Recovering a linear combination of exponential signals characterized by parameters is highly significant in many MR imaging applications such as parameter mapping and spectroscopy. The parameters carry useful clinical information and can act as biomarkers for various cardiovascular and neurological disorders. However, their accurate estimation requires a large number of high spatial resolution images, resulting in long scan time. One of the ways to reduce scan time is by acquiring undersampled measurements. The recovery of images is usually posed as an optimization problem, which is regularized by functions enforcing sparsity, smoothness or low rank structure. Recently structured matrix priors have gained prominence in many MRI applications because of their superior performance over the aforementioned conventional priors. However, none of them are designed to exploit the smooth exponential structure of the 3D dataset. In this thesis, we exploit the exponential structure of the signal at every pixel location and the spatial smoothness of the parameters to derive a 3D annihilation relation in the Fourier domain. This relation translates into a product of a Hankel/Toeplitz structured matrix, formed from the k-t samples, and a vector of filter coefficients. We show that this matrix has a low rank structure, which is exploited to recover the images from undersampled measurements. We demonstrate the proposed method on the problem of MR parameter mapping. We compare the algorithm with the state-of-the-art methods and observe that the proposed reconstructions and parameter maps have fewer artifacts and errors. We extend the structured low rank framework to correct field inhomogeneity artifacts in MR images. We introduce novel approaches for field map compensation for data acquired using Cartesian and non-Cartesian trajectories. We adopt the time segmentation approach and reformulate the artifact correction problem into a recovery of time series of images from undersampled measurements. Upon recovery, the first image of the series will correspond to the distortion-free image. With the above re-formulation, we can assume that the signal at every pixel follows an exponential signal characterized by field map and the damping constant R2*. We exploit the smooth exponential structure of the 3D dataset to derive a low rank structured matrix prior, similar to the parameter mapping case. We demonstrate the algorithm on spherical MR phantom and human data and show that the artifacts are greatly reduced compared to the uncorrected images. Finally, we develop a structured matrix recovery framework to accelerate cardiac breath-held MRI. We model the cardiac image data as a 3D piecewise constant function. We assume that the zeros of a 3D trigonometric polynomial coincides with the edges of the image data, resulting in a Fourier domain annihilation relation. This relation can be compactly expressed in terms of a structured low rank matrix. We exploit this low rank property to recover the cardiac images from undersampled measurements. We demonstrate the superiority of the proposed technique over conventional sparsity and smoothness based methods. Though the model assumed here is not exponential, yet the proposed algorithm is closely related to that developed for parameter mapping. The direct implementation of the algorithms has a high memory demand and computational complexity due to the formation and storage of a large multi-fold Toeplitz matrix. Till date, the practical utility of such algorithms on high dimensional datasets has been limited due to the aforementioned reasons. We address these issues by introducing novel Fourier domain approximations which result in a fast and memory efficient algorithm for the above-mentioned applications. Such approximations allow us to work with large datasets efficiently and eliminate the need to store the Toeplitz matrix. We note that the algorithm developed for exponential recovery is general enough to be applied to other applications beyond MRI.

The aim of this thesis was to investigate some methods for quantitative analysis of bone structure, particularly techniques which might ultimately be applied post-operatively following orthopaedic reconstruction operations. Initially it was decided to explore the efficacy of MRI in quantifying the bone structure at high resolution by comparing high resolution MRI against 'gold standards' such as Scanning Electron Microscopy (SEM) and optical histology. This basic study provided a measure of the distortions in the morphological bone parameters derived from MR images due to susceptibility artefacts and partial volume effects. The study of bone architecture was then extended to a model of advanced renal osteodystrophy in a growing rat. For this study, high-resolution micro computed tomography (microCT) was used and as a result of the high resolution images obtained, three new bone morphological parameters were introduced to characterise the bone structure. The desire to study bone architecture post-operatively in hip replacements led to a preliminary study on ex-vivo sheep acetabulae following total hip replacement, to determine the extent that the bone architecture could be investigated around the acetabulum. The motivation for studying the acetabulum was based on the high occurrence of debonding at the bone / prosthesis interface. This study demonstrated the superior nature of 3D MRI over conventional x-ray radiographs in early quantitation of fibrous membranes located between the host bone and the non-metallic implant and/or the bone cement. The presence of such fibrous membranes is strongly indicative of failure of the prosthesis. When using clinical MRI to image post-operative hip replacement, the image quality is severely affected by the presence of the metallic implant. The head of the prosthesis is shaped like a metal sphere and is located in the acetabular cup. This problem was investigated by performing simulations of MR images in the presence of the field perturbation induced by the presence of a metal sphere, with the effects of slice excitation and frequency encoding incorporated into the simulations. The simulations were compared with experimental data obtained by imaging a phantom comprising a stainless steel ball bearing immersed in agarose gel. The simulations were used to predict the effects of changing imaging parameters that influence artefact size and also to show how current metal artefact reduction techniques such as view angle tilting (VAT) work and to identify their limitations. It was shown that 2D SE and VAT imaging techniques should not be used when metallic prosthesis are present due to extreme slice distortion, whereas 3D MRI provided a method that has no slice distortion, although the effects of using a frequency encoding gradient still remain.

