In rigid registration, the patient is assumed to be a rigid object, but this is generally not the reality, especially when anatomical sites are affected by inter- and intra-fractional variations. Therefore, the most common registration type is DIR (figure 3.2). To capture the possible deformations in space, such transformation models consist of many more parameters, making convergence difficult and optimisation execution longer. Generally, two non-rigid transformation models are formulated: parametric and non-parametric. Non-parametric models describe the deformation at every voxel by directly calculating a dense 3D deformable vector field (DVF), resulting in a vast number of parameters (∼80 million for an image with 512×512×100 voxels). Several methods formulate the registration as a motion problem and solve it as a fluid or elastic deformation (Horn and Schunck 1981, Bajcsy and Kovačič 1989, Beg et al 2005). It can also be considered as a diffusion process, better known as the demons algorithm (Thirion 1998). In contrast, the parametric deformation models use a parameterised deformation space. Thus, the algorithms predict the values for these parameters instead of directly calculating the deformation for each voxel. A typical parameterisation uses a limited set of basis functions, such as Fourier series, splines or thin plate splines (radial basis functions). The advantage of the parametric model is the fewer degrees of freedom (i.e. the weights of the basis functions), but it is also associated with lower flexibility. A trendy model is a free-form deformation with B-splines (Rueckert et al 1999). The deformation is estimated at a limited number of control points, and the deformation between the control points is calculated with B-splines. The advantage of this approach is the local support, i.e. the deformation of a control point only affects a local region around itself instead of the whole grid, making it a computationally efficient algorithm (Shackleford et al 2010). However, since the B-spline-based DVF