Registration method

We propose to adapt the above method so that it becomes fully automated. To achieve this, we define the matching criterion as the distance between the cortical and ventricular system surfaces and the equivalent structures in the CBA database. The CBA surface is used as the reference surface from which the distance transformation is generated. The MRI surface is used as the mobile surface, on which the transformation is applied in order to fit the reference surface.
First, we segment the cortex from the MRI, using a variant of the directional watershed transform described by Warscotte [177]. The resulting object is simplified using a mathematical morphology closing that merges the sulci with the cortex.
The set of possible transformations $\overrightarrow{p}=(p_x,p_y,p_z) \rightarrow T(\overrightarrow{p})$ is defined using N basis functions f_j and 3.N parameters $\alpha_{ij}$.

$$T_i(\overrightarrow{p}) \ = \ \sum_{j=0}^{N-1} \alpha _{ij} . f_j(\overrightarrow{p})$$ (7.2)

with $i \in \{x,y,z\}$. The affine transform is represented with N=4, $f_j(\overrightarrow{p}) = 1$, p_x, p_y, p_z, and 12 coefficients $\alpha_{ij}$. The 3D polynomial second degree transform uses N=10, $f_j(\overrightarrow{p}) = 1$, p_x, p_y, p_z, p_x², p_y², p_z², p_xp_y, p_xp_z, p_yp_z, and 30 coefficients $\alpha_{ij}$. The effects of some of these elementary transformations in 2D are illustrated at figure .

  

Figure 7.5: Set of elementary first and second degree transformations in the direction of the x-axis.

The matching itself is performed in two steps. First, the best affine transform is found using the cortical surface only in the matching criterion. Then, the second degree coefficients are optimized using both the cortical surface and the ventricular system as matching criterion. In both cases, the minimization of the criterion is performed using a gradient-descent algorithm in the 3N-dimensional parameter space, after ortho-normalizing the functions f_j relatively to the mobile surface.