In general , a watermark system must satisfy the two basic requirements : imperceptibility and robustness . in this thesis , we proposed a frequency - based watermarking algorithm using toral automorphisms , in which a kind of toral automorphism is employed to mix a recognizable binary image ( bw ) , which contains the copyright information , into a noise - like signal 算法简单描述如下：首先用一个环形自同构映射把要嵌入的数字水印映射成伪随机噪声，然后把该伪随机噪声嵌入到宿主数据频域的低频段中，接着把得到的频域系数进行相应的逆变换得到嵌入水印的图象，最后对该嵌入水印的图象进行视觉掩蔽处理。
Firstly , some basic properties of the limit shadowing are given ; secondly , we give the characterization of both lin - ear automorphisms and linear flows on r ~ ( n ) with the limit shadowing property ; thirdly , as applications we prove that the hyperbolic endomorphisms on t ~ ( n ) have the limit shadowing properties , smale " horseshoes " have the same prop - erties on their invariant sets 首先，给出了极限跟踪性的一些基本性质；其次，得到了n维欧氏空间上线性自同构及线性流具有极限跟踪性的特征；最后，作为应用证明了双曲环面自同态以及smale “马蹄”在其不变集上具有极限跟踪性。
The embedded bw also successfully survives cropping , filtering , noise and multi - watermarking etc . the new ideas the thesis proposed are : first , frequency - based watermarking idea using toral automorphisms is really a novel one , although , in the literature , there are some waterniarking schemes using toral autpmorphisms , all of which are based on space ; second , some improvements of the quantization function contribute to the algorithm ' s performances to some degree 另外该算法对图象裁剪、滤波、噪声和多重水印处理等都具有较强的鲁棒性。本文提出了两个新思想：第一，虽然现有文献已经提出了利用环形自同构映射的数字水印处理算法，但这种算法是基于空域的，而利用环形自同构映射基于频域的水印处理算法，目前文献上未见报道。
