## 2 Results and Discussion
The processed GM and FA images were derived from a dataset composed by a group of 17 patients affected by Alzheimer's disease and a control group of 11 unities. MRI images were acquired using a 3 Tesla Siemens Impact scanner at the Neuroradiology Unit of the Ospedale Maggiore Borgo Trento, Verona, Italy (2008) with a standard head coil. Axial diffusion tensor images (DTI) were acquired with diffusion encoding gradients applied in thirty noncollinear directions and a b-factor of 1000 s/mm\({}^{2}\), plus five images without diffusion weighting. The following parameters were used: TR = 5000 ms, TE = 118 ms, number of slices = 23, slice thickness = 5 mm, slice gap = 1 mm, acquisition matrix = 128 \(\times\)128, FOV = 220 mm, the scan duration was 3 min. FA and directional maps for two subjects are shown in Figure 9, the images were realized by using the Software Tools FSL 4.0 (Figures 9 and 10) (Functional Software Library of the FMRIB Centre, Oxford Centre for Functional Magnetic Resonance Imaging of the Brain-Oxford University) [18,19]. Figure 6:2 (**a**) Divergence map of AVF field; (**b**) Divergence map of AVF field after 60 iterations after 180 iterations. The first processed GM image (Figure 11) refers to a subject of the control group (see the first image of Figure 12). The force field has been evaluated from Equation (9) after eight iterations, the overlapping green curves represent initial contours that are automatically selected from the contour map of divergence. In Figures 13 and 14 we have treated GM images derived from two patients affected by Alzheimer's disease and they refer respectively to the second and third images which are shown in Figure 6:3 Once edges are detected, we produce a boundary representation of gray matter that can be used for an automatic analysis of shapes from the geometrical, metrical or morphological point of view. Figure 6:4 Segmentation of a gray matter image (GM), derived by the Software Tools FSL. Figure 6:5 FA and directional Maps derived by the Software Tools FSL. Figure 6:6 Edge extraction with AVF force field after 8 iterations. Figure 6:7 GM images for a subject of the control group and two patients affected by Alzheimer’s disease. Figure 6:8 Edge extraction with AVF force field after 8 iterations. Figure 6:9 Edge extraction with AVF force field after 8 iterations. Moreover, we test the AVF method with T1-weighted MRI images of our dataset. The initial positions of the active contours are shown in green overlaid on the real image, whereas the outlines of the final contours are drawn in red. Many details of borders are captured both for gray matter and lateral ventricle profiles, whereas the interhemispheric fissure contours results are not completely retrieved, as we can see in \[\vec{v}_{i,j}^{t+1}=\vec{v}_{i,j}^{t}+\frac{1}{4}\Big{[}g_{N}\
abla_{N}\vec{v}_{i,j }^{t}+g_{S}\
abla_{S}\vec{v}_{i,j}^{t}+g_{E}\
abla_{E}\vec{v}_{i,j}^{t}+g_{W} \
abla_{W}\vec{v}_{i,j}^{t}\Big{]}+F\Big{(}\vec{v}_{i,j}^{t}\Big{)}\]0  
For what concerns DTI images, this approach is addressed to demarcate the areas with higher diffusivity of water molecules. We could note the high level of noise present in them; in order to perform a noise reduction, images are initially pre-processed applying a median digital filter (Figure 16). As previously mentioned, the most significant information of a grayscale FA map are taken from zones with high intensity levels corresponding to preferential directions of water diffusion, presumably along axonal bundles. Thus after analyzing the intensity distribution of image pixels (Figures 17 and 18), we have obtained an image segmentation by using thresholds ranging from 180 to 210. Subsequently, edge extraction has been performed through an automatic selection of an initial contour from the level curves of the divergence map (Figures 18-20).