二维双尺度矩形映射及其在图像置乱上的应用

2D Bi-scale Rectangular Mapping and its Application in Image Scrambling

传统的二维等尺度矩阵变换不能直接对矩形图像进行置乱,对于任意矩形图像需通过正方形扩展或分块处理,由此带来了一次迭代过程中额外的运算代价.为此,提出一种用于任意矩形图像置乱的二维双尺度矩形映射,并证明了双尺度矩形映射是任意二维双模线性映射满足一一映射的充分必要条件;然后结合二维等尺度矩阵变换和二维双尺度三角映射2类特殊映射的逆映射,给出了双尺度矩形映射的逆映射.实验结果表明了文中映射对任意矩形图像置乱的有效性,对矩形图像置乱和恢复的低代价性,以及对剪切、擦除和JPEG有损压缩攻击的鲁棒性.与已有方法相比,文中映射具有最小的一次置乱和恢复的代价、更大的变换阵生成空间,且无需计算最小可恢复周期.

The conventional 2D equiscale matrix transform is unable to directly scramble rectangle images. For a rectangle image, it needs to expand it into a square image or divide it into several square images, which brings extra computational cost. To address this problem, this study proposes a 2D bi- scale rectangular mapping for rectangle image scrambling, and it is proved that the proposed 2D biscale rectangular mapping is the necessary and sufficient condition for any 2D dual-module mapping to be a one-one mapping. The inverse mapping of the 2D bi-scale rectangular mapping is also proposed by combining two kinds of special inverse mappings of 2D equiscale matrix transform and 2D bi-scale triangular mapping. The experimental results show that the proposed mappings are of validity in scrambling any rectangle images, of low cost in scrambling and recovering rectangle image, and of robustness to erasing, cropping and compressing attacks. In contrast to the conventional scrambling methods, the proposed mappings no longer need to compute the minimum recovering period, and have the least iteration cost and the largest transform matrix generation space.