Accelerated gradient methods for the X-ray imaging of solar flares

In this paper we present new optimization strategies for the reconstruction of X-ray images of solar flares by means of the data collected by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). The imaging concept of the satellite is based of rotating modulation collimator instruments, which allow the use of both Fourier imaging approaches and reconstruction techniques based on the straightforward inversion of the modulated count profiles. Although in the last decade a greater attention has been devoted to the former strategies due to their very limited computational cost, here we consider the latter model and investigate the effectiveness of different accelerated gradient methods for the solution of the corresponding constrained minimization problem. Moreover, regularization is introduced through either an early stopping of the iterative procedure, or a Tikhonov term added to the discrepancy function, by means of a discrepancy principle accounting for the Poisson nature of the noise affecting the data

A Novel Iterative Structure for Online Calibration of M-Channel Time-Interleaved ADCs

published_or_final_versio

Image reconstruction from incomplete information

Imperial Users onl

Numerical methods for phase retrieval

In this work we consider the problem of reconstruction of a signal from the magnitude of its Fourier transform, also known as phase retrieval. The problem arises in many areas of astronomy, crystallography, optics, and coherent diffraction imaging (CDI). Our main goal is to develop an efficient reconstruction method based on continuous optimization techniques. Unlike current reconstruction methods, which are based on alternating projections, our approach leads to a much faster and more robust method. However, all previous attempts to employ continuous optimization methods, such as Newton-type algorithms, to the phase retrieval problem failed. In this work we provide an explanation for this failure, and based on this explanation we devise a sufficient condition that allows development of new reconstruction methods---approximately known Fourier phase. We demonstrate that a rough (up to $\pi/2$ radians) Fourier phase estimate practically guarantees successful reconstruction by any reasonable method. We also present a new reconstruction method whose reconstruction time is orders of magnitude faster than that of the current method-of-choice in phase retrieval---Hybrid Input-Output (HIO). Moreover, our method is capable of successful reconstruction even in the situations where HIO is known to fail. We also extended our method to other applications: Fourier domain holography, and interferometry. Additionally we developed a new sparsity-based method for sub-wavelength CDI. Using this method we demonstrated experimental resolution exceeding several times the physical limit imposed by the diffraction light properties (so called diffraction limit).Comment: PhD. Thesi

Under-Sampled Reconstruction Techniques for Accelerated Magnetic Resonance Imaging

Due to physical and biological constraints and requirements on the minimum resolution and SNR, the acquisition time is relatively long in magnetic resonance imaging (MRI). Consequently, a limited number of pulse sequences can be run in a clinical MRI session because of constraints on the total acquisition time due to patient comfort and cost considerations. Therefore, it is strongly desired to reduce the acquisition time without compromising the reconstruction quality. This thesis concerns under-sampled reconstruction techniques for acceleration of MRI acquisitions, i.e., parallel imaging and compressed sensing. While compressed sensing MRI reconstructions are commonly regularized by penalizing the decimated wavelet transform coefficients, it is shown in this thesis that the visual artifacts, associated with the lack of translation-invariance of the wavelet basis in the decimated form, can be avoided by penalizing the undecimated wavelet transform coefficients, i.e., the stationary wavelet transform (SWT). An iterative SWT thresholding algorithm for combined SWT-regularized compressed sensing and parallel imaging reconstruction is presented. Additionally, it is shown that in MRI applications involving multiple sequential acquisitions, e.g., quantitative T1/T2 mapping, the correlation between the successive acquisitions can be incorporated as an additional constraint for joint under-sampled reconstruction, resulting in improved reconstruction performance. While quantitative measures of quality, e.g., reconstruction error with respect to the fully-sampled reference, are commonly used for performance evaluation and comparison of under-sampled reconstructions, this thesis shows that such quantitative measures do not necessarily correlate with the subjective quality of reconstruction as perceived by radiologists and other expert end users. Therefore, unless accompanied by subjective evaluations, quantitative quality measurements/comparisons will be of limited clinical impact. The results of experiments aimed at subjective evaluation/comparison of different under-sampled reconstructions for specific clinical neuroimaging MRI applications are presented in this thesis. One motivation behind the current work was to reduce the acquisition time for relaxation mapping techniques DESPOT1 and DESPOT2. This work also includes a modification to the Driven Equilibrium Single Pulse Observation of T1 with high-speed incorporation of RF field inhomogeneities (DESPOT1-HIFI), resulting in more accurate estimation of T1 values at high strength (3T and higher) magnetic fields

