The goal of image registration is to align two or more images of the same scene obtained at different times, from different perspectives, or sensors such as MRI, X-ray and CT. This step is required to facilitate automatic segmentation for tumour detection or to inform further decisions in treatment planning. It is an important and challenging subject which usually involves high storage, computational cost and dealing with distorted and occluded data. The paradigm behind image registration is to find a reasonable transformation so that the template image becomes similar to the so-called given reference image. Through such transformation, information from these images can be compared or combined. This thesis deals with the mathematical modelling of image registration by way of energy minimisation of a functional. We propose a new decomposition model for image registration which combines parametric transformation and non-parametric deformation. The first category of methods is based on a small number of parameters and for the second category the transformation is based on a functional map (or discretely a large number of parameters) with a regularisation term. We choose one cubic B-spline based model and the linear curvature model for the parametric and non-parametric parts respectively where the overall deformation consists of both global and local displacement for effective image registration. Some results for synthetic and real images will be presented to illustrate the effectiveness of the new model in contrast with the individual models. We then propose a novel variational model for image registration which employs Gaussian curvature as a regulariser. The model is motivated by the surface restoration work in geometric processing [21]. An effective numerical solver is provided for the model using an augmented Lagrangian method. Numerical experiments show that the new model outperforms three competing models based on, respectively, the linear curvature [24], the mean curvature [19] and the diffeomorphic demon models [93] in terms of robustness and accuracy. Finally, we present an improved model for joint segmentation and registration based on active contour without edges. The proposed model is motivated by an earlier model [58] and the linear curvature model [24]. Numerical results show that the new model outperforms the existing model for registration and segmentation of one or multiple objects in the image. The proposed model also leads to improved registration results when features exist inside the object.
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