A Flexible Representation of Quantum Images (FRQI) is proposed to provide a representation for images on quantum computers in the form of a normalized state which captures information about colors and their corresponding positions in the images. A constructive polynomial preparation for the FRQI state from an initial state, an algorithm for quantum image compression (QIC), and processing operations for quantum images are combined to build the whole process for quantum image processing on FRQI. The simulation experiments on FRQI include storing, retrieving of images and a detection of a line in binary images by applying quantum Fourier transform as a processing operation. The compression ratios of QIC between groups of same color positions range from 68.75 to 90.63% on single digit images and 6.67---31.62% on the Lena image. The FRQI provides a foundation not only to express images but also to explore theoretical and practical aspects of image processing on quantum computers.

A flexible representation of quantum images for polynomial preparation, image compression, and processing operations

Quantum image processing (QIMP) is devoted to utilizing the quantum computing technologies to capture, manipulate, and recover quantum images in different formats and for different purposes. Logically, percolating this requires that representations to encode images based on the quantum mechanical composition of any potential quantum computing hardware be conjured. This paper gathers the current mainstream quantum image representations (QIRs) and discusses the advances made in the area. Some similarities, differences, and likely applications for some of the available QIRs are reviewed. We believe this compendium will provide the readership an overview of progress witnessed in the area of QIMP while also simulating further interest to pursue more advanced research in it.

A survey of quantum image representations

Watermarking and authentication of quantum images based on restricted geometric transformations

Although image scaling algorithms in classical image processing have been extensively studied and widely used as basic image transformation methods, the quantum versions do not exist. Therefore, this paper proposes quantum algorithms and circuits to realize the quantum image scaling based on the improved novel enhanced quantum representation (INEQR) for quantum images. It is necessary to use interpolation in image scaling because there is an increase or a decrease in the number of pixels. The interpolation method used in this paper is nearest neighbor which is simple and easy to realize. First, NEQR is improved into INEQR to represent images sized $$2^{n_{1}} \times 2^{n_{2}}$$2n1×2n2. Based on it, quantum circuits for image scaling using nearest neighbor interpolation from $$2^{n_{1}} \times 2^{n_{2}}$$2n1×2n2 to $$2^{m_{1}} \times 2^{m_{2}}$$2m1×2m2 are proposed. It is the first time to give the quantum image processing method that changes the size of an image. The quantum strategies developed in this paper initiate the research about quantum image scaling.

Quantum image scaling using nearest neighbor interpolation

Strategies for designing geometric transformations on quantum images

In this paper, the quantum circuit design of an 8/spl times/8 discrete cosine transform (DCT) is investigated. The proposed design procedure can be divided into the following three steps. First, the DCT matrix is decomposed into the product of sparse matrices, based on its fast computation flow graph. Second, each sparse matrix is implemented by elementary quantum gates. Third, the sparse matrix circuits are cascaded to obtain the final circuit. The proposed method makes the fast DCT computation algorithm in digital signal processing suitable for implementation in a quantum computer.

Quantum circuit design of 8/spl times/8 discrete cosine transform using its fast computation flow graph

In this paper, the 8 /spl times/ 8 discrete Hartley transform will be implemented by using quantum elementary gates such as controlled NOT gates and Hadamard gates Two steps involved are as follows: first, the discrete Hartley transform matrix is decomposed into the product of several sparse unitary matrices using the structure of fast Hartley transform algorithm Then, each sparse matrix is implemented by elementary quantum gates and cascade them to obtain the final circuit The design results show that the proposed design method has less circuit complexity than the conventional Cybenko's method

Quantum circuit design of 8 /spl times/ 8 discrete Hartley transform

In this paper, the quantum circuit design of a discrete Hartley transform (DHT) is investigated. The proposed design procedure can be divided into the following two steps. First, the N-point transform matrix is decomposed into the combination of N/2-point transform matrices and several sparse matrices such that any 2/sup n/-point transform can be computed recursively from the 2-point transform. Second, starting from the 2-point Hadamard transform, the quantum circuit of 2/sup n/-point transform can be designed recursively by using elementary quantum gates. The design results are compared with those of the conventional factorization method to demonstrate the effectiveness of proposed method.

Tsung-Ming Hwang

Papers

Quantum circuit design of 8/spl times/8 discrete cosine transform using its fast computation flow graph

Quantum circuit design of 8 /spl times/ 8 discrete Hartley transform

Quantum circuit design of discrete Hartley transform using recursive decomposition formula