Sampling strategies and reconstruction techniques for magnetic resonance imaging

In magnetic resonance imaging (MRI), samples of the object's spectrum are measured in the spatial frequency domain (k-space). For a number of reasons there is a desire to reduce the time taken to gather measurements. The approach considered is to sample below the Nyquist density, using prior knowledge of the object's support in the spatial domain to enable full reconstruction. The two issues considered are where to position the samples (sampling strategies) and how to form an image (reconstruction techniques). Particular attention is given to a special case of irregular sampling, referred to as Cartesian sampling, in which the samples are located on a Cartesian grid but only constitute a subset of the full grid. A further special case is considered where the sampling scheme repeats periodically, referred to as periodic Cartesian sampling. These types of sampling schemes are applicable to 3-D Cartesian MRI, MRSI, and other modalities that measure a single point in 2-D k-space per echo. The case of general irregular sampling is also considered, which is applicable to spiral sampling, for example. A body of theory concerning Cartesian sampling is developed that has practical implications for how to approach the problem and provides intuition about its nature. It is demonstrated that periodic Cartesian sampling effectively decomposes the problem into a number of much smaller subproblems, which leads to the development of a reconstruction algorithm that exploits these computational advantages. An additional algorithm is developed to predict the regions that could be reconstructed from a particular sampling scheme and support; it can be used to evaluate candidate sampling schemes before measurements are obtained. A number of practical issues are also discussed using illustrative examples. Sample selection algorithms for both Cartesian and periodic Cartesian sampling are developed using heuristic metrics that are fast to compute. The result is a significant reduction in selection time at the expense of a slightly worse conditioned system. The reconstruction problem for a general irregular sampling scheme is also analysed and a reconstruction algorithm developed that trades off computation time for better image quality

Nowe metody przetwarzania losowo próbkowanych wielowymiarowych eksperymentów NMR

The topic of this dissertation is a new algorithm for processing of sparsely sampled data sets from multidimensional nuclear magnetic resonance (NMR) experiments. NMR remains one of the major experimental technique for studying biological macromolecules. However, increasing size of investigated objects poses a challenge for NMR due to rapidly decreasing sensitivity and increasing signal crowding. The first chapter focuses on recent advances in sensitivity enhancements and summarises a few solutions for resolution of spectral overlap. Subsequently, one describes the crucial and limiting problem of signal sampling in multidimensional NMR, which, up to recently, has impeded the widespread use of high-dimensional NMR methods. Major fast acquisition and non-uniform sampling (NUS) approaches are presented. The particular emphasis was put on detailed discussion of competetive approaches to processing of data from NUS experiments. In chapter 3 the new iterative algorithm is proposed for artefact suppression in high-resolution NMR spectra. The detailed description of its design and implementation is given, and followed by comparison with selected processing methods. The efficacy of the algorithm is demonstrated on model synthetic and experimental data. The last chapter of the thesis shows various applications of the proposed method to existing and new four- and five-dimensional NMR experiments. The algorithm is proven most beneficial in challenging applications including spectra for assignment of sidechain resonances in protein and nucleic acids, NOESY spectra for structural analysis, and cross-correlated relaxation measurements for proteins. // Niniejsza praca jest poświecona nowej metodzie przetwarzania danych pochodzących z oszczędnie próbkowanych wielowymiarowych eksperymentów jądrowego rezonansu magnetycznego (ang. Nuclear Magnetic Resonance, NMR). Technika ta jest, obok krystalografii rentgenowskiej, główną eksperymentalną metodą badawczą pozwalającą na określenie struktury i dynamiki makromolekuł o znaczeniu biologicznym. Jednakże NMR napotyka dwie istotne przeszkody w odniesieniu do dużych biomolekuł, a mianowicie gwałtownie pogarszającą się czułość oraz krytyczne zatłoczenie sygnałów w widmach. W rozdziale pierwszym przedstawiono ostatnie osiagnięcia w poprawie czułości technik NMR oraz rozwiązania służące podniesieniu rozdzielczości widm. Następnie opisano kluczowy problem próbkowania wielowymiarowych sygnałów NMR, który do niedawna uniemożliwiał wykorzystanie pełnego potencjału tych technik do rozdzielenia sygnałów. Omówiono pokrótce współczesne podejścia do szybkiej akwizycji i oszczędnego próbkowania sygnałów NMR (ang. non-uniform sampling, NUS). Szczególny nacisk położono na porównanie i dyskusje wad i zalet stosowanych obecnie metod przetwarzania sygnałów niejednorodnie próbkowanych. W rozdziale 3-cim opisano nowy iteracyjny algorytm oparty o transformacje Fouriera, usuwający artefakty oszczędnego próbkowania w wysokorozdzielczych widmach NMR. Szczegółowo omówiono schemat algorytmu oraz jego programową implementację. Rozdział uzupełnia porównanie wyników algorytmu oraz wybranych metod przetwarzania na wysymulowanych oraz modelowych danych eksperymentalnych. W ostatnim rozdziale pracy zademonstrowano użyteczność nowej metody do literaturowych oraz nowych cztero- i pieciowymiarowych eksperymentów NMR. Wśród proponowanych zastosowań wymienić można widma do przypisania sygnałów w łańcuchach bocznych aminokwasów (w białkach) i pierścieniach rybozy (w kwasach rybonukleinowych), widma NOESY służące określeniu struktury trójwymiarowej biomolekuł, oraz pomiary szybkości relaksacji skorelowanej w łańcuchach głównych białek

A unified approach to sparse signal processing

A unified view of the area of sparse signal processing is presented in tutorial form by bringing together various fields in which the property of sparsity has been successfully exploited. For each of these fields, various algorithms and techniques, which have been developed to leverage sparsity, are described succinctly. The common potential benefits of significant reduction in sampling rate and processing manipulations through sparse signal processing are revealed. The key application domains of sparse signal processing are sampling, coding, spectral estimation, array processing, compo-nent analysis, and multipath channel estimation. In terms of the sampling process and reconstruction algorithms, linkages are made with random sampling, compressed sensing and rate of innovation. The redundancy introduced by channel coding i