The restoration of degraded images is one important application of image processing. The classical approach of image restoration, such as low-pass filter method, is usually stressed on the numerical error but with a disadvantage in visual quality of blurred texture. Therefore, a new method of image restoration, based upon image model by Compound Gauss-Markov(CGM) Random Fields, using MAP(maximum a posteriori probability) approach focused on image texture effect has been proved to be helpful. However, the contour of the restored image and numerical error for the method is poor because the conventional CGM model uses fixed global parameters for the whole image. To improve these disadvantages, we adopt the adjustable parameters method to estimate model parameters and restore the image. But the parameter estimation for the CGM model is difficult since the CGM model has 80 interdependent parameters. Therefore, we first adopt the parameter reduction approach to reduce the complexity of parameter estimation. Finally, the initial value set of the parameters is important. The different initial value might produce different results. The experiment results show that the proposed method using adjustable parameters has good numerical error and visual quality than the conventional methods using fixed parameters.

This Compound Gauss-Markov image model has been proven helpful in image restoration. In this model, a pixel in the image random field is determined by the surrounding pixels according to a predetermined line field. In this thesis, we restored the noisy image based upon the traditional Compound Gauss-Markov image field without the constraint of the model parameters introduced in the original work. The image is restored in two steps iteratively: restoring the line field by the assumed image field and restoring the image field by the just computed line field. Two methods are proposed to replace the traditional method in solving for the line field. They are probability method and vector method. In probability method, we break away from the limitation of the energy function Vcl(L) and the mystical system parameters Ckll(m,n) and£mw2. In vector method, the line field appears more reasonable than the original method. The image restored by our methods has a similar visual quality but a better numerical value than the original method.

Nanomanufacturing is critical to the future growth of U.S. manufacturing. Yet the process yield of current nanodevices is typically 10% or less. Particularly in nanomaterials growth, there may exist large variability across the sites on a substrate, which could lead to variability in properties. Essential to the reduction of variability is to mathematically describe the spatial variation of nanostructure. This research therefore aims at a method of modeling and estimating nanostructure morphology variation for process yield improvement. This method consists of (1) morphology variation modeling based on Gaussian Markov random field (GMRF) theory, and (2) maximum likelihood estimation (MLE) of morphology variation model based on measurement data. The research challenge lies in the proper definition and estimation of the interactions among neighboring nanostructures. To model morphology variation, nanostructures on all sites are collectively described as a GMRF. The morphology variation model serves for the space-time growth model of nanostructures. The probability structure of the GMRF is specified by a so-called simultaneous autoregressive scheme, which defines the neighborhood systems for any site on a substrate. The neighborhood system characterizes the interactions among adjacent nanostructures by determining neighbors and their influence on a given site in terms of conditional auto-regression. The conditional auto-regression representation uniquely determines the precision matrix of the GMRF. Simulation of nanostructure morphology variation is conducted for various neighborhood structures. Considering the boundary effects, both finite lattice and infinite lattice models are discussed. The simultaneous autoregressive scheme of the GMRF is estimated via the maximum likelihood estimation (MLE) method. The MLE estimation of morphology variation requires the approximation of the determinant of the precision matrix in the GMRF. The absolute term in the double Fourier expansion of a determinant function is used to approximate the coefficients in the precision matrix. Since the conditional MLE estimates of the parameters are affected by coding the date, different coding schemes are considered in the estimation based on numerical simulation and the data collected from SEM images. The results show that the nanostructure morphology variation modeling and estimation method could provide tools for yield improvement in nanomanufacturing.

