The Convolutional Neural Network (CNN) has been successfully applied in many fields during recent decades; however it lacks the ability to utilize prior domain knowledge when dealing with many realistic problems. We present a framework called Geometric Operator Convolutional Neural Network (GO-CNN) that uses domain knowledge, wherein the kernel of the first convolutional layer is replaced with a kernel generated by a geometric operator function. This framework integrates many conventional geometric operators, which allows it to adapt to a diverse range of problems. Under certain conditions, we theoretically analyze the convergence and the bound of the generalization errors between GO-CNNs and common CNNs. Although the geometric operator convolution kernels have fewer trainable parameters than common convolution kernels, the experimental results indicate that GO-CNN performs more accurately than common CNN on CIFAR-10/100. Furthermore, GO-CNN reduces dependence on the amount of training examples and enhances adversarial stability. Convolutional Neural Networks have been successfully applied in many fields during recent decades, but the theoretical understanding of the deep neural network is still in the preliminary stages. Although CNNs have strong expressive abilities, they have two clear deficiencies. First, as complex functional mappings, CNNs, like black boxes, cannot take full advantage of domain knowledge and prior information. Second, when little data is available for a certain task, CNNs' generalization ability weakens. This is due to overfitting, which may occur due to the large number of parameters and the large model size. Stemming from these two defects, a great deal of research has been done to modify CNNs BID7 Wang et al., 2018; Sarwar et al., 2017) .Before CNNs were applied, traditional geometric operators had developed quite well. Each geometric operator represents the precipitation of domain knowledge and prior information. For example, the Sobel operator (Works) is a discrete difference operator, which can extract image edge information for edge detection. The Schmid operator (Schmid, 2001 ) is an isotropic circular operator, which extracts texture information from images for face recognition. The Histogram of Oriented Gradients (HOG) BID8 ) is a statistic operator of gradient direction, which extracts edge direction distributions from images for pedestrian detection and other uses.Many computer vision tasks require domain knowledge and prior information. For example, in BID2 , the texture information from the image is used for an auxiliary diagnosis of a fracture. Geometric operators can make use of domain knowledge and prior information, but cannot automatically change parameter values by learning from data. Convolutional Neural Networks have strong data expression abilities and learning abilities, but they struggle to make use of domain knowledge. For better data learning, we have combined the two. It is natural to directly use geometric operators for pre-processing, and then classify the data through a Convolutional Neural Network (Yao et al., 2016) . However, this method uses human experience to select geometric operator parameter values, and then carries out the Convolutional Neural Network learning separately. This method is a kind of two-stage technique, and without reducing parameter redundancy in a Convolutional Neural Network, it is difficult to achieve global optimization. The method proposed in this paper directly constructs geometric operator convolution and then integrates geometric operator convolution into a Convolutional Neural Network to form a new framework -the Geometric Operator Convolutional Neural Network. This method achieves global optimizations and utilizes the properties of geometric operators.In summary, the contributions of this work are as follows:• This framework can integrates many conventional geometric operators, which reveals its broad customization capabilities when handling diverse problems.• In theory, the same approximation accuracy and generalization error bounds are achieved when geometric operators meet certain conditions.• The Geometric Operator Convolutional Neural Network not only reduces the redundancy of the parameters, but also reduces the dependence on the amount of the training samples.• The Geometric Operator Convolutional Neural Network enhances adversarial stability. In this paper, we present a novel framework named the Geometric Operator Convolution Neural Network, where the kernel in the first convolutional layer is replaced with kernels generated by geometric operator functions. This new network boasts several contributions. Firstly, the GO-CNN is customizable for diverse situations. Secondly, there is a theoretical guarantee in the learning framework of the GO-CNN. Thirdly, the GO-CNN reduces the dependence on training samples. Lastly, the GO-CNN enhances adversarial stability. In the future, we can explore a more appropriate geometric operator convolution block.
Traditional image processing algorithms are combined with Convolutional Neural Networks，a new neural network.
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