Image segmentation is a fundamental task in image analysis that aims at partitioning an image into sub-regions or at representing the image into something that is more meaningful and easier to analyse. Variational image segmentation models have become very popular in recent years, especially global image segmentation models which aim to segment all objects in an image. Given a set of user-defined prior points, selective variational models aim to segment selectively one object only. Time marching methods with semi-implicit schemes (gradient descents) or additive operator splitting (AOS) are used frequently to solve the resulting partial differential equation (PDE) of Euler Lagrange equations derived from these models. For images of moderate size, such methods are effective. However, to segment images of large size, urgent need exists to develop fast iterative solvers. We first propose an optimisation based multilevel algorithm for efficiently solving a class of nonconvex selective segmentation models. In literature, the models are originally solved based on optimise-discretise scheme where the PDE derived from the models are numerically solved using AOS method. The multilevel method we use is based on discretise-optimise scheme where minimisation of a variational problem is solved directly without using a PDE. Numerical results on synthetic and real medical data show that good segmentation quality is obtained and, as expected, excellent efficiency is observed in reducing computational time compare to AOS method. Secondly, we propose a new convex selective segmentation model, allowing a global minimiser to be found independently of initialisation. The existing convex segmentation model however is expensive to solve and suffers from parameter sensitivity due to existence of a highly nonlinear term in the formulation. Our new formulation using primal-dual framework is able to reduce the complexity of the existing model. To speed up the segmentation process, we also develop a multilevel algorithm for the new convex segmentation model. Experiments using synthetic and real medical data shows that the new model is less sensitive to parameters compared to the existing convex model and an optimal computational time is achieved. Many application fields such as medical imaging, geological surveying and computational fluid dynamics can greatly benefit from 3-D selective segmentation as a 3-D representation carries more information compared to 2-D representation. This information is highly useful for example in medical surgery planning. However, there exist little literature addressing selective segmentation in 3-D and their formulations are nonconvex. Hence, we present an extension of the multilevel algorithm and convex selective variational image segmentation model above into 3-D framework. Numerical tests show that the proposed model is effective and the algorithm is efficient in locally segmenting 3-D complex image structures. Due to natural complexity of real images, the targeted objects might be occluded by other ones or some parts of them may not be distinguished from the background. For example, in medical image analysis, targeted tumour might be blended by other ones or some part of them may be occluded by other organs or tumour or some object boundaries even missing due to imaging conditions. The grey intensity selective based segmentation models might not be well suited to do the segmentation task as the models heavily rely on grey intensity values of the given image. In the final study, we develop two new selective segmentation models which impose curvature constraints on the formulations to restore those boundaries that are missing or not well defined by the grey intensity images. On top of that, we develop multilevel algorithms to solve these higher order minimisation problems. Numerical tests demonstrated that the models give satisfactory results in optimum computational time when compared to other existing models.