The morphology observed in the brains of patients affected by Alzheimer's disease (AD) is a combination of different biological processes, such as normal aging and the pathological matter loss specific to AD. The ability to differentiate between these biological factors is fundamental to reliably evaluate pathological AD-related structural changes, especially in the earliest phase of the disease, at prodromal and preclinical stages. Here we propose a method based on non-linear image registration to estimate and analyze from observed brain morphologies the relative contributions from aging and pathology. In particular, we first define a longitudinal model of the brain's normal aging process from serial T1-weight magnetic resonance imaging scans of 65 healthy participants. The longitudinal model is then used as a reference for the cross-sectional analysis. Given a new brain image, we then estimate its anatomical age relative to the aging model; this is defined as a morphological age shift with respect to the average age of the healthy population at baseline. Finally, we define the specific morphological process as the remainder of the observed anatomy after the removal of the estimated normal aging process. Experimental results from 105 healthy participants, 110 subjects with mild cognitive impairment (MCI), 86 with MCI converted to AD, and 134 AD patients provide a novel description of the anatomical changes observed across the AD time span: normal aging, normal aging at risk, conversion to MCI, and the latest stages of AD. More advanced AD stages are associated with an increased morphological age shift in the brain and with strong disease-specific morphological changes affecting mainly ventricles, temporal poles, the entorhinal cortex, and hippocampi. Our model shows that AD is characterized by localized disease-specific brain changes as well as by an accelerated global aging process. This method may thus represent a more precise instrument to identify potential clinical outcomes in clinical trials for disease modifying drugs.