Browsing by Subject "Inverse problem"
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Item Boundary inverse problems for networks of vibrating strings with attached masses(University of Minnesota. Institute for Mathematics and Its Applications, 2015-11) Avdonin, Sergei; Avdonina, Nina; Edward, JulianItem Existence, uniqueness and stability of solutions of generalized Tikhonov-Phillips functionals(University of Minnesota. Institute for Mathematics and Its Applications, 2011-08) Mazzieri, G.L.; Spies, Ruben D.; Temperini, Karina G.Item Functional Mapping of Three-Dimensional Electrical Activation in Ventricles(2010-02) Liu, ChenguangVentricular arrhythmias account for nearly 400,000 deaths per year in the United States alone. Electrical mapping of the ventricular activation could facilitate the diagnosis and treatment of arrhythmias, e.g. guiding catheter ablation. To date, both direct mapping and non-contact mapping techniques have been routinely used in electrophysiology labs for obtaining the electrical activity on the endocardial surface. Non-invasive functional mapping methods are also developed to estimate the electrical activity on the epicardium or on both epicardium and endocardium from the body surface measurements. Though successful, the results using above methods are all limited on the surface of the heart and thus cannot directly characterize the cardiac events originating within the myocardial wall. Our group's goal is to develop a functional mapping method to estimate the three-dimensional cardiac electrical activity from either non-invasive body surface potential maps or minimally-invasive intracavitary potential maps, by solving the so-called "inverse problem". Hence the information under the surface of the heart could be revealed to better characterize the cardiac activation. In the present thesis study, the previously developed three-dimensional cardiac electrical imaging (3DCEI) approach has been further investigated. Its function is expanded for not only estimating the global activation sequence but also reconstructing the potential at any myocardial site throughout the ventricle. New algorithms under the 3DCEI scheme are also explored for more powerful mapping capability. The performance of the enhanced 3DCEI approach is rigorously evaluated in both control and diseased swine models when the clinical settings are mimicked. The promising results validate the feasibility of estimating detailed three-dimensional cardiac activation by using the 3DCEI approach, and suggest that 3DCEI has great potential of guiding the clinical management of cardiac arrhythmias in a more efficient way.Item Global saturation of regularization methods for inverse ill-posed problems(University of Minnesota. Institute for Mathematics and Its Applications, 2008-07) Spies, Ruben D.; Temperini, Karina G.Item Hydro-meteorological inverse problems via sparse regularization: advanced frameworks for rainfall spaceborne estimation(2013-09) Ebtehaj, MohammadThe past decades have witnessed a remarkable emergence of new spaceborne and ground-based sources of multiscale remotely sensed geophysical data. Apart from applications related to the study of short-term climatic shifts, availability of these sources of information has improved dramatically our real-time hydro-meteorological forecast skills. Obtaining improved estimates of hydro-meteorological states from a single or multiple low-resolution observations and assimilating them into the background knowledge of a prognostic model have been a subject of growing research in the past decades. In this thesis, with particular emphasis on precipitation data, statistical structure of rainfall images have been thoroughly studied in transform domains (i.e., Fourier and Wavelet). It is mainly found that despite different underlying physical structure of storm events, there are general statistical signatures that can be robustly characterized and exploited as a prior knowledge for solving hydro-meteorological inverse problems such rainfall downscaling, data fusion, retrieval and data assimilation. In particular, it is observed that in the wavelet domain or derivative space, rainfall images are sparse. In other words, a large number of the rainfall expansion coefficients are very close to zero and only a small number of them are significantly non-zero, a manifestation of the non-Gaussian probabilistic structure of rainfall data. To explain this signature, relevant family of probability models including Generalized Gaussian Density (GGD) and a specific class of conditionally linear Gaussian Scale Mixtures (GSM) are studied. Capitalizing on this important but overlooked property of precipitation, new methodologies are proposed to optimally integrate and improve resolution of spaceborne and ground-based precipitation data. In particular, a unified framework is proposed that ties together the problems of downscaling, data fusion and data assimilation via a regularized variational approach, while taking into account the underlying sparsity in an appropriately chosen transform domain. This framework seeks solutions beyond the paradigm of the classic least squares by imposing a proper regularization. The results suggest that sparsity-promoting regularization can reduce uncertainty of estimation in hydro-meteorological inverse problems of downscaling, data fusion, and data assimilation. In continuation of the proposed methodologies, we also introduce a new data driven framework for multisensor spaceborne rainfall retrieval problem and present some preliminary and promising results.Item Magnetic Resonance Based Electrical Properties Tomography (Ept) Using Multi-Channel Transmission For Imaging Human Brain And Animal Cancer Models(2018-05) Wang, YicunElectrical properties (EPs) of biological tissues are determined by tissue constituents, and therefore may provide novel biomarkers for characterization of diseased tissues such as cancer. In addition, accurate quantification of tissue EPs is essential for understanding the biological effects of electromagnetic radiation involved in MRI exams as well as wireless communication. In this dissertation, non-invasive EP imaging methods are proposed based on inverse problems using a plurality of radiofrequency electromagnetic field maps (B1 maps) acquired with ultra-high-field MRI. For human brain imaging, an automatic seed selection strategy is developed for gradient-based Electrical Properties Tomography (gEPT) to provide objective EP values. Reconstruction results of twelve healthy subjects demonstrate that considerable intra- and inter- subject EP heterogeneity resides in the normal brain, which may provide rationale for subject-specific mapping of EPs for improved accuracy in electromagnetic safety evaluation. Furthermore, a generalized technology called “CONtrast Conformed Electrical Properties Tomography (CONCEPT)” is developed based on transmit B1 maps and image sparsity. Numerical simulations and phantom experiments have been performed to quantify its accuracy and sensitivity. For rodent cancer model imaging, Boundary Informed Electrical Properties Tomography (BIEPT) technology is proposed based on a constrained inverse problem that exploits prior information and image sparsity. The imaging platform and BIEPT reconstruction method have been evaluated using simulations, phantom experiments and in vivo cancer xenograft imaging experiments. The reconstructed EPs are compared to multiple conventional MR contrasts as well as histopathology slides to demonstrate their potential value for cancer diagnosis and staging.Item Robust Group Synchronization via Quadratic Programming(2023-06) Wyeth, ColeWe review existing methods for the group synchronization problem and discuss ournovel quadratic programming formulation for estimating the corruption levels in group synchronization, and use these estimates to solve this problem. Our objective function exploits the cycle consistency of the group and we thus refer to our method as detection and estimation of structural consistency (DESC). This general framework can be extended to other algebraic and geometric structures. Our formulation has the following advantages: it can tolerate corruption as high as the information-theoretic bound, it does not require a good initialization for the estimates of group elements, it has a simple interpretation, and under some mild conditions the global minimum of our objective function exactly recovers the corruption levels. We demonstrate the competitive accuracy of our approach on both synthetic and real data experiments of rotation averaging.