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arXiv:2305.05953

Quantum Fourier Transform for Image Processing

Zhang Yu, Weibo Gao

Published May 10, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2305.05953

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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.
In this paper we address the emerging field of Quantum Image Processing. We investigate the use of quantum computing systems to represent and manipulate images. In particular, we consider the basic task of image filtering. We prove that a quantum version for this operation can be achieved, even though the quantum convolution of two sequences is physically impossible. In our approach we use the principle of the quantum oracle to implement the filter function. We provide the quantum circuit that implements the filtering task and present the results of several simulation experiments on grayscale images. There are important differences between the classical and the quantum implementations for image filtering. We analyze these differences and show that the major advantage of the quantum approach lies in the exploitation of the efficient implementation of the quantum Fourier 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.
Quantum information processing can overcome the limitations of classical computation.Consequently, image filtering using quantum computation has become a research hotspot.Here, a quantum algorithm is presented on the basis of the classical image filtering principle to detect and cancel the noise of an image.To this end, a quantum algorithm that completes the image filtering task is proposed and implemented.The novel enhanced quantum representation of digital images is introduced.Then, four basic modules, namely, position-shifting, parallel-CNOT, parallel-swap, and compare the max, are demonstrated.Two composite modules that can be utilised to realise the reversible logic circuit of the proposed quantum algorithm are designed on the basis of these basic modules.Simulation-based experimental results show the feasibility and the capabilities of the proposed quantum image filtering scheme.In addition, our proposal has outperformed its classical counterpart and other existing quantum image filtering schemes supported by detailed theoretical analysis of the computational complexity.Thus, it can potentially be used for highly efficient image filtering in a quantum computer age.
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.
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.
Quantum image processing is paramount in image processing-based research areas nowadays due to the less computational time. This paper presents the quantum mechanic-based image denoising model. Also, This article discusses the quantum boolean mean filter and new quantum-boolean image denoising that are connected to computational basis states. This paper describes basic quantum information processing principles, new and better image representation approaches, quantum image processing implementation problems, quantum boolean mean filtering approach for image denoising observation using appropriate matrice, and finally, a work proposal for future work and conclusion. A comparison table of different classification techniques is also included in the paper.