Statistiska modeller och intelligenta datainsamlingsmetoder för MRI och PET mätningar med tillämpning för monitoring av cancerbehandling

Title [en]

Statistical modelling and intelligent data sampling in MRI and PET measurements for cancer therapy assessment

Abstract [sv]

In general, most bio-imaging (imaging resulting in images that represent actual biological quantities, e.g., perfusion) is limited by noise, resolution, motion artifacts etc., but at the same time heavily oversampled with respect to the information relevant for the actual purpose. The purpose of this project is to develop new statistical and computational methodology for intelligent data sampling and uncertainty analysis of MRI and PET measurements. More specific, statistical spatiotemporal models to characterize stochastic noise in parametric imaging based on MRI and PET will be developed and, for the same techniques, intelligent data sampling based on Compressed Sensing will be investigated. Focus will be on the statistical and computational challenges arising from uncertainty analysis and error versus speed optimization for high-dimensional data. This project should contribute to the general understanding of optimised data sampling in bio-imaging and to efficient noise reduction for improved quality of the estimated parametric images. When applied in therapy response imaging this project should result in significantly shorter imaging time and more reliable quantitative information which are two important steps in bringing bio-imaging towards a more widespread clinical use.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Yu, Jun

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

2019 (English)In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 48, no 16, p. 4034-4050Article in journal (Refereed) Published

Abstract [en]

In this paper, the adaptive estimation for varying coefficient models proposed by Chen, Wang, and Yao (2015) is extended to allowing for nonstationary covariates. The asymptotic properties of the estimator are obtained, showing different convergence rates for the integrated covariates and stationary covariates. The nonparametric estimator of the functional coefficient with integrated covariates has a faster convergence rate than the estimator with stationary covariates, and its asymptotic distribution is mixed normal. Moreover, the adaptive estimation is more efficient than the least square estimation for non normal errors. A simulation study is conducted to illustrate our theoretical results.

Place, publisher, year, edition, pages

Taylor & Francis, 2019

Keywords

Varying coefficient model, adaptive estimation, local linear fitting, non stationary covariates

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Zhou, Zhiyong

Department of Statistics, Zhejiang University City College, China.

Yu, Jun

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

2019 (English)In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139Article in journal (Refereed) Epub ahead of print

Abstract [en]

We study the recovery conditions of weighted mixed ℓ2/ℓp minimization for block sparse signal reconstruction from compressed measurements when partial block supportinformation is available. We show theoretically that the extended block restricted isometry property can ensure robust recovery when the data fidelity constraint is expressed in terms of an ℓq norm of the residual error, thus establishing a setting wherein we arenot restricted to Gaussian measurement noise. We illustrate the results with a series of numerical experiments.

Place, publisher, year, edition, pages

Global Science Press, 2019

Keywords

Compressed sensing, block sparsity, partial support information, signal reconstruction, convex optimization

National Category

Signal Processing Probability Theory and Statistics Computational Mathematics

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Zhou, Zhiyong

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Garpebring, Anders

Umeå University, Faculty of Medicine, Department of Radiation Sciences.

Yu, Jun

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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 ∈ C^{N} 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.

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.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Zhou, Zhiyong

Department of Statistics, Zhejiang University City College, China.

Yu, Jun

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

2019 (English)In: EURASIP Journal on Advances in Signal Processing, ISSN 1687-6172, E-ISSN 1687-6180, Vol. 57Article in journal (Refereed) Epub ahead of print

Abstract [en]

In this paper, we introduce the q-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 ℓ_{2}/ℓ_{q} and the mixed ℓ_{2}/ℓ_{1} 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, the q-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.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

2019 (English)In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 165, p. 128-132Article in journal (Refereed) Published

Abstract [en]

As a kind of computable incoherence measure of the measurement matrix, q-ratio constrained minimal singular values (CMSV) was proposed in Zhou and Yu (2019) to derive the performance bounds for sparse recovery. In this paper, we study the geometrical properties of the q-ratio CMSV, based on which we establish new sufficient conditions for signal recovery involving both sparsity defect and measurement error. The ℓ_{1}-truncated set q-width of the measurement matrix is developed as the geometrical characterization of q-ratio CMSV. In addition, we show that the q-ratio CMSVs of a class of structured random matrices are bounded away from zero with high probability as long as the number of measurements is large enough, therefore these structured random matrices satisfy those established sufficient conditions. Overall, our results generalize the results in Zhang and Cheng (2012) from q=2 to any q ∈ (1, ∞] and complement the arguments of q-ratio CMSV from a geometrical view.

Department of Statistics, Zhejiang University City College, China.

Yu, Jun

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

2019 (English)In: Optimization Letters, ISSN 1862-4472, E-ISSN 1862-4480Article in journal (Refereed) Epub ahead of print

Abstract [en]

We study the recovery conditions of weighted ℓ1 minimization for real-valued signal reconstruction from phaseless compressive sensing measurements when partial support information is available. A strong restricted isometry property condition is provided to ensure the stable recovery. Moreover, we present the weighted null space property as the sufficient and necessary condition for the success of k-sparse phaseless recovery via weighted ℓ1 minimization. Numerical experiments are conducted to illustrate our results.

Place, publisher, year, edition, pages

Springer, 2019

Keywords

Phaseless compressive sensing, Partial support information, Strong restricted isometry property, Weighted null space property

National Category

Signal Processing Probability Theory and Statistics Computational Mathematics

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Zhou, Zhiyong

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Statistics, Zhejiang University City College, China.

Yu, Jun

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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