基于统计数学的伪随机序列与图像正交性分析

在数字图像水印、图像加密等领域，经常需要一个与图像正交的伪随机序列。为了获得具有较好正交效果的伪随机序列，基于统计数学的方法，推导出了一个自然图像与伪随机序列两者互相关的二阶数字特征在空域和频域的表达式。结果显示，具有高通型频谱特征的伪随机序列，例如游程受限(run-length limited，RLL)序列，在与图像的正交性方面，比目前广泛使用的白噪声序列性能更优。统计实验也证实了上述结论和该数学模型的有效性。另外，为了快速生成2维RLL序列，还给出了一种简便易行的2维RLL序列生成算法。

Statistical Analysis of the Orthogonality Between Imagesand Pseudo-random Sequences

In digital image watermarking and image encryption, researchers often require a pseudo random sequence which is statistically orthogonal to an image. The white noise sequences, such as m sequence, are often used in these situations. But our study shows they are not optimum on orthogonal property. To get a better orthogonality, we studied the cross correlation of unartificial images and pseudo random sequences by means of statistical mathematics method. The second order expectation of the cross correlation value was determined in both space and frequency domains. The expression suggests that sequences which are high pass in frequency domain, like Run Length Limited (RLL) sequences, have better orthogonal character with unartificial images than white noise sequences which are widely used at present. The conclusion and the validity of our mathematical model were also proved by the result of statistical experiments. In order to generate 2D RLL sequences rapidly, we developed a simple and convenient algorithm. The experiments that confirmed 2D RLL sequences have better orthogonality with natural images than m sequences.