As the special imaging principle of the interference hyperspectral image data, there are lots of vertical interference stripes in every frames. The stripes’ positions are fixed, and their pixel values are very high. Horizontal displacements also exist in the background between the frames. This special characteristics will destroy the regular structure of the original interference hyperspectral image data, which will also lead to the direct application of compressive sensing theory and traditional compression algorithms can’t get the ideal effect. As the interference stripes signals and the background signals have different characteristics themselves, the orthogonal bases which can sparse represent them will also be different. According to this thought, in this paper the morphological component analysis (MCA) is adopted to separate the interference stripes signals and background signals. As the huge amount of interference hyperspectral image will lead to slow iterative convergence speed and low computational efficiency of the traditional MCA algorithm, an improved MCA algorithm is also proposed according to the characteristics of the interference hyperspectral image data, the conditions of iterative convergence is improved, the iteration will be terminated when the error of the separated image signals and the original image signals are almost unchanged. And according to the thought that the orthogonal basis can sparse represent the corresponding signals but cannot sparse represent other signals, an adaptive update mode of the threshold is also proposed in order to accelerate the computational speed of the traditional MCA algorithm, in the proposed algorithm, the projected coefficients of image signals at the different orthogonal bases are calculated and compared in order to get the minimum value and the maximum value of threshold, and the average value of them is chosen as an optimal threshold value for the adaptive update mode. The experimental results prove that whether LASIS and LAMIS image data, the traditional MCA algorithm can separate the interference stripes signals and background signals very well, and make the interference hyperspectral image decomposition perfectly, and the improved MCA algorithm not only keep the perfect results of the traditional MCA algorithm, but also can reduce the times of iteration and meet the iterative convergence conditions much faster than the traditional MCA algorithm, which will also provide a very good solution for the new theory of compressive sensing.