In this paper, we propose a new two-dimensional Sine improved Logistic iterative chaotic map with infinite collapse (ICMIC) modulation map (2D-SLIM) based on the improved two-dimensional closed-loop modulation coupling model. The performance analysis results show that it has a large parameter space, large Lyapunov exponents and high complexity. Combining this map with compressive sensing , a fast image encryption algorithm is proposed. In this algorithm, a plain image is measured by two circular measurement matrices from two directions, and then the image is re-encrypted by employing the row and column encryption to simultaneously achieve permutation and diffusion. In addition, the SHA-512 hash values of the plain image are generated to calculate the initial conditions of the chaotic map with secret keys, which greatly improve the ability to withstand the known plaintext and chosen plaintext attacks . Simulation and performance analysis verify that this algorithm has acceptable compression, high security and low time complexity.