In today’s machine learning research, probabilistic graphical models are used extensively to model complicated systems with uncertainty, to help understanding of the problems, and to help inference and predict unknown events. For inference tasks, exact inference methods such as junction tree algorithms exist, but they suffer from exponential growth of cluster size and thus is not able to handle large and highly connected graphs. Approximate inference methods do not try to find exact probabilities, but rather give results that improve as algorithm runs. Gibbs sampling, as one of the approximate inference methods, has gained lots of traction and is used extensively in inference tasks, due to its ease of understanding and implementation. However, as problem size grows, even the faster algorithm needs a speed boost to meet application requirement. The number of variables in an application graphical model can range from tens of thousands to billions, depending on problem domain. The original sequential Gibbs sampling may not return satisfactory result in limited time. Thus, in this thesis, we investigate in ways to speed up Gibbs sampling. We will study ways to do better initialization, blocking variables to be sampled together, as well as using simulated annealing. These are the methods that modifies the algorithm itself. We will also investigate in ways to parallelize the algorithm. An algorithm is parallelizable if some steps do not depend on other steps, and we will find out such dependency in Gibbs sampling. We will discuss how the choice of different hardware and software architecture will affect the parallelization result. We will use optical flow problem as an example to demonstrate the various speed up methods we investigated. An optical flow method tries to find out the movements of small image patches between two images in a temporal sequence. We demonstrate how we can model it using probabilistic graphical model, and solve it using Gibbs sampling. The result of using sequential Gibbs sampling is demonstrated, with comparisons from using various speed up methods and other optical flow methods.

One of the main classes of spatial epidemiological studies is disease mapping, where the main aim is to describe the overall disease distribution on a map, for example, to highlight areas of elevated or lowered mortality or morbidity risk, or to identify important social or environmental risk factors adjusting for the spatial distribution of the disease. This thesis focused and proposed solutions to two most commonly seen obstacles in disease mapping applications, the changing census boundaries due to long study period and data aggregation for patients' confidentiality. In disease mapping, when target diseases have low prevalence, the study usually covers a long time period to accumulate sufficient cases. However, during this period, numerous irregular changes in the census regions on which population is reported may occur, which complicates inferences. A new model was developed for the case when the exact location of the cases is available, consisting of a continuous random spatial surface and fixed effects for time and ages of individuals. The process is modelled on a fine grid, approximating the underlying continuous risk surface with Gaussian Markov Random Field and Bayesian inference is performed using integrated nested Laplace approximations. The model was applied to clinical data on the location of residence at the time of diagnosis of new Lupus cases in Toronto, Canada, for the 40 years to 2007, with the aim of finding areas of abnormally high risk. Predicted risk surfaces and posterior exceedance probabilities are produced for Lupus and, for comparison, Psoriatic Arthritis data from the same clinic. Simulation studies are also carried out to better understand the performance of the proposed model as well as to compare with existing methods. When the exact locations of the cases are not known, inference is complicated by the uncertainty of case locations due to data aggregation on census regions for confidentiality. Conventional modelling relies on the census boundaries that are unrelated to the biological process being modelled, and may result in stronger spatial dependence in less populated regions which dominate the map. A new model was developed consisting of a continuous random spatial surface with aggregated responses and fixed covariate effects on census region levels. The continuous spatial surface was approximated by Markov random field, greatly reduces the computational complexity. The process was modelled on a lattice of fine grid cells and Bayesian inference was performed using Markov Chain Monte Carlo with data augmentation. Simulation studies were carried out to assess performance of the proposed model and to compare with the conventional Besag-York-Molli\'e model as well as model assuming exact locations are known. Receiver operating characteristic curves and Mean Integrated Squared Errors were used as measures of performance. For the application, surveillance data on the locations of residence at the time of diagnosis of syphilis cases in North Carolina, for the 9 years to 2007 are modelled with the aim of finding areas of abnormally high risk. Predicted risk surfaces and posterior exceedance probabilities are also produced, identifying Lumberton as a ``syphilis hotspot".

