Block compressed sensing with random permutation (BCS-RP) has been shown to be very effective for image Encryption-then-Compression (ETC) applications. However, in the BCS-RP scheme, the statistical information of the blocks is disclosed, because the encryption is conducted within each small block of the image. To solve this problem, a two-dimension compressed sensing (2DCS) with 2D random permutation (2DRP) strategy for image ETC applications is proposed in this letter, where the 2DRP strategy is used for encrypting the image and the 2DCS scheme is used for compressing the encrypted image. Compared with the BCS-RP scheme, the proposed approach has two benefits. Firstly, it offers better security. Secondly, it obtains a significant gain of peak signal-to-noise ratio (PSNR) of the reconstructed-images.

Block compressed sensing (CS) with optimal permutation is a promising method to improve sampling efficiency in CS-based image compression. However, the existing optimal permutation scheme brings a large amount of extra data to encode the permutation information because it needs to know the permutation information to accomplish signal reconstruction. When the extra data is taken into consideration, the improvement in sampling efficiency of this method is limited. In order to solve this problem, a new optimal permutation strategy for block CS (BCS) is proposed. Based on the proposed permutation strategy, an improved optimal permutation based BCS method called BCS-NOP (BCS with new optimal permutation) is proposed in this paper. Simulation results show that the proposed approach reduces the amount of extra data to encode the permutation information significantly and thereby improves the sampling efficiency compared with the existing optimal permutation based BCS approach.