Thesis (M.S.)--Oregon State University, 2011. / Printout. Includes bibliographical references (leaves 57-59). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 116-124).

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Variational methods, which have proven to be very useful to solve the ill-posed inverse problems, have been generating a lot of research interest in the image restoration problem. It transforms the restoration problem into the optimization of a well-designed variational model. While the designed model is convex, the recovered image is the global solution found by an appropriate numerical algorithm and the quality of the restored image depends on the accuracy of the designed model. Thus, a lot of efforts have been put to propose a more precise model that can produce a result with more pleasing visual quality. Besides, due to the high- dimension and the nonsmoothness of the imaging model, an efficient algorithm to find the exact solution of the variational model, is also of the research interest, since it influences the efficiency of the restoration techniques in the practical applications. In this thesis, we are interested in the designing of both the variational models for image restoration problems and the numerical algorithms to solve these models. The first objective of this thesis is to make improvements on two models for image denoising. For the multiplicative noise removal, we designed a regularizer based on the statistical property of the speckle noise, which can transform the traditional model (named by AA) into a convex one. Therefore, a global solution can be found independent of the initialization of the numerical algorithm. Moreover, the regularization term added on the AA model can help produce a sharper result. The second model is improved on the traditional ROF model by adding an edge regularization which incorporates an edge prior obtained from the observed image. Extensive experiments show that designed edge regularization has superiority to increase the texture of the recovered result and remove the staircase artifacts in the meanwhile. It is also presented that the edge regularization designed can be easily adapted into other restoration task, such as image deblurring. The second objective of this thesis is to study the numerical algorithms for a general nonsmooth imaging restoration model. As the imaging models are usually high-dimensional, the existing algorithms usually only use the first-order information of the image. Differently, a novel numerical algorithm based on the inexact Lagrangian function is proposed in this thesis, which exploits the second-order information to reach a superlinear convergence rate. Experiments show that the proposed algorithm is able to efficiently reach the solution with higher accuracy compared to the state-of-the-art algorithm

Variational methods, which have proven to be very useful to solve the ill-posed inverse problems, have been generating a lot of research interest in the image restoration problem. It transforms the restoration problem into the optimization of a well-designed variational model. While the designed model is convex, the recovered image is the global solution found by an appropriate numerical algorithm and the quality of the restored image depends on the accuracy of the designed model. Thus, a lot of efforts have been put to propose a more precise model that can produce a result with more pleasing visual quality. Besides, due to the high- dimension and the nonsmoothness of the imaging model, an efficient algorithm to find the exact solution of the variational model, is also of the research interest, since it influences the efficiency of the restoration techniques in the practical applications. In this thesis, we are interested in the designing of both the variational models for image restoration problems and the numerical algorithms to solve these models. The first objective of this thesis is to make improvements on two models for image denoising. For the multiplicative noise removal, we designed a regularizer based on the statistical property of the speckle noise, which can transform the traditional model (named by AA) into a convex one. Therefore, a global solution can be found independent of the initialization of the numerical algorithm. Moreover, the regularization term added on the AA model can help produce a sharper result. The second model is improved on the traditional ROF model by adding an edge regularization which incorporates an edge prior obtained from the observed image. Extensive experiments show that designed edge regularization has superiority to increase the texture of the recovered result and remove the staircase artifacts in the meanwhile. It is also presented that the edge regularization designed can be easily adapted into other restoration task, such as image deblurring. The second objective of this thesis is to study the numerical algorithms for a general nonsmooth imaging restoration model. As the imaging models are usually high-dimensional, the existing algorithms usually only use the first-order information of the image. Differently, a novel numerical algorithm based on the inexact Lagrangian function is proposed in this thesis, which exploits the second-order information to reach a superlinear convergence rate. Experiments show that the proposed algorithm is able to efficiently reach the solution with higher accuracy compared to the state-of-the-art algorithm

Single photon emission computed tomography (SPECT) is used in a variety of medical procedures, including myocardial perfusion, bone metabolism, and thyroid function studies. In SPECT, the emissions of a radionuclide within a patient are counted at a gamma camera to form a 2-dimensional projection of the 3-dimensional radionuclide distribution within the patient. This unknown 3-dimensional source distribution can be reconstructed from many 2-dimensional projections obtained at different angles around the patient. This reconstruction can be improved by properly modeling the physics in the patient, i.e., particle absorption and scattering. Currently, such modeling is done using statistical Monte Carlo methods, but deterministic codes have the potential to offer fast computation speeds while fully modeling particle interactions within the patient. Deterministic codes are not susceptible to statistical uncertainty, but have been over-looked for applications to nuclear medicine, most likely due to their own limitations, including discretization and large memory requirements. A novel deterministic reconstruction methodology for SPECT (DRS) has been developed to apply the advantages of deterministic algorithms to SPECT iterative image reconstruction. Using a maximum likelihood expectation maximization (ML-EM) algorithm, a deterministic code can fully model particle transport in the patient in the forward projection step, without the need of a large system matrix. The TITAN deterministic transport code has a SPECT formulation that allows for fast simulation of SPECT projection images and has been benchmarked through comparison with results from the SIMIND and MCNP5 Monte Carlo codes in this dissertation. The TITAN SPECT formulation has been improved through a modified collimator representation and full parallelization. The DRS methodology has been implemented in the TITAN code to create TITAN with Image Reconstruction (TITAN-IR). The TITAN-IR code has been used to successfully reconstruct the source distribution from SPECT data for the Jaszczak and NCAT phantoms. Extensive studies have been conducted to examine the sensitivity of TITAN-IR image quality to deterministic parameter selection as well as collimator blur and noise in the projection data being reconstructed. The TITAN-IR reconstruction has also been compared with other reconstruction algorithms. This novel image reconstruction methodology has been shown to reconstruct images in short computation times, demonstrating its potential in a clinical setting with further development. / Ph. D.

published_or_final_version / abstract / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Mathematics / Doctoral / Doctor of Philosophy