Although there is no cure for Alzheimer's disease (AD), an accurate early diagnosis is essential for health and social care, and will be of great significance when the course of the disease could be reversed through treatment options. Florbetapir positron emission tomography (¹⁸F-AV-45 PET) is proven to be the most powerful imaging technique to investigate the deposition of amyloid plaques, one of the potential hallmarks of AD, signifying the onset of AD before it changes the brains structure. In this paper, we propose a novel classification algorithm to discriminate the patients having AD, early mild cognitive impairment (MCI), late MCI, and normal control in ¹⁸F-AV-45 PET using shearlet based deep convolutional neural network (CNN). It is known that the conventional CNNs involve convolution and pooling layers, which in fact produce the smoothed representation of data, and this results in losing detailed information. In view of this fact, the conventional CNN is integrated with shearlet transform incorporating the multiresolution details of the data. Once the model is pretrained to transform the input data into a better stacked representation, the resulting final layer is passed to softmax classifier, which returns the probabilities of each class. Through experimental results, it is shown that the performance of the proposed classification framework is superior to that of the traditional CNN in Alzheimer's disease neuroimaging initiative (ADNI) database in terms of classification accuracy. As a result, it has the potential to distinguish the different stages of AD progression with less clinical prior information.