A reversible blind image hiding algorithm (ReBIHA) with help of an adaptive embedding mechanism and a compressive sensing technology is presented in this paper, of which can effectively hide the secret image through a natural meaningful carrier image. In the first phase, a new key generation model (KeyGM) is built by combining random numbers and the plaintext, followed by a newly designed four dimensional chaotic map (NewCM) with stronger pseudo-randomness. Then, the generated chaotic sequence is taken to produce a measurement matrix for compressive sensing. In the second phase, the plain image is sparsely processed, confused, encrypted and compressed to get measurements. Afterward, these measurements are subjected to diffusion and binary exchange to achieve a middle cipher image (MCI). In the third stage, a meaningful carrier image is randomly selected, and integer wavelet transform (IWT) is applied to it, getting one low-frequency and three high-frequency coefficients. Then, the pixels of MCI are subjected by splitting and symmetrically shifting transformation for three element matrices. Finally, these matrices are respectively embedded into the high-frequency coefficients through an adaptive mechanism to find replaceable elements, and a final carrier image hiding secrets (CiHS) can be achieved after performing an inverse IWT. The experimental results show that the proposed algorithm can avoid the extra transmission of original carrier image. Moreover, peak signal-to-noise ratio (PSNR) of reconstructed image can surpass 32 dB at compression ratio 0.5. Especially, the proposed adaptive embedding mechanism can not only achieve blind extraction of secrets well, but also ensure the visual quality for the original carrier image.