Sparse Reconstruction Models And Algorithms For Tomographic Imaging

Tomographic reconstructions such as computer tomography (CT), magnetic resonance imaging (MRI) and positron emission tomography (PET) have radical impact on diverse fields ranging from the medical radio-diagnosis to the industry inspection. Although analytic algorithm (such as filtered backprojection, FBP) is widely-used for tomographic reconstruction because of its simplification, it is sensitive to noise due to the high-pass filtering and suffers from streak artifacts under the condition of fewer projections. However, in practice, always only limited number even noisy projections are acquired because of limited scanning time, the amount of radiation exposure or the specific imaging environment. In these decades, many iterative algorithms have been used to overcome data insufficiency and suppress noise. These algorithms differ in many ways such as the constraints imposed on the image function, the cost function, and the actual implementation of the iterative scheme. The sparse representation is one of the latest new image models, which represents images in a compact and efficient way. Only few coefficients are big, most atom coefficients are zero, and the nonzero coefficients can reveal the intrinsic structures and essential properties of images. For these reasons, sparse representation is robust to noise and calculation error and then beneficial to subsequent image processing applications. This paper focuses on the tomographic reconstruction algorithm based on sparse representation model for two dimension images. And the main achievements include:(1) Two iterative algorithms are developed for solving the recently popular TV-regularized CT optimization problem. The most existing schemes use the gradient of TV norm as ’prediction-compensation’of the POCS (projections onto convex sets) reconstruction framework, which can not produce the exact solution of the target problem. In order to find the stable solution of this model, we use Chambolle’s scheme to overcome the numerical difficulty due to the non differentiability of the TV norm and use Bregman scheme to accelerate the iteration. In order to divide the sum minimization scheme into minimizing weighted least square function and TV denoising with weighted norm iteratively, we propose two methods:surrogate function and forward-backward operator splitting. Experimental results show that the proposed approach outperforms some existing TV tomography methods based on the gradient descent algorithms.(2) A new optimization model of tomography reconstruction for texture-rich object is proposed based on morphological components analysis. These images often contain different morphological structural components, such as piecewise smooth component, edge structure and texture. Most of existing methods based on TV regularization are appropriate to piecewise smooth object, but do not behave very well on texture-rich object While in our method, compound regularizations are exploited for the different morphological components. Furthermore, an alternating iterative algorithm is presented to solve the relevant optimization problem. We compare its numerical performance with two recently algorithms. The experimental results demonstrate that our proposed method is highly efficient especially to reconstruct texture-rich component as well as piecewise, smooth ones.(3) A new sparsity regularized convex functional is proposed to reconstruct nonnegative image intensities from linear projections contaminated with Poisson noise. Due to the limited nuclear elements half-life time and the limited safety nuclear doses, PET imaging often faces the problems of incomplete and noisy measurement data. So the PET image reconstruction is always ill-posed problem which need some appropriate prior informations for the suitable solution. Adopting Bayesian-MAP estimation framework, the negative-log Poisson likelihood functional is used as data fidelity term and sparse image representation overcomplete dictionary is used as regularization term. An indicating function is also added into the functional to ensure the non-negative of the reconstructed image intensities. Through introducing an intermediate variable and Bergman distance, the original problem is transformed into solving two simple sub-problems iteratively. And a multi-step fast iterative algorithm is proposed. The experimental results demonstrate the effectiveness of our model and numerical iteration algorithm.(4) A TV-based optimization model for Micro-CT super reconstruction is proposed based on upsampling of reconstruction grid with original detector and X-ray dose. In Micro-CT system, further spatial resolution improvement of the reconstructed images is limited by X-ray dose level, the pixel pitch, the aperture of the detector element. Utilized extension of gradient projection method, an alternating minimization algorithm is employed to solve the corresponding energy function. On the process of the minimization, the treatment is separated into gradient step of the fit-to-data term, total variation (TV) denoising, and specific linear combination of the previous two points. Experiments were run on simulated data as well as real Micro-CT data The results show that our proposed approach can dramatically improve the spatial resolution of the reconstructed images compared to the conventional FBP algorithm.