm for processing information, such as one-dimensional time series and two-dimensional images, in the frequency domain. The information of interest is encoded into the magnitude of probability amplitude or the coefficient of each basis state. The oracle for filtering operates based on postselection results, and its explicit circuit design is presented. This oracle is versatile enough to perform all basic filtering, including high pass, low pass, band pass, band stop, and many other processing techniques. Finally, we present two novel schemes for transposing matrices in this paper. They use similar encoding rules but with deliberate choices in terms of selecting basis states. These schemes could potentially be useful for other quantum information processing tasks, such as edge detection. The proposed techniques are implemented on the IBM Qiskit quantum simulator. Some results are compared with traditional information processing results to verify their correctness and are presented in this paper.
The Quantum Image Processing (QIP, based on developing along quantum computing and digital image analysis) is an emerging field. Due to the growing needs for image processing with high speed and efficiency in applications including medical imaging, satellite surveillance, autonomous vehicle, and big data analytics, there are limitations of scalability and speed with classical methods. By exploring the very principles on which the modern world operates, from superposition to entanglement, quantum computing provides an altogether transformative perspective on how to process and analyze images exponentially faster. QIP is the topic of this paper and how it might change the way that images are stored, processed, and interpreted; that is, we look at the theoretical foundations and practical considerations of QIP. Quantum representation of images, quantum Fourier transforms, and quantum edge detection are discussed. The challenges to technical and implementable aspects of QIP need not preclude the next generation of high speed computation. The objectives of this research include discovering the potential of QIP, and the weaknesses and future occurrence of the QIP technology.