Discrete Cosine Transform (DCT) is very important in image compression. Classical 1-D DCT and 2-D DCT has time complexity O(NlogN) and O(N²logN) respectively. This paper presents a quantum DCT iteration, and constructs a quantum 1-D and 2-D DCT algorithm for image compression by using the iteration. The presented 1-D and 2-D DCT has time complexity O(sqrt(N)) and O(N) respectively. In addition, the method presented in this paper generalizes the famous Grover's algorithm to solve complex unstructured search problem.
Discrete Cosine Transform (DCT) is very important in image compression. Classical 1-D DCT and 2-D DCT has time complexity O(NlogN) and O(N²logN) respectively. This paper presents a quantum DCT iteration, and constructs a quantum 1-D and 2-D DCT algorithm for image compression by using the iteration. The presented 1-D and 2-D DCT has time complexity O(sqrt(N)) and O(N) respectively. In addition, the method presented in this paper generalizes the famous Grover's algorithm to solve complex unstructured search problem.
The Discrete Cosine Transform (DCT) is an integral part of classical image compression, which becomes the basis of the JPEG standard. With advances in quantum computing, there is growing interest in exploring quantum analogues of classical algorithms for such transforms. This paper presents a simulation of the Quantum Discrete Cosine Transform (QDCT) using the Qiskit Python library, applied to standard grayscale images segmented into 8x8 and 4x4 blocks. The QDCT algorithm is formulated as a unitary operator and evaluated using state-vector simulation. Due to the current constraints of quantum hardware and challenges in circuit synthesis, this study focused on simulation results, as direct quantum circuit measurement leads to amplitude collapse and unusable output. The performance of QDCT is quantitative compared to classical DCT using Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index (SSIM), and compression ratio metrics. The result of this study reveals the feasibility and current limitations of QDCT-based image compression while providing a reproducible benchmark for future quantum image processing research. This study offers a realistic assessment of the potential and technical limitations of QDCT in the context of emerging quantum technologies.
Abstract In recent years, quantum image processing got a lot of attention in the field of image processing due to the opportunity to place huge image data in quantum Hilbert space. Hilbert space or Euclidean space has infinite dimensions to locate and process the image data faster. Moreover, several types of research show that the computational time of the quantum process is faster than classical computers. Encoding and compressing the image in the quantum domain is still a challenging issue. From the literature survey, we have proposed a DCT-EFRQI (Direct Cosine Transform Efficient Flexible Representation of Quantum Image) algorithm to represent and compress gray image efficiently which save computational time and minimize the complexity of preparation. This work aims to represent and compress various gray image sizes in quantum computers using the DCT (Discrete Cosine Transform) and EFRQI (Efficient Flexible Representation of Quantum Image) approaches together. The Quirk simulation tool is used to design the corresponding quantum image circuit. Due to the limitation of qubits, a total of 16 numbers of qubits is used to represent the coefficient and its position of the grayscale image. Among those, 8 number of qubits are used to map the coefficient values and the rest are used to generate the corresponding coefficient XY-coordinate position. Theoretical analysis and experimental results show that the proposed DCT-EFRQI scheme provides better representation and compression compared to DCT-GQIR, DWT-GQIR, and DWT-EFRQI in terms of PSNR(Peak Signal to Noise Ratio), and bit rate.
In recent year, quantum image processing got a lot of attention in the field of image processing due to opportunity to place huge image data in quantum Hilbert space. Hilbert space or Euclidean space has infinite dimension to locate and process the image data faster. Moreover, several researches show that, the computational time of quantum process is faster than classical computer. By encoding and compressing the image in quantum domain is still challenging issue. From literature survey, we have proposed a DCTEFRQI (Direct Cosine Transform Efficient Flexible Representation of Quantum Image) algorithm to represent and compress gray image efficiently which save computational time and minimize the complexity of preparation. The objective of this work is to represent and compress various gray image size in quantum computer using DCT(Discrete Cosine Transform) and EFRQI (Efficient Flexible Representation of Quantum Image) approach together. Quirk simulation tool is used to design corresponding quantum image circuit. Due to limitation of qubit, total 16 numbers of qubit are used to represent the gray scale image among those 8 are used to map the coefficient values and the rest 8 are used to generate the corresponding coefficient position. Theoretical analysis and experimental result show that, proposed DCTEFRQI scheme provides better representation and compression compare to DCT-GQIR, DWT-GQIR and DWT-EFRQI in terms of PSNR(Peak Signal to Noise Ratio) and bit rate..
Quantum computing has emerged as a revolutionary paradigm in numerous fields owing to its basis in quantum mechanical concepts such as quantum entanglement and parallelism. This has potential in various applications including computer vision, particularly, image processing and compression. Applications that depend on largescale image data for storing, processing, and transmission have to rely on compression methods for real-time processing. While the classical methods based on discrete cosine transformations (DCT used in JPEG) and discrete wavelet transformations (DWT) perform image compression efficiently, they suffer from trade-off between loss-less compression and compression ratios. In the current work, we would like to explore the usefulness of implementing quantum wavelet transforms for image compression for better PSNR values and execution times. We compare the application of quantum Haar wavelet transform (QHWT) with different quantum encoding methods such as NASS and FRQI and find that NASS-encoded images on application of QHWT gives better PSNR values and shorter runtime with visibly closer details to the original images, showcasing potential for efficient and faster image compression methods.
The Quantum Signal Induced Heap Transform (QsiHT) is an emerging transform in the field of quantum computing, designed to enhance efficiency in signal processing tasks. This transform offers a novel approach for handling large-scale data structures in quantum algorithms, particularly in applications related to quantum image processing and signal decomposition. In this paper, we explore the use of QsiHT in generating various types of the quantum discrete cosine transforms (DCT), including the II-DCT and III-DCT, commonly used in image compression and signal processing. In QsiHT, different paths can be used in the computation, and in this work, we analyze a specific path to make the transformation computation most efficient. The proposed quantum circuits and algorithms for the QsiHT with 3 qubits are described in detail.
In recent years, quantum image computing draws a lot of attention due to storing and processing image data faster compared to classical computers. A number of approaches have been proposed to represent the quantum image inside a quantum computer. Representing and compressing medium and big-size images inside the quantum computer is still challenging. To address this issue, we have proposed a block-wise DCT-EFRQI (Direct Cosine Transform Efficient Flexible Representation of Quantum Image) approach to represent and compress the gray-scale image efficiently to save computational time and reduce the quantum bits (qubits) for the state preparation. In this work, we have demonstrated the capability of block-wise DCT and DWT transformation inside the quantum domain to investigate their relative performances. The Quirk simulation tool is used to design the corresponding quantum image circuit. In the proposed DCT-EFRQI approach, a total of 17 qubits are used to represent the coefficients, the connection between coefficients and state (i.e., auxiliary), and their position for representing and compressing grayscale images inside a quantum computer. Among those, 8 qubits are used to map the coefficient values and the rest are used to generate the corresponding coefficient XY-coordinate position including one auxiliary qubit. Theoretical analysis and experimental results show that the proposed DCT-EFRQI scheme provides better representation and compression compared to DCT-GQIR, DWT-GQIR, and DWT-EFRQI in terms of rate-distortion performance.