In this paper, we provide some safe ways to transfer images securely by using cryptography and steganography methods. In order to enhance the security of the image transmission, we introduce a new type of uniformly distributed 2Dhybrid chaos map based on Logistic, Sine and Tent maps, and use the cellular automata and discrete framelet transform in the proposed algorithms and also mix the position of the image pixels by apply kinds of shifts. To show that the proposed algorithms are able to resist various attacks, different types of simulation results and security analysis are used.

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The results in this paper can be divided into three parts. First, we generalize the recent results of Benavente, Christensen and Zakowicz on approximately dual generalized shift-invariant frames on the real line to generalized translation-invariant (GTI) systems on locally compact abelian (LCA) groups. Second, we explain in detail how GTI frames can be realized as [Formula: see text]-frames. Finally, the known results on perturbation of [Formula: see text]-frames and results on perturbation of generalized shift-invariant systems are applied and extended to GTI systems on LCA groups.