To address the consumption and security of color images for transmission and storage, a cross-channel color image encryption scheme based on a discrete memristive coupled neuron model and DWT compression is designed in this article. Firstly, the dynamics of the discrete memristive coupled neuron system are analyzed and found to possess the hyperchaotic phenomenon, which provides sufficient security for the encryption scheme. Secondly, the color image processed by discrete wavelet transform (DWT) has a quarter of the previous capacity. Then, the color image is combined with a Hash function, and the resulting Hash sequence is given the initial value of the hyperchaotic system. Next, a particle swarm foraging algorithm (PSFA) is designed to better disrupt the correlation in the RGB channel. Finally, a complementary DNA coding rule is implemented for the further encryption of color images. Simulation results show that even with DWT lossy compression, the recovered image can be clearly seen. The performance analysis illustrates that under the hyperchaotic system, the proposed encryption algorithm brings higher security for color images.