A Survey on the New Generation of Deep Learning in Image Processing <s> 2) CV-CNN <s> A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors, followed by (2) taking the absolute value of every entry of the resulting vectors, followed by (3) local averaging. For processing real-valued random vectors, complex-valued convnets can be viewed as data-driven multiscale windowed power spectra, data-driven multiscale windowed absolute spectra, data-driven multiwavelet absolute values, or (in their most general configuration) data-driven nonlinear multiwavelet packets. Indeed, complex-valued convnets can calculate multiscale windowed spectra when the convnet filters are windowed complex-valued exponentials. Standard real-valued convnets, using rectified linear units (ReLUs), sigmoidal (e.g., logistic or tanh) nonlinearities, or max pooling, for example, do not obviously exhibit the same exact correspondence with data-driven wavelets (whereas for complex-valued convnets, the correspondence is much more than just a vague analogy). Courtesy of the exact correspondence, the remarkably rich and rigorous body of mathematical analysis for wavelets applies directly to (complex-valued) convnets. <s> BIB001 </s> A Survey on the New Generation of Deep Learning in Image Processing <s> 2) CV-CNN <s> Following the great success of deep convolutional neural networks (CNNs) in computer vision, this paper proposes a complex-valued CNN (CV-CNN) specifically for synthetic aperture radar (SAR) image interpretation. It utilizes both amplitude and phase information of complex SAR imagery. All elements of CNN including input-output layer, convolution layer, activation function, and pooling layer are extended to the complex domain. Moreover, a complex backpropagation algorithm based on stochastic gradient descent is derived for CV-CNN training. The proposed CV-CNN is then tested on the typical polarimetric SAR image classification task which classifies each pixel into known terrain types via supervised training. Experiments with the benchmark data sets of Flevoland and Oberpfaffenhofen show that the classification error can be further reduced if employing CV-CNN instead of conventional real-valued CNN with the same degrees of freedom. The performance of CV-CNN is comparable to that of existing state-of-the-art methods in terms of overall classification accuracy. <s> BIB002 </s> A Survey on the New Generation of Deep Learning in Image Processing <s> 2) CV-CNN <s> In this paper, complex-valued convolutional neural networks are presented, by giving the full deduction of the gradient descent algorithm for training this type of networks. The performances of convolutional neural networks in the real-valued domain for image classification gave rise to the idea of extending them to the complex-valued domain, also. Real-valued image classification experiments done using the MNIST and CIFAR-10 datasets have shown an improvement in performance of complex-valued convolutional neural networks over their real-valued counterparts. <s> BIB003
To fully explore the rich information embedded in complexvalued (CV) images, CV-CNN has to be developed BIB002 , BIB003 , BIB001 . The architecture of a CV-CNN can regard VOLUME 7, 2019 as a variant of the deep neural networks, which not only takes complex data as input but also propagates the phase information through all layers. For CV-CNN, all the layers of the networks, including the convolutional layer, pooling layer, fully connected layer, should be fully CV. The details of each layer in its CV version are presented in the following. Firstly, for the convlutional layer, the complex output feature maps O (l) k are computed by the convolution between all the previous layers input feature maps O (l−1) ∈ C n×m and a set of filters { (l) k ∈ C v×v , k = 1, 2, · · · , K }, and then add a bias {γ (l) k ∈ C, k = 1, 2, · · · , K }. where 'C' denotes the complex domain and the superscript is its dimension. where σ (·) is an element-wise non-linear function, and ∀z ∈ C. We have where z = R(z) + jI(z). And this CV-convolution operation can be calculated by Secondly, for the pooling layer, In other words, pooling layers can regard as sub-sampling layers, and pooling helps to make the representation invariant to small shifts and distortions of the input. Finally, after several CV convolutional and CV pooling layers, CV fully-connected layers are usually added to act as classification layers. The expected outputỹ and predictive output t can be written as, where f is the non-linear function, and real-valued vector y ∈ R c is the label of input sample, and CV vector v can be obtained by flattening the CV matrix V (l) . If the softmax is applied to CV-CNN, the result is not a probability due to its CV input. Therefore, the final output is the classifier, and the least-squares loss function is adopted in CV-CNN. Experiments show that the classification error can further reduce if employing CV-CNN instead of conventional realvalued CNN with the same degrees of freedom BIB003 ,  .