Speckle Noise Reduction via Nonconvex High Total Variation Approach

We address the problem of speckle noise removal. The classical total variation is extensively used in this field to solve such problem, but this method suffers from the staircase-like artifacts and the loss of image details. In order to resolve these problems, a nonconvex total generalized variation (TGV) regularization is used to preserve both edges and details of the images. The TGV regularization which is able to remove the staircase effect has strong theoretical guarantee by means of its high order smooth feature. Our method combines the merits of both the TGV method and the nonconvex variational method and avoids their main drawbacks. Furthermore, we develop an efficient algorithm for solving the nonconvex TGV-based optimization problem. We experimentally demonstrate the excellent performance of the technique, both visually and quantitatively

Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise

Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. On the basis of previous work, we further present a tight and a refined version of the corresponding multiplicative MHDM. We discuss existence and uniqueness of solutions for the proposed models, and additionally, provide convergence properties. Moreover, we present a discrepancy principle stopping criterion which prevents recovering excess noise in the multiscale reconstruction. Through comprehensive numerical experiments and comparisons, we qualitatively and quantitatively evaluate the validity of all proposed models for denoising and deblurring images degraded by multiplicative noise. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image denoising

Stable Backward Diffusion Models that Minimise Convex Energies

The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy to handle numerical scheme. So far, existing stabilisation strategies in literature require sophisticated numerics to solve the underlying initial value problem. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images

A Multiplicative Regularisation for Inverse Problems

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, 201