Li Hai-Sheng, Fan Ping, Peng Huiling, Song Shuxiang, Long Gui-Lu
IEEE Trans Cybern. 2022 Aug;52(8):8467-8480. doi: 10.1109/TCYB.2021.3049509. Epub 2022 Jul 19.
Wavelet transform is being widely used in classical image processing. One-dimension quantum wavelet transforms (QWTs) have been proposed. Generalizations of the 1-D QWT into multilevel and multidimension have been investigated but restricted to the quantum wavelet packet transform (QWPTs), which is the direct product of 1-D QWPTs, and there is no transform between the packets in different dimensions. A 2-D QWT is vital for image processing. We construct the multilevel 2-D QWT's general theory. Explicitly, we built multilevel 2-D Haar QWT and the multilevel Daubechies D4 QWT, respectively. We have given the complete quantum circuits for these wavelet transforms, using both noniterative and iterative methods. Compared to the 1-D QWT and wavelet packet transform, the multilevel 2-D QWT involves the entanglement between components in different degrees. Complexity analysis reveals that the proposed transforms offer exponential speedup over their classical counterparts. Also, the proposed wavelet transforms are used to realize quantum image compression. Simulation results demonstrate that the proposed wavelet transforms are significant and obtain the same results as their classical counterparts with an exponential speedup.
小波变换在经典图像处理中得到了广泛应用。一维量子小波变换（QWT）已被提出。一维QWT到多级和多维的推广已经得到研究，但仅限于量子小波包变换（QWPT），它是一维QWPT的直积，并且不同维度的包之间没有变换。二维QWT对图像处理至关重要。我们构建了多级二维QWT的一般理论。具体而言，我们分别构建了多级二维哈尔QWT和多级达布西耶D4 QWT。我们使用非迭代和迭代方法给出了这些小波变换的完整量子电路。与一维QWT和小波包变换相比，多级二维QWT涉及不同程度分量之间的纠缠。复杂度分析表明，所提出的变换比其经典对应物提供指数级加速。此外，所提出的小波变换用于实现量子图像压缩。仿真结果表明，所提出的小波变换具有重要意义，并且以指数级加速获得了与经典对应物相同的结果。
