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Enhanced block sparse signal recovery and bayesian hierarchical models with applications
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is carried out within two projects ‘Statistical modelling and intelligentdata sampling in Magnetic resonance imaging (MRI) and positron-emission tomography(PET) measurements for cancer therapy assessment’ and ‘WindCoE -Nordic Wind Energy Center’ during my PhD study. It mainly focuses on applicationsof Bayesian hierarchical models (BHMs) and theoretical developments ofcompressive sensing (CS). Under the first project, Paper I improves the quantityestimation of MRI parametric imaging by utilizing inherent dependent structure inthe image through BHMs; Paper III constructs a theoretically unbiased and asymptoticallynormal estimator of sparsity of a sparsified MR image by using a BHM;Paper IV extends block sparsity estimation from real-valued signal recovery tocomplex-valued signal recovery. It also demonstrates the importance of accuratelyestimating the block sparsity through a sensitivity analysis; Paper V proposes anew measure, i.e. q-ratio block constrained minimal singular value, of measurementmatrix for block sparse signal recovery. An algorithm for computing thisnew measure is also presented. In the second project, Paper II estimates the uncertaintyof Weather Research and Forecasting (WRF) model’s daily-mean 2-metertemperature in a cold region by using a BHM. It is a computationally cheaper andfaster alternative to traditional ensemble approach. In summary, this thesis makessignificant contributions in improving and optimizing the estimation proceduresof parameters of interest in MRI and WRF in practice, and developing the novelestimators and measure under the framework of CS in theory.

Place, publisher, year, edition, pages
Umeå: Umeå University , 2019. , p. 35
Series
Research report in mathematical statistics, ISSN 1653-0829 ; 69
Keywords [en]
Magnetic resonance imaging, Bayesian hierarchical models, Weather Research and Forecasting, Compressive sensing, Block sparsity, Multivariate isotropic symmetric a-stable distribution, q-ratio block constrained minimal singular value
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-165285ISBN: 978-91-7855-148-4 (print)OAI: oai:DiVA.org:umu-165285DiVA, id: diva2:1371147
Public defence
2019-12-17, N450, Naturvetarhuset, Umeå University, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2019-11-26 Created: 2019-11-19 Last updated: 2019-11-26Bibliographically approved
List of papers
1. Contrast agent quantification by using spatial information in dynamic contrast enhanced MRI
Open this publication in new window or tab >>Contrast agent quantification by using spatial information in dynamic contrast enhanced MRI
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2016 (English)Manuscript (preprint) (Other academic)
Abstract [en]

The purpose of this study is to investigate a method, using simulations, toimprove contrast agent quantication in Dynamic Contrast Enhanced MRI.Bayesian hierarchical models (BHMs) are applied to smaller images such that spatial information can be incorporated. Then exploratory analysisis done for larger images by using maximum a posteriori (MAP).

For smaller images: the estimators of proposed BHMs show improvementsin terms of the root mean squared error compared to the estimators in existingmethod for a noise level equivalent of a 12-channel head coil at 3T. Moreover,Leroux model outperforms Besag models. For larger images: MAP estimatorsalso show improvements by assigning Leroux prior.

Publisher
p. 12
Keywords
Contrast agent quantication, BHM, Besag, Leroux, INLA, MAP
National Category
Probability Theory and Statistics Medical Image Processing
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-141525 (URN)
Projects
Statistical modelling and intelligent data sampling in MRI and PET measurements for cancer therapy assessment
Funder
Swedish Research Council, 340-2013-5342
Available from: 2017-11-07 Created: 2017-11-07 Last updated: 2019-11-19
2. Weather Simulation Uncertainty Estimation using Bayesian Hierarchical Model
Open this publication in new window or tab >>Weather Simulation Uncertainty Estimation using Bayesian Hierarchical Model
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2019 (English)In: Journal of Applied Meteorology and Climatology, ISSN 1558-8424, E-ISSN 1558-8432, Vol. 58, no 3, p. 585-603Article in journal (Refereed) Published
Abstract [en]

Estimates of the uncertainty of model output fields (e.g. 2-meter temperature, surface radiation fluxes or wind speed) are of great value to the weather and climate communities. The traditional approach for the uncertainty estimation is to conduct an ensemble of simulations where the model configuration is perturbed, and/or different models are considered. This procedure is very computationally expensive and may not be feasible in particular for higher resolution experiments. In this paper a new method based on Bayesian Hierarchical Models (BHM) that requires just one model run is proposed. It is applied to the Weather Research and Forecasting (WRF) model’s 2-meter temperature in the Botnia-Atlantica region in Scandinavia for a 10-day period in the winter and summer seasons. For both seasons, the estimated uncertainty using the BHM is found to be comparable to that obtained from an ensemble of experiments in which different Planetary Boundary Layer (PBL) schemes are employed. While WRF-BHM is not capable of generating the full set of products obtained from an ensemble of simulations, it can be used to extract commonly used diagnostics including the uncertainty estimation which is the focus of this work. The methodology proposed here is fully general and can easily be extended to any other output variable and numerical model.

