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Modern Electronics Technique

Image Compression Based on Quantum Haar Wavelet Transform

Song Hong Yan

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

The Haar wavelet algorithm based on quantum computing and its application for color image compression are discussed.Firstly,classical Haar wavelet operator is decomposed into direct sums,direct products and dot products of unitary matrices,so that shuffle permutation matrices is found.Secondly,based on the generalized Kronecker product,Harr functions are decomposed into a sequence of W-H and shuffle permutation matrices.Thirdly,the efficient,complete,quantum circuits for representative quantum wavelet transform are given.Finally,algorithm for color image compression by quantum Haar wavelet is presented,and experiment results are given.The experiment results show that the quantum Haar wavelet has very good feasibility in image compression.

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The Haar wavelet algorithm based on quantum computing and its application for color image compression are discussed.Firstly,classical Haar wavelet operator is decomposed into direct sums,direct products and dot products of unitary matrices,so that shuffle permutation matrices is found.Secondly,based on the generalized Kronecker product,Harr functions are decomposed into a sequence of W-H and shuffle permutation matrices.Thirdly,the efficient,complete,quantum circuits for representative quantum wavelet transform are given.Finally,algorithm for color image compression by quantum Haar wavelet is presented,and experiment results are given.The experiment results show that the quantum Haar wavelet has very good feasibility in image compression.
Because of the prominent advantages of quantum computation compared to classic computation,implementation of quantum wavelet transforms has profound significance to its completion and application.After the circuit for perfect shuffle permutation matrices is finished,the unitary matrices for Harr functions are decomposed into a sequence of W-H and perfect shuffle permutation matrices,based on the generalized Kronecker product.Then the quantum circuit for Harr matrices is provided,its realizable complexity and pos-sibility of physical implementation is analyzed.
The quantum Fourier transform (QFT), a quantum analog of the classical Fourier transform, has been shown to be a powerful tool in developing quantum algorithms. However, in classical computing there is another class of unitary transforms, the wavelet transforms, which are every bit as useful as the Fourier transform. Wavelet transforms are used to expose the multi-scale structure of a signal and are likely to be useful for quantum image processing and quantum data compression. In this paper, we derive efficient, complete, quantum circuits for two representative quantum wavelet transforms, the quantum Haar and quantum Daubechies D(4)D^{(4)}D(4) transforms. Our approach is to factor the operators for these transforms into direct sums, direct products and dot products of unitary matrices. In so doing, we find that permutation matrices, a particular class of unitary matrices, play a pivotal role. Surprisingly, we find that operations that are easy and inexpensive to implement classically are not always easy and inexpensive to implement quantum mechanically, and vice versa. In particular, the computational cost of performing certain permutation matrices is ignored classically because they can be avoided explicitly. However, quantum mechanically, these permutation operations must be performed explicitly and hence their cost enters into the full complexity measure of the quantum transform. We consider the particular set of permutation matrices arising in quantum wavelet transforms and develop efficient quantum circuits that implement them. This allows us to design efficient, complete quantum circuits for the quantum wavelet transform.
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.
An algorithm of quantum image encryption based on a normal arbitrary superposition state (NASS) is proposed by using quantum geometric transform, phase-shift transform, and quantum Haar wavelet packet transform (QHWPT) in this paper. Firstly, to scramble an image, we design an implementation circuit of quantum image pixel exchanges along diagonal lines. Next, [Formula: see text] keys randomly distributed between 0 and [Formula: see text] are encoded in the encryption algorithm by using phase-shift transform. The huge key space makes it difficult to attack encrypted images. Then, we store image information in low frequency using QHWPT to improve encryption effect. After that, the encryption algorithms of gray image and color image are simulated by using MATLAB, respectively. Meanwhile, evaluation indexes such as histogram, complexity and correlation of adjacent pixels are analyzed. Simulation experiments show that the proposed algorithm is valid.
In this paper, a novel frequency domain quantum watermarking scheme is proposed based on the Flexible Representation of Quantum Images, which can embed a 2n1× 2n1binary watermark image into a 2n× 2ngrayscale carrier image. The quantum Haar wavelet transform is developed and used to decompose quantum images. The diagonal detail coefficients of the carrier image are obtained from the image decomposition. Then, according to the watermark image information, the diagonal wavelet coefficients are either unchanged or slightly modified. Since all of the used quantum operations are invertible, extraction of the watermark image is performed in a straightforward manner by reversing the watermarking embedding process. Finally, the proposed quantum image watermarking scheme is simulated on a classical computer and evaluated under different carrier and watermark images. The simulation results and performance analyses indicate the high performance of the presented watermarking scheme in terms of the similarity between the watermarked and carrier images.
The quantum image encryption and authentication algorithm are proposed based on quantum affine transformation and quantum Haar wavelet transform. This algorithm is divided into two main steps: encryption and embedding. During the encryption phase, affine transform and Hénon map are combined to carry out scrambling-diffusion processing on the plaintext image. In the phase of embedding, the color carrier image undergoes decomposition using the quantum Haar wavelet transform. Then the ciphertext image information and identity information are hidden into the wavelet domain of the color carrier image, and the final transmitted image has visual significance. The suggested algorithm significantly minimizes the risk of image data compromise during transmission and realizes the dual security of encryption and authentication. The security and the robustness of the algorithm are analyzed by using the evaluation indexes of image encryption and information hiding. The findings indicate that the introduced algorithm offers superior security compared to alternative methods, ensuring a more dependable and safeguarded transmission of image data.