This thesis considers self-adaptive regularisation methods, focusing particularly on new, multiplicative methods, in which the cost functional is constructed as a product of two terms, rather than the more usual sum of a fidelity term and a regularisation term. By re-formulating the multiplicative regularisation model in the framework of the alternating minimisation algorithm, we were able to obtain a series of rigorous theoretical results, as well as formulating a number of new models in both multiplicative and additive form. The first two chapters of my thesis set the scene of my research. Chapter 1 gives a general review of the field of inverse problems and common regularisation strategies, while Chapter 2 provides relevant technical details as mathematical preliminaries. The multiplicative regularisation model by Abubakar et al (2004) falls into the category of self-adaptive methods, where the regularisation strength is automatically adjusted in the model. By investigating the model and implementing it on various examples, I demonstrated its power for deblurring piecewise constant images with the presence of noise with high amplitude and various distributions (Chapter 3). I also discovered a possible improvement of this model by the introduction an extra parameter μ, and came up with a formula to determine its most appropriate value in a straightforward manner. The derivation and numerical validation or this formula is presented in Chapter 4. This parameter μ supplements Abubakar’s multiplicative method, and plays an important role in the model: it enables the multiplicative model to reach its full potential, without adding any significant effort in parameter tuning. Despite its numerical strength, there are barely any theoretical results regarding the multiplicative type of regularisation, which motivates me to carry out further research in this aspect. Inspired by Charbonnier et al (1997) who provided an additive model with regularisation strength spatially controlled by a sequence of self-adapted weight functions bn, I re-formulated the multiplicative regularisation model in the framework of alternating minimisation algorithm. This results in a series of new models of the multiplicative type. In Chapter 5 I presented two new models MMR and MSSP equipped with two-step and three-step alternating minimization algorithm respectively. The scaling parameter δ is fixed in the former model while it is self-adaptive based on an additional recurrence relation in the latter model. In both models, the objective cost functional Cn is monotonically decreasing and convergent, while the image intensity un exhibits semi-convergence nature. Both models are capable of incorporating different potential functions in the objective cost functional, and require no extra tuning parameter μ in the algorithm. Numerically they exhibit similar behaviours as Abubakar’s multiplicative method in terms of high noise level tolerance and robustness over different noise distributions. In Chapter 6 I presented a third, enhanced multiplicative model (EMM), which employs not only a three-step minimisation with self-adaptive weight function bn and scaling parameter δn, but also the same augmented recurrence relation as discussed in Chapter 4 with steering parameter μ. This model leads to promising results both theoretically and numerically. It is a novel approach with enhanced performance exceeding all the multiplicative type of models presented in this dissertation.

Cambridge Trust Scholarship awarded by Cambridge Commonwealth, European and International Trust, 2014 Trinity Hall Research Studentship awarded by Trinity Hall College, 2014