Convolutional neural networks -CNNs- have become an essential tool for solving many machine vision and machine learning problems. A major element of these networks is the convolution operator which essentially computes the inner product between a weight vector and the vectorized image patches extracted by sliding a window in the image planes of the previous layer. In this paper, we propose two classes of surrogate functions for the inner product operation inherent in the convolution operator and so attain two generalizations of the convolution operator. The first one is based on the class of positive definite kernel functions where their application is justified by the kernel trick. The second one is based on the class of similarity measures defined according to some distance function. We justify this by tracing back to the basic idea behind the neocognitron which is the ancestor of CNNs. Both of these methods are then further generalized by allowing a monotonically increasing function -possibly depending on the weight vector- to be applied subsequently. Like any trainable parameter in a neural network, the template pattern and the parameters of the kernel/distance function are trained with the back-propagation algorithm. As an aside, we use the proposed framework to justify the use of sine activation function in CNNs. Additionally, we discovered a family of generalized convolution operators which is based on the convex combination of the dot-product and the negative squared Euclidean distance functions. Our experiments on the MNIST dataset show that the performance of ordinary CNNs can be achieved by generalized CNNs based on weighted L1/L2 distances, proving the applicability of the proposed generalization of the convolutional neural networks.

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