Place, publisher, year, edition, pages
American Meteorological Society, 2019
Keywords
WRF, Uncertainty, Bayesian Hierarchical Model, Matérn Covariance, Planetary Boundary Layer, Botnia-Atlantica
National Category
Probability Theory and Statistics Meteorology and Atmospheric Sciences
Research subject
Mathematical Statistics; Meteorology
Identifiers
urn:nbn:se:umu:diva-155617 (URN)10.1175/JAMC-D-18-0018.1 (DOI)000460652900002 ()
Projects
WindCoE
Available from: 2019-01-24 Created: 2019-01-24 Last updated: 2019-11-19Bibliographically approved
3. Bayesian sparsity estimation in compressive sensing with application to MR images
Open this publication in new window or tab >>Bayesian sparsity estimation in compressive sensing with application to MR images
2019 (English)In: Communications in Statistics: Case Studies, Data Analysis and Applications, ISSN 2373-7484Article in journal (Refereed) Epub ahead of print
Abstract [en]

The theory of compressive sensing (CS) asserts that an unknownsignal x ∈ CN can be accurately recovered from m measurements with m « N provided that x is sparse. Most of the recovery algorithms need the sparsity s = ||x||0 as an input. However, generally s is unknown, and directly estimating the sparsity has been an open problem. In this study, an estimator of sparsity is proposed by using Bayesian hierarchical model. Its statistical properties such as unbiasedness and asymptotic normality are proved. In the simulation study and real data study, magnetic resonance image data is used as input signal, which becomes sparse after sparsified transformation. The results from the simulation study confirm the theoretical properties of the estimator. In practice, the estimate from a real MR image can be used for recovering future MR images under the framework of CS if they are believed to have the same sparsity level after sparsification.

Place, publisher, year, edition, pages
Taylor & Francis Group, 2019
Keywords
Compressive sensing; sparsity; Bayesian hierarchical model; Matérn covariance; MRI
National Category
Probability Theory and Statistics Signal Processing Medical Image Processing
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-164952 (URN)10.1080/23737484.2019.1675557 (DOI)
Available from: 2019-11-05 Created: 2019-11-05 Last updated: 2019-11-20
4. Statistical inference for block sparsity of complex signals
Open this publication in new window or tab >>Statistical inference for block sparsity of complex signals
2019 (English)Manuscript (preprint) (Other academic)
Abstract [en]

Block sparsity is an important parameter in many algorithms to successfully recover block sparse signals under the framework of compressive sensing. However, it is often unknown and needs to beestimated. Recently there emerges a few research work about how to estimate block sparsity of real-valued signals, while there is, to the best of our knowledge, no investigation that has been conductedfor complex-valued signals. In this paper, we propose a new method to estimate the block sparsity of complex-valued signal. Its statistical properties are obtained and verified by simulations. In addition,we demonstrate the importance of accurately estimating the block sparsity in signal recovery through asensitivity analysis.

Keywords
Block sparsity, Complex-valued signals, Multivariate isotropic symmetric α-stable distribution
National Category
Signal Processing Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-162998 (URN)
Available from: 2019-09-04 Created: 2019-09-04 Last updated: 2019-11-19
5. Enhanced block sparse signal recovery based on q-ratio block constrained minimal singular values
Open this publication in new window or tab >>Enhanced block sparse signal recovery based on q-ratio block constrained minimal singular values
2019 (English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper we introduce theq-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed ℓ2/ℓq and the mixed ℓ2/ℓ1 norms of the reconstruction errors for stable and robust recovery using block Basis Pursuit (BBP), the block Dantzig selector (BDS) and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, theq-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV based error bounds are tighter than the block restricted isotropic constant based bounds.

Publisher
p. 20
Keywords
Compressive sensing;q-ratio block sparsity;q-ratio block constrained minimal singularvalue; Convex-concave procedure
National Category
Signal Processing Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-162953 (URN)
Available from: 2019-09-03 Created: 2019-09-03 Last updated: 2019-11-19

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