Mathematical Modeling And Algorithms For Medical Image Reconstruction

Medical image reconstruction is an important non-invasion technique of creating visual representation of the interior of a body.With the development of medical imaging techniques and applications,more requirements are put forward for image reconstruction,such as in dynamic image and ultra low dose computed tomography(ULDCT)reconstruction.High spatial and temporal resolution and low acquisition time are the basic requirements,in addition to reducing patient exposure to radiation.In dynamic single-photon emission computed tomography(SPECT)reconstruction,high image quality becomes more and more important with the requirement of more effective methods for disease diagnosis.For the existence of dynamic change and limitation of devices,we can only acquire few projection data in practice.This pose a challenge for almost all kinds of imaging devices to process the acquired data.Moreover,due to the noise and motion of objects or organs,more difficulties arise for dynamic image reconstruction.Furthermore,exposure to X-rays for long period of time or high dose X-rays can cause great harm to humans.To reduce X-Ray radiation,ULDCT is a very popular research area.This dissertation aims to propose new models for dynamic SPECT and ULDCT reconstruction to obtain high quality images while reducing X-ray damage to human body.SPECT allows us to monitoring the biological processes.However,reconstruction of the images from dynamic SPECT data is still a challenge problem.With the fast decay of radioisotope by time,we can only collect few projection data of every image.The projection data contain very little information of the images,especially when noise and tissue deformation present.Firstly,we propose a variational model,namely sparsity enforced matrix factorization(SEMF),based on low rank matrix factorization of unknown images and enforced sparsity constraints for both representing coefficients and bases.SEMF model is solved via an alternating iterative scheme,for which each subproblem is convex and involves efficient alternating direction method of multiplier(ADMM).The convergence of the overall alternating scheme for the nonconvex problem relies upon Kurdyka-?ojasiewicz property,studied in [1,2].Our simulation on2D/3D dynamic images shows the advantage of the proposed method compared to conventinal methods.To overcome the drawbacks and improve the results of SEMF model,we propose another variational model using the infimal convolution of Bregman distance with respect to total variation to model edge dependence of sequential frames.The proposed model is solved with an alternating iterative scheme,for which each subproblem is convex and can be solved by existing algorithms.The proposed model is formulated under both Gaussian and Poisson noise assumption and the simulation on two sets of dynamic images shows the advantage of the proposed method compared to SEMF methods.Statistical image reconstruction(SIR)methods for X-ray CT produce high-quality and accurate images,while greatly reducing patient exposure to radiation.When further reducing X-ray dose to an ultra-low level by lowering the tube current,photon starvation happens and electronic noise starts to dominate,which introduces negative or zero values into the raw measurements.These non-positive values pose challenges to post-log SIR methods that require taking the logarithm of the raw data,and causes artifacts in the reconstructed images if simple correction methods are used to process these non-positive raw measurements.The raw data at ultra-low dose deviates significantly from Poisson or shifted Poisson statistics for pre-log data and from Gaussian statistics for post-log data.This thesis proposes a novel SIR method called MPG(mixed Poisson-Gaussian).MPG models the raw noisy measurements using a mixed Poisson-Gaussian distribution that accounts for both the quantum noise and electronic noise.MPG is able to directly use the negative and zero values in raw data without any pre-processing.MPG cost function contains a reweighted least square data-fit term,an edge preserving regularization term and a non-negativity constraint term.We use ADMM to separate the MPG optimization problem into several sub-problems that are easier to solve.Our results on 3D simulated cone-beam data set and synthetic helical data set generated from clinical data indicate that the proposed MPG method reduces noise and decreases bias in the reconstructed images,comparing with the conventional filtered back projection(FBP),penalized weighted least-square(PWLS)and shift Poisson(SP)method for ultra-low dose CT(ULDCT)imaging.