Tato disertační práce analyzuje artefakty v NMR obrazech. V lékařské praxi mohou artefakty zabránit diagnostice patologických tkání, a proto je třeba tyto nežádoucí jevy odstranit. Vzhledem k nutnosti vyloučit artefakty způsobené nehomogenitou statického magnetického pole jsou v dané souvislosti diskutovány také nové možnosti měření deformací. Práce popisuje metodu pro měření základního pole i RF pole a jejich nehomogenit na základě měření MR obrazů T2 a T2*. Metoda kombinuje akviziční postupy pro spinové a gradientní echo za účelem rozlišení relaxačních časů T2 a T2*. V této souvislosti také vyvstává potřeba provést exponenciální aproximaci relaxačního procesu. Experimentální výsledky pro plastové i měděné vzorky jsou prezentovány v příslušné části práce. Pokud jde o vlastní členení popisovaných jevů, je třeba uvést, že významný problém představují artefakty způsobené magnetickou susceptibilitou. Různé hodnoty magnetické vodivosti na rozhraní mezi dvěma materiály mohou způsobit deformaci magnetického pole, přičemž může dojít i k úplné ztrátě signálu.

The presented dissertation thesis analyses artefacts in diffusion-weighted images. In medical practice, the artefacts can impede the diagnostics of pathological tissues and, therefore, need to be eliminated. As the first step within the thesis, an analysis of the most frequent artefacts in diffusion-weighted images is performed, and the hitherto known approaches to artefact elimination are described. In order to facilitate the reduction of artefacts caused by the inhomogeneity of the static magnetic field and induced by eddy currents, a novel three-measurement method is shown. This technique will find application especially in measuring the diffusion coefficient of isotropic materials. At this point, it is important to note that a significant and commonly found problem is the magnetic susceptibility artefact; different magnetic susceptibility values at the boundary between two materials can cause magnetic field inhomogeneities and even complete loss of the signal. Therefore, we designed a novel method for the measurement of magnetic susceptibility in various samples of magnetically incompatible materials, which do not produce any MR signal. The technique was experimentally verified using a set of differently shaped diamagnetic and paramagnetic samples. In addition to the magnetic susceptibility problem, the thesis presents artefacts such as noise, motion-induced items, hardware limitations, chemical shift, and the dependence of the diffusion coefficient on the temperature. To enable precise measurement of the diffusion coefficient, we proposed a thermal system; in the experiment, it was determined that when the measurement error does not exceed 5%, the temperature change should not be higher than 0,1 °C. In the final sections of the thesis, practical application examples involving the designed methods are shown.

This thesis is concerned with problems of measuring and mapping of magnetic field in MR tomograph, for purpose of magnetic flux density homogeneity optimization. Attention is paid to mapping techniques on rotary symmetric volume and to ways of magnetic fields optimization with utilization of passive and active correction systems. Theoretical analysis of magnetic field decomposition with utilization of spherical harmonics and numerical decomposition is made. Mapping and approximation techniques of basic magnetic field are verified by experiments in the laboratory at the Institute of Scientific Instruments AS CR in Brno.