Large Scale Inverse Problems

This book is thesecond volume of a three volume series recording the "Radon Special Semester 2011 on Multiscale Simulation &amp Analysis in Energy and the Environment" that took placein Linz, Austria, October 3-7, 2011. This volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications. The solution of inverse problems is fundamental to a wide variety of applications such as weather forecasting, medical tomography, and oil exploration. Regularisation techniques are needed to ensure solutions of sufficient quality to be useful, and soundly theoretically based. This book addresses the common techniques required for all the applications, and is thus truly interdisciplinary. This collection of survey articles focusses on the large inverse problems commonly arising in simulation and forecasting in the earth sciences

Signal processing with Fourier analysis, novel algorithms and applications

Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions, also analogously known as sinusoidal modeling. The original idea of Fourier had a profound impact on mathematical analysis, physics and engineering because it diagonalizes time-invariant convolution operators. In the past signal processing was a topic that stayed almost exclusively in electrical engineering, where only the experts could cancel noise, compress and reconstruct signals. Nowadays it is almost ubiquitous, as everyone now deals with modern digital signals. Medical imaging, wireless communications and power systems of the future will experience more data processing conditions and wider range of applications requirements than the systems of today. Such systems will require more powerful, efficient and flexible signal processing algorithms that are well designed to handle such needs. No matter how advanced our hardware technology becomes we will still need intelligent and efficient algorithms to address the growing demands in signal processing. In this thesis, we investigate novel techniques to solve a suite of four fundamental problems in signal processing that have a wide range of applications. The relevant equations, literature of signal processing applications, analysis and final numerical algorithms/methods to solve them using Fourier analysis are discussed for different applications in the electrical engineering/computer science. The first four chapters cover the following topics of central importance in the field of signal processing: • Fast Phasor Estimation using Adaptive Signal Processing (Chapter 2) • Frequency Estimation from Nonuniform Samples (Chapter 3) • 2D Polar and 3D Spherical Polar Nonuniform Discrete Fourier Transform (Chapter 4) • Robust 3D registration using Spherical Polar Discrete Fourier Transform and Spherical Harmonics (Chapter 5) Even though each of these four methods discussed may seem completely disparate, the underlying motivation for more efficient processing by exploiting the Fourier domain signal structure remains the same. The main contribution of this thesis is the innovation in the analysis, synthesis, discretization of certain well known problems like phasor estimation, frequency estimation, computations of a particular non-uniform Fourier transform and signal registration on the transformed domain. We conduct propositions and evaluations of certain applications relevant algorithms such as, frequency estimation algorithm using non-uniform sampling, polar and spherical polar Fourier transform. The techniques proposed are also useful in the field of computer vision and medical imaging. From a practical perspective, the proposed algorithms are shown to improve the existing solutions in the respective fields where they are applied/evaluated. The formulation and final proposition is shown to have a variety of benefits. Future work with potentials in medical imaging, directional wavelets, volume rendering, video/3D object classifications, high dimensional registration are also discussed in the final chapter. Finally, in the spirit of reproducible research we release the implementation of these algorithms to the public using Github

Development of registration methods for cardiovascular anatomy and function using advanced 3T MRI, 320-slice CT and PET imaging

Different medical imaging modalities provide complementary anatomical and functional information. One increasingly important use of such information is in the clinical management of cardiovascular disease. Multi-modality data is helping improve diagnosis accuracy, and individualize treatment. The Clinical Research Imaging Centre at the University of Edinburgh, has been involved in a number of cardiovascular clinical trials using longitudinal computed tomography (CT) and multi-parametric magnetic resonance (MR) imaging. The critical image processing technique that combines the information from all these different datasets is known as image registration, which is the topic of this thesis. Image registration, especially multi-modality and multi-parametric registration, remains a challenging field in medical image analysis. The new registration methods described in this work were all developed in response to genuine challenges in on-going clinical studies. These methods have been evaluated using data from these studies. In order to gain an insight into the building blocks of image registration methods, the thesis begins with a comprehensive literature review of state-of-the-art algorithms. This is followed by a description of the first registration method I developed to help track inflammation in aortic abdominal aneurysms. It registers multi-modality and multi-parametric images, with new contrast agents. The registration framework uses a semi-automatically generated region of interest around the aorta. The aorta is aligned based on a combination of the centres of the regions of interest and intensity matching. The method achieved sub-voxel accuracy. The second clinical study involved cardiac data. The first framework failed to register many of these datasets, because the cardiac data suffers from a common artefact of magnetic resonance images, namely intensity inhomogeneity. Thus I developed a new preprocessing technique that is able to correct the artefacts in the functional data using data from the anatomical scans. The registration framework, with this preprocessing step and new particle swarm optimizer, achieved significantly improved registration results on the cardiac data, and was validated quantitatively using neuro images from a clinical study of neonates. Although on average the new framework achieved accurate results, when processing data corrupted by severe artefacts and noise, premature convergence of the optimizer is still a common problem. To overcome this, I invented a new optimization method, that achieves more robust convergence by encoding prior knowledge of registration. The registration results from this new registration-oriented optimizer are more accurate than other general-purpose particle swarm optimization methods commonly applied to registration problems. In summary, this thesis describes a series of novel developments to an image registration framework, aimed to improve accuracy, robustness and speed. The resulting registration framework was applied to, and validated by, different types of images taken from several ongoing clinical trials. In the future, this framework could be extended to include more diverse transformation models, aided by new machine learning techniques. It may also be applied to the registration of other types and modalities of imaging data

Combinatorial optimisation for arterial image segmentation.

