Hierarchical Bayesian sparse image reconstruction with application to MRFM

This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hierarchical Bayesian model. To overcome the complexity of the posterior distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be used to estimate the image to be recovered, e.g. by maximizing the estimated posterior distribution. In our fully Bayesian approach the posteriors of all the parameters are available. Thus our algorithm provides more information than other previously proposed sparse reconstruction methods that only give a point estimate. The performance of our hierarchical Bayesian sparse reconstruction method is illustrated on synthetic and real data collected from a tobacco virus sample using a prototype MRFM instrument.Comment: v2: final version; IEEE Trans. Image Processing, 200

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

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

Fast, Iterative Image Reconstruction for MRI in the Presence of Field Inhomogeneities

In magnetic resonance imaging, magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast Fourier transform (FFT) methods. Several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities, but these methods assume that the field map that characterizes the off-resonance frequencies is spatially smooth. Recently, iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off-resonance effects. However, straightforward implementations of such iterative methods suffer from inconveniently long computation times. This paper describes a tool for accelerating iterative reconstruction of field-corrected MR images: a novel time-segmented approximation to the MR signal equation. We use a min-max formulation to derive the temporal interpolator. Speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast Fourier transform with normalized root mean squared approximation errors of 0.07%. The proposed method provides fast, accurate, field-corrected image reconstruction even when the field map is not smooth.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86010/1/Fessler69.pd