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In this paper, a novel shift-invariant dual-tree complex shearlet transform (SIDCST) is constructed and applied to infrared and visible image fusion. Firstly, the mathematical morphology is used for the source images. Then, the images are decomposed by SIDCST to obtain the low frequency sub-band coefficients and high frequency sub-band coefficients. For the low frequency sub-band coefficients, a novel sparse representation (SR)-based fusion rule is presented. For the high frequency sub-band coefficients, a scheme based on the theory of adaptive dual-channel pulse coupled neural network (2APCNN) is presented, and the energy of edge is used for the external input of 2APCNN. Finally, the fused image is obtained by performing the inverse SIDCST. The experimental results show that the proposed approach can obtain state-of-the-art performance compared with conventional image fusion methods in terms of both objective evaluation criteria and visual quality. [ABSTRACT FROM AUTHOR]