In this article, we propose a novel wavelet convolution unit for the image-oriented neural network to integrate wavelet analysis with a vanilla convolution operator to extract deep abstract features more efficiently. On one hand, in order to acquire non-local receptive fields and avoid information loss, we define a new convolution operation by composing a traditional convolution function and approximate and detailed representations after single-scale wavelet decomposition of source images. On the other hand, multi-scale wavelet decomposition is introduced to obtain more comprehensive multi-scale feature information. Then, we fuse all these cross-scale features to improve the problem of inaccurate localization of singular points. Given the novel wavelet convolution unit, we further design a network based on it for fine-grained Alzheimer's disease classifications (i.e., Alzheimer's disease, Normal controls, early mild cognitive impairment, late mild cognitive impairment). Up to now, only a few methods have studied one or several fine-grained classifications, and even fewer methods can achieve both fine-grained and multi-class classifications. We adopt the novel network and diffuse tensor images to achieve fine-grained classifications, which achieved state-of-the-art accuracy for all eight kinds of fine-grained classifications, up to 97.30%, 95.78%, 95.00%, 94.00%, 97.89%, 95.71%, 95.07%, 93.79%. In order to build a reference standard for Alzheimer's disease classifications, we actually implemented all twelve coarse-grained and fine-grained classifications. The results show that the proposed method achieves solidly high accuracy for them. Its classification ability greatly exceeds any kind of existing Alzheimer's disease classification method.