One of the main classes of spatial epidemiological studies is disease mapping, where the main aim is to describe the overall disease distribution on a map, for example, to highlight areas of elevated or lowered mortality or morbidity risk, or to identify important social or environmental risk factors adjusting for the spatial distribution of the disease. This thesis focused and proposed solutions to two most commonly seen obstacles in disease mapping applications, the changing census boundaries due to long study period and data aggregation for patients' confidentiality. In disease mapping, when target diseases have low prevalence, the study usually covers a long time period to accumulate sufficient cases. However, during this period, numerous irregular changes in the census regions on which population is reported may occur, which complicates inferences. A new model was developed for the case when the exact location of the cases is available, consisting of a continuous random spatial surface and fixed effects for time and ages of individuals. The process is modelled on a fine grid, approximating the underlying continuous risk surface with Gaussian Markov Random Field and Bayesian inference is performed using integrated nested Laplace approximations. The model was applied to clinical data on the location of residence at the time of diagnosis of new Lupus cases in Toronto, Canada, for the 40 years to 2007, with the aim of finding areas of abnormally high risk. Predicted risk surfaces and posterior exceedance probabilities are produced for Lupus and, for comparison, Psoriatic Arthritis data from the same clinic. Simulation studies are also carried out to better understand the performance of the proposed model as well as to compare with existing methods. When the exact locations of the cases are not known, inference is complicated by the uncertainty of case locations due to data aggregation on census regions for confidentiality. Conventional modelling relies on the census boundaries that are unrelated to the biological process being modelled, and may result in stronger spatial dependence in less populated regions which dominate the map. A new model was developed consisting of a continuous random spatial surface with aggregated responses and fixed covariate effects on census region levels. The continuous spatial surface was approximated by Markov random field, greatly reduces the computational complexity. The process was modelled on a lattice of fine grid cells and Bayesian inference was performed using Markov Chain Monte Carlo with data augmentation. Simulation studies were carried out to assess performance of the proposed model and to compare with the conventional Besag-York-Molli\'e model as well as model assuming exact locations are known. Receiver operating characteristic curves and Mean Integrated Squared Errors were used as measures of performance. For the application, surveillance data on the locations of residence at the time of diagnosis of syphilis cases in North Carolina, for the 9 years to 2007 are modelled with the aim of finding areas of abnormally high risk. Predicted risk surfaces and posterior exceedance probabilities are also produced, identifying Lumberton as a ``syphilis hotspot".

Random heterogeneous, scale-dependent structures can be observed from many image sources, especially from remote sensing and scientific imaging. Examples include slices of porous media data showing pores of various sizes, and a remote sensing image including small and large sea-ice blocks. Meanwhile, rather than the images of phenomena themselves, there are many image processing and analysis problems requiring to deal with \emph{discrete-state} fields according to a labeled underlying property, such as mineral porosity extracted from microscope images, or an ice type map estimated from a sea-ice image. In many cases, if discrete-state problems are associated with heterogeneous, scale-dependent spatial structures, we will have to deal with complex discrete state fields. Although scale-dependent image modeling methods are common for continuous-state problems, models for discrete-state cases have not been well studied in the literature. Therefore, a fundamental difficulty will arise which is how to represent such complex discrete-state fields. Considering the success of hidden field methods in representing heterogenous behaviours and the capability of hierarchical field methods in modeling scale-dependent spatial features, we propose a Hidden Hierarchical Markov Field (HHMF) approach, which combines the idea of hierarchical fields with hidden fields, for dealing with the discrete field modeling challenge. However, to define a general HHMF modeling structure to cover all possible situations is difficult. In this research, we use two image application problems to describe the proposed modeling methods: one for scientific image (porous media image) reconstruction and the other for remote-sensing image synthesis. For modeling discrete-state fields with a spatially separable complex behaviour, such as porous media images with nonoverlapped heterogeneous pores, we propose a Parallel HHMF model, which can decomposes a complex behaviour into a set of separated, simple behaviours over scale, and then represents each of these with a hierarchical field. Alternatively, discrete fields with a highly heterogeneous behaviour, such as a sea-ice image with multiple types of ice at various scales, which are not spatially separable but arranged more as a partition tree, leads to the proposed Tree-Structured HHMF model. According to the proposed approach, a complex, multi-label field can be repeatedly partitioned into a set of binary/ternary fields, each of which can be further handled by a hierarchical field.

