X-ray computed tomography (CT) is one of widely used diagnostic tools for medical and dental tomographic imaging of the human body. However, the standard filtered back-projection reconstruction method requires the complete knowledge of the projection data. In the case of limited data, the inverse problem of CT becomes more ill-posed, which makes the reconstructed image deteriorated by the artifacts. In this paper, we consider two dimensional CT reconstruction using the projections truncated along the spatial direction in the Radon domain. Over the decades, the numerous results including the sparsity model based approach has enabled the reconstruction of the image inside the region of interest (ROI) from the limited knowledge of the data. However, unlike these existing methods, we try to reconstruct the entire CT image from the limited knowledge of the sinogram via the tight frame regularization and the simultaneous sinogram extrapolation. Our proposed model shows more promising numerical simulation results compared with the existing sparsity model based approach.

A practical image reconstruction method for multi-source quantitative photoacoustic tomography (QPAT) is proposed in this work with the consideration of detector response function and limited-view scanning. First, the correct detector response function, i.e., spatial impulse response (SIR) and acousto-electric impulse response (EIR), is considered for the ultrasonic transducer to accurately model the acoustic measurement; second, acoustic data is only measured near optical sources with meaningful signal-to-noise ratio (SNR), i.e., the limited-view scanning, which also reduces the data acquisition time for point transducer. However, due to the incomplete limited-view data, a two-step image reconstruction method (i.e., to first reconstruct initial acoustic pressure and then reconstruct optical coefficients) no longer applies, since it is neither possible nor necessary to robustly reconstruct initial acoustic pressure with limited-view data. Therefore, here we propose a direct image reconstruction method that incorporates SIR, EIR and limited-view scanning in a coupled opto-acoustic forward model, regularizes the nonlinear QPAT data fidelity with tensor framelet sparsity, and then solves the QPAT problem with Quasi-Newton method based alternating direction method of multipliers.

The implicit convex feasibility problem attempts to find a point in the intersection of a finite family of convex sets, some of which are not explicitly determined but may vary. We develop simultaneous and sequential projection methods capable of handling such problems and demonstrate their applicability to image denoising in a specific medical imaging situation. By allowing the variable sets to undergo scaling, shifting and rotation, this work generalizes previous results wherein the implicit convex feasibility problem was used for cooperative wireless sensor network positioning where sets are balls and their centers were implicit.

The purpose of this paper is to investigate the ability of the infimal convolution regularization in curing the staircasing artifacts of the TV model in the SPECT reconstruction. We formulate the problem of SPECT reconstruction with the infimal convolution regularization as a convex three-block optimization problem and characterize its solution by a system of fixed-point equations in terms of the proximity operator of the functions involved in its objective function. We then develop a novel fixed-point proximity algorithm based on the fixed-point equations. Moreover, we introduce a preconditioning matrix motivated by the classical MLEM (maximum-likelihood expectation maximization) algorithm. We prove convergence of the proposed algorithm. The numerical results are included to show that the infimal convolution regularization is capable of effectively reducing the staircasing artifacts, while maintaining comparable image quality in terms of the signal-to-noise ratio and coefficient recovery contrast.

In the bioluminescence tomography (BLT) problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this paper, a parameter-dependent coupled complex boundary method (CCBM) based Tikhonov regularization is applied to the BLT problem governed by the radiative transfer equation (RTE). By properly adjusting the parameter in the Robin boundary condition, we achieve one important property: the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The discrete-ordinate finite-element method is used to compute numerical solutions. Numerical results are provided to illustrate the performance of the proposed method.

High-attenuation object-induced streaking and shadow artifacts in computerized tomography (CT) are somewhat connected to the misfit of the X-ray projection data to the range space of the Radon transform. This misfit is mainly due to the beam hardening factor of the projection data which is unavoidable for polychromatic sources. The major difficulty in dealing with the beam hardening-induced streaking and shadow artifacts comes from its highly nonlinear nature depending on geometries of high attenuation objects. In this work, we investigate the mathematical characteristics of those streaking and shadow artifacts from the structure of the projection data. We also proposed a metal artifacts reduction method by incorporating the recent technique of the nonlinear beam-hardening corrector. Numerical simulations show that the proposed method effectively alleviates the streaking artifacts without changing the background images.

As X-ray computed tomography (CT) is widely used in diagnosis and radiotherapy, it is important to reduce the radiation dose as low as reasonably achievable. For this purpose, one may use the TV based methods or wavelet frame based methods to reconstruct high quality images from reduced number of projections. Furthermore, by using the interior tomography scheme which only illuminates a region-of-interest (ROI), one can save more radiation dose. In this paper, a robust wavelet frame regularization based model is proposed for both global reconstruction and interior tomography. The model can help to reduce the errors caused by mismatch of the huge sparse projection matrix. A three-system decomposition scheme is applied to decompose the reconstructed images into three different parts: cartoon, artifacts and noise. Therefore, by discarding the estimated artifacts and noise parts, the reconstructed images can be obtained with less noise and artifacts. Similar to other frame based image restoration models, the model can be efficiently solved by the split Bregman algorithm. Numerical simulations show that the proposed model outperforms the FBP and SART+TV methods in terms of preservation of sharp edges, mean structural similarity (SSIM), contrast-to-noise ratio, relative error and correlations. For example, for real sheep lung reconstruction, the proposed method can reach the mean structural similarity as high as 0.75 using only 100 projections while the FBP and the SART+TV methods need more than 200 projections. Additionally, the proposed robust method is applicable for interior and exterior tomography with better performance compared to the FBP and the SART+TV methods.

Image Quality Assessment (IQA) is a fundamental problem in image processing. It is a common principle that human vision is hierarchical: we first perceive global structural information such as contours then focus on local regional details if necessary. Following this principle, we propose a novel framework for IQA by quantifying the degenerations of structural information and region content separately, and mapping both to obtain the objective score. The structural information can be obtained as contours by contour detection techniques. Experiments are conducted to demonstrate its performance in comparison with multiple state-of-the-art methods on two large scale datasets.

We have proposed a primal-dual fixed point algorithm (PDFP) for solving minimization of the sum of three convex separable functions, which involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function. Compared with similar works, the parameters in PDFP are easier to choose and are allowed in a relatively larger range. We will extend PDFP to solve two kinds of separable multi-block minimization problems, arising in signal processing and imaging science. This work shows the flexibility of applying PDFP algorithm to multi-block problems and illustrates how practical and fully splitting schemes can be derived, especially for parallel implementation of large scale problems. The connections and comparisons to the alternating direction method of multiplier (ADMM) are also presented. We demonstrate how different algorithms can be obtained by splitting the problems in different ways through the classic example of sparsity regularized least square model with constraint. In particular, for a class of linearly constrained problems, which are of great interest in the context of multi-block ADMM, can be also solved by PDFP with a guarantee of convergence. Finally, some experiments are provided to illustrate the performance of several schemes derived by the PDFP algorithm.