We describe a fully automatic framework for classification of two types of dementia based on the differences in the shape of brain structures. We consider Alzheimer's disease (AD), mild cognitive impairment of individuals who converted to AD within 18 months (MCIc), and normal controls (NC). Our approach uses statistical learning and a feature space consisting of projection-based shape descriptors, allowing for canonical representation of brain regions. Our framework automatically identifies the structures most affected by the disease. We evaluate our results by comparing to other methods using a standardized data set of 375 adults available from the Alzheimer's Disease Neuroimaging Initiative (ADNI). Our framework is sensitive to identifying the onset of Alzheimer's disease, achieving up to 88.13% accuracy in classifying MCIc versus NC, outperforming previous methods.