Neurosurgery is one of the most demanding specialities with regard to accuracy in surgical procedures. To perform these surgical procedures, neurosurgeons can use brain atlases. A brain atlas contains brain images in which experts have identified anatomical and/or functional structures. Since no patients' brains are the same, it is necessary to adapt the brain atlas to the specific patient's anatomy. This is possible using registration. The applications of registration in medical imaging are numerous. Some of these are: preoperative planning and simulation, interventional radiology, radiological diagnosis, minimally invasive surgery, radiotherapy, intraoperative navigation, computer-assisted surgery, etc. Neurosurgeons and/or neuroradiologists can identify anatomical and functional structures that are difficult to see in most types of medical imaging using registration.
This thesis presents a new method for the registration between brain atlases and Magnetic Resonance images. The work hypotheses are:  
    1.  A deformable brain atlas that uses radial basis functions with compare support and Appearance Active Models will locate anatomical and/or functional brain structures in Magnetic Resonance images with a margin of error from 1 to 2 mm. which is an error of the same magnitude as current surgical procedures. 
    2.  Radial basis functions with compact support have advantages over radial basis functions such as the Thin Plate Spline in the image registration problem. These advantages are: locality of the transformation, control over the nature of the transformation, lower computational cost, and greater numerical stability. 

To verify these hypotheses, the following have been developed:  
    • A new non-rigid registration method of brain atlases in Magnetic Resonance images based on the use of different radial basis functions with compact support. Of these, Wu's radial basis function with compact support has been used for the first time in the field of registration of medical images. 
    • A segmentation method of the cortex and ventricles in Magnetic Resonance images based on the Talairach-Tournoux brain atlas and Appearance Active Models. 
    • An algorithm that allows the 3D reconstruction of the Talairach-Tournoux brain atlas. 

The registration method was quantitatively validated with two structures that are used in the surgical procedure known as deep brain stimulation. In particular, it was validated with the subthalamic nucleus and the red nucleus. Deep brain stimulation is a surgical procedure that is used for treating several disabling neurological symptoms, most commonly the debilitating symptoms of Parkinson's disease, such as trembling, rigidity, strangling, slow movement, and walking problems. The success of deep brain stimulation depends critically on the accuracy in which the electrode is placed over the chosen structure (the subthalamic nucleus, the ventro-intermediate nucleus, or the internal part of the globlus pallidus). These structures are not easily identifiable in most types of medical images. In this thesis, the error offered by the developed system was evaluated with patients to whom deep brain stimulation has been applied (subthalamic nucleus); and in T2-weighted Magnetic Resonance images in which an expert identified the contour of the red nucleus.
This validation was also performed to determine which radial basis function offered the best results. The results indicate that the Wendland function of class 0 and dimension 3 offered the best results, with a mean error of 1.885±0.995  mm. for the subthalamic nucleus and 0.902±0.569  mm. for the red nucleus.
If the results of other authors are compared, it is possible to affirm that the error offered by the developed system when the subthalamic nucleus is located is of the same order of magnitude as most of the works (from 1.5 to 2 mm.) or even lower [244, 193, 109]. On the other hand, the error offered by the system when the red nucleus is located was lower than the method developed by Stancello et al. [243] (the only reference found). The mean error offered by the Stancello et al. method was 1.39 mm. All of these results corroborate the first hypothesis of this thesis.
After a detailed analysis of several radial basis functions in the field of image registration, it is possible to conclude that:
 
    • Radial basis functions with compact support allow the transformation of the registration to be affine, elastic and local. These aspects provide advantages over the traditional radial basis functions such as the Thin Plate Spline. 
    • In relation to the computational aspect, the use of radial basis functions with compact support offers advantages when calculating the transformation: the matrix is disperse, positive-defined (when polynomial accuracy is not used) and well-conditioned. If conditionally positive-defined radial basic functions are used, the matrix is dense, not positive-defined, and very badly conditioned. 

These conclusions corroborate the second hypothesis of this thesis.