Present image encryption algorithms ignore the protection of the visual security of encrypted images. Therefore, it is valuable to combine a new cosine chaotic map (CCM) with Bayesian compressive sensing (BCS) and thus propose a visually meaningful image encryption (VMIE) algorithm. Firstly, a new one-dimensional chaotic map based on the cosine function is proposed to construct a controlled measurement matrix. In addition, the proposed new CCM can better disrupt the strong correlation of images. Secondly, the wavelet packet coefficient matrix of a plain image is scrambled by 2D Arnold scrambling algorithm. Then, the scrambled secret image is compressed and encrypted by a chaotic measurement matrix and bidirectional modulo-adding diffusion strategy. Finally, a visually meaningful ciphertext image is obtained by embedding the secret image into the carrier image after game-of-life (GOL) mixed scrambling through the least significant bit embedding algorithm. Simulation results and security analysis show that the proposed algorithm is feasible and efficient on the premise of ensuring visual security and decryption quality.