Patients who develop Alzheimer's disease (AD) have passed through a prodromal stage called mild cognitive impairment (MCI) but, not every MCI patient will develop AD. Structural magnetic resonance imaging (MRI) has the potential to capture the differences between the groups of MCI: patients who converted to dementia due to AD (MCIc) and those who remained stable (MCInc). Sulci and folding structures are affected due to the neurodegenerarive nature of AD. A characterization method for these structures is proposed by performing a multi-spherical mapping and a curvelet decomposition at different levels of the brain. 829 MCI subjects from the ADNI initiative were studied at 24, 36 and 60 months of conversion. A random forest classification model was used to validate the methodology achieving an AUC = 0.77 with high-dimensional feature vectors and AUC = 0.70 when 5% of the features are selected. This suggests discriminating differences in sulci and gyri depth between non-converters and converters to AD.