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The Gaussian geostatistical model has been widely used in Bayesian modeling of spatial data. A core difficulty for this model is at inverting the n x n covariance matrix, where n is a sample size. The computational complexity of matrix inversion increases as O(n3). This difficulty is involved in almost all statistical inferences approaches of the model, such as Kriging and Bayesian modeling. In Bayesian inference, the inverse of covariance matrix needs to be evaluated at each iteration in posterior simulations, so Bayesian approach is infeasible for large sample size n due to the current computational power limit. In this dissertation, we propose two approaches to address this computational issue, namely, the auxiliary lattice model (ALM) approach and the Bayesian site selection (BSS) approach. The key feature of ALM is to introduce a latent regular lattice which links Gaussian Markov Random Field (GMRF) with Gaussian Field (GF) of the observations. The GMRF on the auxiliary lattice represents an approximation to the Gaussian process. The distinctive feature of ALM from other approximations lies in that ALM avoids completely the problem of the matrix inversion by using analytical likelihood of GMRF. The computational complexity of ALM is rather attractive, which increase linearly with sample size. The second approach, Bayesian site selection (BSS), attempts to reduce the dimension of data through a smart selection of a representative subset of the observations. The BSS method first split the observations into two parts, the observations near the target prediction sites (part I) and their remaining (part II). Then, by treating the observations in part I as response variable and those in part II as explanatory variables, BSS forms a regression model which relates all observations through a conditional likelihood derived from the original model. The dimension of the data can then be reduced by applying a stochastic variable selection procedure to the regression model, which selects only a subset of the part II data as explanatory data. BSS can provide us more understanding to the underlying true Gaussian process, as it directly works on the original process without any approximations involved. The practical performance of ALM and BSS will be illustrated with simulated data and real data sets.

Changes in retinal structure are related to different eye diseases. Various retinal imaging techniques, such as fundus imaging and optical coherence tomography (OCT) imaging modalities, have been developed for non-intrusive ophthalmology diagnoses according to the vasculature changes. However, it is time consuming or even impossible for ophthalmologists to manually label all the retinal structures from fundus images and OCT images. Therefore, computer aided diagnosis system for retinal imaging plays an important role in the assessment of ophthalmologic diseases and cardiovascular disorders. The aim of this PhD thesis is to develop segmentation methods to extract clinically useful information from these retinal images, which are acquired from different imaging modalities. In other words, we built the segmentation methods to extract important structures from both 2D fundus images and 3D OCT images. In the first part of my PhD project, two novel level set based methods were proposed for detecting the blood vessels and optic discs from fundus images. The first one integrates Chan-Vese's energy minimizing active contour method with the edge constraint term and Gaussian Mixture Model based term for blood vessels segmentation, while the second method combines the edge constraint term, the distance regularisation term and the shape-prior term for locating the optic disc. Both methods include the pre-processing stage, used for removing noise and enhancing the contrast between the object and the background. Three automated layer segmentation methods were built for segmenting intra-retinal layers from 3D OCT macular and optic nerve head images in the second part of my PhD project. The first two methods combine different methods according to the data characteristics. First, eight boundaries of the intra-retinal layers were detected from the 3D OCT macular images and the thickness maps of the seven layers were produced. Second, four boundaries of the intra-retinal layers were located from 3D optic nerve head images and the thickness maps of the Retinal Nerve Fiber Layer (RNFL) were plotted. Finally, the choroidal layer segmentation method based on the Level Set framework was designed, which embedded with the distance regularisation term, edge constraint term and Markov Random Field modelled region term. The thickness map of the choroidal layer was calculated and shown.