We describe a new approach to estimating the cortical thickness of human brains using magnetic resonance imaging data. Our algorithm is part of a processing chain consisting of a brain segmentation (skull stripping), as well as white and grey matter segmentation procedures. In this paper, only the grey matter segmentation together with the cortical thickness estimation is described. In contrast to many existing methods, our estimation method is voxel-based and does not use any surface meshes. While this fact poses a principal limit on the accuracy that can be achieved by our method, it offers tremendous advantages with respect to practical applicability. In particular, it is applicable to data sets showing severe cortical atrophies that involve areas of high curvature and extremely thin gyral stalks. In contrast to many other methods, it is entirely automatic and very fast with computation times of a few minutes. Our method has been used in two clinical studies involving a total of 27 patients and 23 healthy subjects.