In order to distinguish tissue specific gray-level transitions, such as occur at the bounding surfaces of the cortical layer, we have designed a gray-level-based local operator to obtain the likelihood of each voxel lying on the surface separating specific tissue types, such as gray and white matter, instead of using the gradient information alone.
We make the assumption of an image I, in which voxels belonging to tissue A are independently drawn from a Gaussian distribution G(pA, aA), and voxels belonging to tissue B are independently drawn from , °B). Thus, at a possible boundary with normal direction 6 dividing the neighborhood of this site into parts R1 and R2, we have:
The direction and magnitude of the estimate are given by the maximum of pAB .
In Fig. 1, we show an example of the result from our local operator, applied to the gray-white transition and to the gray-CSF transition. These results can be used for detecting, for example, the outer and inner cortical surfaces respectively [43-45].
We represent curves  and surfaces  using a Fourier parameterization. It is a strong model concisely represented in terms of parameters and easily allows the incorporation of prior shape information. Associated parameter probability distributions are used to introduce a bias toward an expected range of shapes. A surface is represented by three coordinate functions of two surface parameters as x(u, v) = (x(u, v), y(u, v), z(u, v)) where each is represented by a Fourier parametrization:
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