Cardiovascular disease is one of the leading causes of the mortality in the western world. Many imaging modalities have been used to diagnose cardiovascular diseases. However, each has different forms of noise and artifacts that make the medical image analysis field important and challenging. This thesis is concerned with developing fully automatic segmentation methods for cross-sectional coronary arterial imaging in particular, intra-vascular ultrasound and optical coherence tomography, by incorporating prior and tracking information without any user intervention, to effectively overcome various image artifacts and occlusions. Combinatorial optimisation methods are proposed to solve the segmentation problem in polynomial time. A node-weighted directed graph is constructed so that the vessel border delineation is considered as computing a minimum closed set. A set of complementary edge and texture features is extracted. Single and double interface segmentation methods are introduced. Novel optimisation of the boundary energy function is proposed based on a supervised classification method. Shape prior model is incorporated into the segmentation framework based on global and local information through the energy function design and graph construction. A combination of cross-sectional segmentation and longitudinal tracking is proposed using the Kalman filter and the hidden Markov model. The border is parameterised using the radial basis functions. The Kalman filter is used to adapt the inter-frame constraints between every two consecutive frames to obtain coherent temporal segmentation. An HMM-based border tracking method is also proposed in which the emission probability is derived from both the classification-based cost function and the shape prior model. The optimal sequence of the hidden states is computed using the Viterbi algorithm. Both qualitative and quantitative results on thousands of images show superior performance of the proposed methods compared to a number of state-of-the-art segmentation methods

Dynamic Thermal Imaging for Intraoperative Monitoring of Neuronal Activity and Cortical Perfusion

Neurosurgery is a demanding medical discipline that requires a complex interplay of several neuroimaging techniques. This allows structural as well as functional information to be recovered and then visualized to the surgeon. In the case of tumor resections this approach allows more fine-grained differentiation of healthy and pathological tissue which positively influences the postoperative outcome as well as the patient's quality of life. In this work, we will discuss several approaches to establish thermal imaging as a novel neuroimaging technique to primarily visualize neural activity and perfusion state in case of ischaemic stroke. Both applications require novel methods for data-preprocessing, visualization, pattern recognition as well as regression analysis of intraoperative thermal imaging. Online multimodal integration of preoperative and intraoperative data is accomplished by a 2D-3D image registration and image fusion framework with an average accuracy of 2.46 mm. In navigated surgeries, the proposed framework generally provides all necessary tools to project intraoperative 2D imaging data onto preoperative 3D volumetric datasets like 3D MR or CT imaging. Additionally, a fast machine learning framework for the recognition of cortical NaCl rinsings will be discussed throughout this thesis. Hereby, the standardized quantification of tissue perfusion by means of an approximated heating model can be achieved. Classifying the parameters of these models yields a map of connected areas, for which we have shown that these areas correlate with the demarcation caused by an ischaemic stroke segmented in postoperative CT datasets. Finally, a semiparametric regression model has been developed for intraoperative neural activity monitoring of the somatosensory cortex by somatosensory evoked potentials. These results were correlated with neural activity of optical imaging. We found that thermal imaging yields comparable results, yet doesn't share the limitations of optical imaging. In this thesis we would like to emphasize that thermal imaging depicts a novel and valid tool for both intraoperative functional and structural neuroimaging

Decoding Neural Signals with Computational Models: A Systematic Review of Invasive BMI

There are significant milestones in modern human's civilization in which mankind stepped into a different level of life with a new spectrum of possibilities and comfort. From fire-lighting technology and wheeled wagons to writing, electricity and the Internet, each one changed our lives dramatically. In this paper, we take a deep look into the invasive Brain Machine Interface (BMI), an ambitious and cutting-edge technology which has the potential to be another important milestone in human civilization. Not only beneficial for patients with severe medical conditions, the invasive BMI technology can significantly impact different technologies and almost every aspect of human's life. We review the biological and engineering concepts that underpin the implementation of BMI applications. There are various essential techniques that are necessary for making invasive BMI applications a reality. We review these through providing an analysis of (i) possible applications of invasive BMI technology, (ii) the methods and devices for detecting and decoding brain signals, as well as (iii) possible options for stimulating signals into human's brain. Finally, we discuss the challenges and opportunities of invasive BMI for further development in the area.Comment: 51 pages, 14 figures, review articl

Automatic segmentation of cross-sectional coronary arterial images

We present a novel approach to segment coronary cross-sectional images acquired using catheterization imaging techniques, i.e. intra-vascular ultrasound (IVUS) and optical coherence tomography (OCT). The proposed approach combines cross-sectional segmentation with longitudinal tracking in order to tackle various forms of imaging artifacts and to achieve consistent segmentation. A node-weighted directed graph is constructed on two consecutive cross-sectional frames with embedded shape constraints within individual cross-sections or frames and between consecutive frames. The intra-frame constraints are derived from a set of training samples and are embedded in both graph construction and its cost function. The inter-frame constraints are imposed by tracking the borders of interest across multiple frames. The coronary images are transformed from Cartesian coordinates to polar coordinates. Graph partition can then be formulated as searching an optimal interface in the node-weighted directed graph without user initialization. It also allows efficient parametrization of the border using radial basis function (RBF) and thus reduces the tracking of a large number of border points to a very few RBF centers. Moreover, we carry out supervised column-wise tissue classification in order to automatically optimize the feature selection. Instead of empirically assigning weights to different feature detectors, we dynamically and automatically adapt those weighting depending on the tissue compositions in each individual column of pixels