$C^{2}N^{2}$: Complex-Valued Contourlet Neural Network

Complex-valued convolutional neural networks (CV-CNN) have recently gained recognition in feature representation learning. It implements the repeated application of the operations in convolution, local average pooling, and the absolute value of the resulting vectors. However, it is only conducted in the complex spatial domain, and lacks effective representation of directionality, singularity, and regularity in the complex spectral domain for anomaly detection of images. This is the key to feature learning representation of high-order singularity. To solve this problem, a complex-valued contourlet neural network (C
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N
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) is proposed in this article. It is novel in this sense that, different from the CV-CNN in the spatial domain, the spectral stream of C
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N
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can enhance the multiresolution sparse representation of nonsubsampled contourlet (NSCT) with multiscales and multidirections for images. Furthermore, the spectral feature integration module is proposed to capture the statistical properties of the NSCT coefficients. It is shown that the proposed network can improve the distinguishability of feature learning and classification ability in theoretical analysis and experiments on three benchmark datasets (Flevoland, Xi'an, and Germany) compared with developed methods. Polarimetric synthetic aperture radar image classification is widely used in the fields of agriculture, forestry, and military. It must be emphasized that there is potential in effective feature learning representation and the generalization capability of C
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N
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in deep learning, recognition, and interpretation.