Multi-focus-image-fusion is a crucial embranchment of image processing. Many methods have been developed from different perspectives to solve this problem. Among them, the sparse representation (SR)-based and convolutional neural network (CNN)-based fusion methods have been widely used. Fusing the source image patches, the SR-based model is essentially a local method with a nonlinear fusion rule. On the other hand, the direct mapping between the source images follows the decision map which is learned via CNN. The fusion is a global one with a linear fusion rule. Combining the advantages of the above two methods, a novel fusion method that applies CNN to assist SR is proposed for the purpose of gaining a fused image with more precise and abundant information. In the proposed method, source image patches were fused based on SR and the new weight obtained by CNN. Experimental results demonstrate that the proposed method clearly outperforms existing state-of-the-art methods in addition to SR and CNN in terms of both visual perception and objective evaluation metrics, and the computational complexity is greatly reduced. Experimental results demonstrate that the proposed method not only clearly outperforms the SR and CNN methods in terms of visual perception and objective evaluation indicators, but is also significantly better than other state-of-the-art methods since our computational complexity is greatly reduced.

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By virtue of taking values in a commutative subalgebra [Formula: see text] of Lie algebra [Formula: see text], we construct the [Formula: see text]-Heisenberg ferromagnet model which contains many Heisenberg ferromagnet-type equations. Moreover, we investigate the integrable properties of the [Formula: see text]-Heisenberg ferromagnet model. In terms of the gauge transformation, the gauge equivalent counterpart of the [Formula: see text]-Heisenberg ferromagnet model has been presented. Based on the differential geometry of curves and surfaces, the corresponding geometrical equivalence between the [Formula: see text]-Heisenberg ferromagnet model and [Formula: see text]-nonlinear Schrödinger equation has also been established. Furthermore, we also discuss the [Formula: see text]-generalized inhomogeneous Heisenberg ferromagnet model.

The Heisenberg supermagnet model is an important supersymmetric integrable system which is the super extension of the Heisenberg ferromagnet model. By virtue of introducing the general auxiliary matrix variables, we construct a new [Formula: see text]-dimensional generalized integrable Heisenberg supermagnet models under two constraints. Meanwhile, we establish their corresponding gauge equivalent counterparts. Moreover, we derive new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.