How to achieve the efficient compression and high-quality reconstruction on encrypted signals is a fundamental problem with scientific significance and potential applications for cloud computing in the big-data era. Current researches on this topic mainly exploit statistic properties of cover media in the encryption (i.e., the cloud client) or compression (i.e., the cloud platform) side, which would either increase the computational burden on the encryption side or result in the statistical information leakage and thus would hinder its practical applications. Our preliminary research and analysis indicates that a more practical way for compression of encrypted signals is to fully exploit the statistic property of cover media at the receiver side that has sufficient computational capability and the encryption key. This motivates us to deploy the Markov random field (MRF) to characterize the spatial statistical distribution of cover image, represents the MRF with a factor graph by applying the theory of factor graph, and further integrates it with the factor graph of the low-density parity code (LDPC) decoding, which yields a graph model for image reconstruction. Through this line of thoughts, in this proposal we will exploit statistical characteristics of binary and grey images to construct several reconstruction graph models for lossless compression, in which two/multiple-state MRF and binary/n-ary LDPC are well integrated in an optimized manner. Furthermore, we will also extend our research to the case of color image and to the scenario of lossy compression. It can be expected that the proposal would achieve higher compression ratios at the same reconstruction quality, and thus would well facilitate the development and application of cloud computing in the big-data era and reversible data hiding in the encrypted domain.