We propose a compressive sensing approach for multi-energy computed tomography (CT), namely the prior rank, intensity and sparsity model (PRISM). To further compress the multi-energy image for allowing the reconstruction with fewer CT data and less radiation dose, the PRISM models a multi-energy image as the superposition of a low-rank matrix and a sparse matrix (with row dimension in space and column dimension in energy), where the low-rank matrix corresponds to the stationary background over energy that has a low matrix rank, and the sparse matrix represents the rest of distinct spectral features that are often sparse. Distinct from previous methods, the PRISM utilizes the generalized rank, e.g., the matrix rank of tight-frame transform of a multi-energy image, which offers a way to characterize the multi-level and multi-filtered image coherence across the energy spectrum. Besides, the energy-dependent intensity information can be incorporated into the PRISM in terms of the spectral curves for base materials, with which the restoration of the multi-energy image becomes the reconstruction of the energy-independent material composition matrix. In other words, the PRISM utilizes prior knowledge on the generalized rank and sparsity of a multi-energy image, and intensity/spectral characteristics of base materials. Furthermore, we develop an accurate and fast split Bregman method for the PRISM and demonstrate the superior performance of the PRISM relative to several competing methods in simulations.

The purpose of this paper for four-dimensional (4D) computed tomography (CT) is threefold. (1) A new spatiotemporal model is presented from the matrix perspective with the row dimension in space and the column dimension in time, namely the robust PCA (principal component analysis)-based 4D CT model. That is, instead of viewing the 4D object as a temporal collection of three-dimensional (3D) images and looking for local coherence in time or space independently, we perceive it as a mixture of low-rank matrix and sparse matrix to explore the maximum temporal coherence of the spatial structure among phases. Here the low-rank matrix corresponds to the ‘background’ or reference state, which is stationary over time or similar in structure; the sparse matrix stands for the ‘motion’ or time-varying component, e.g., heart motion in cardiac imaging, which is often either approximately sparse itself or can be sparsified in the proper basis. Besides 4D CT, this robust PCA-based 4D CT model should be applicable in other imaging problems for motion reduction or/and change detection with the least amount of data, such as multi-energy CT, cardiac MRI, and hyperspectral imaging. (2) A dynamic strategy for data acquisition, i.e. a temporally spiral scheme, is proposed that can potentially maintain similar reconstruction accuracy with far fewer projections of the data. The key point of this dynamic scheme is to reduce the total number of measurements, and hence the radiation dose, by acquiring complementary data in different phases while reducing redundant measurements of the common background structure. (3) An accurate, efficient, yet simple-to-implement algorithm based on the split Bregman method is developed for solving the model problem with sparse representation in tight frames.

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Respiration-correlated CBCT, commonly called 4DCBCT, provides respiratory phase-resolved CBCT images. A typical 4DCBCT represents averaged patient images over one breathing cycle and the fourth dimension is actually breathing phase instead of time. In many clinical applications, it is desirable to obtain true 4DCBCT with the fourth dimension being time, i.e., each constituent CBCT image corresponds to an instantaneous projection. Theoretically it is impossible to reconstruct a CBCT image from a single projection. However, if all the constituent CBCT images of a 4DCBCT scan share a lot of redundant information, it might be possible to make a good reconstruction of these images by exploring their sparsity and coherence/redundancy. Though these CBCT images are not completely time resolved, they can exploit both local and global temporal coherence of the patient anatomy automatically and contain much more temporal variation information of the patient geometry than the conventional 4DCBCT. We propose in this work a computational model and algorithms for the reconstruction of this type of semi-time-resolved CBCT, called cine-CBCT, based on low rank approximation that can utilize the underlying temporal coherence both locally and globally, such as slow variation, periodicity or repetition, in those cine-CBCT images.

By effectively utilizing the spatiotemporal coherence of the patient anatomy among different respiratory phases in a multilevel fashion with multibasis sparsifying transform, the proposed STF method potentially enables fast and low-dose 4DCBCT with improved image quality.

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In this paper we study an l1-regularized multilevel approach for bioluminescence tomography based on radiative transfer equation with the emphasis on improving imaging resolution and reducing computational time. Simulations are performed to validate that our algorithms are potential for efficient high-resolution imaging. Besides, we study and compare reconstructions with boundary angular-averaged data, boundary angular-resolved data and internal angular-averaged data respectively.

Purpose: Iterative reconstruction methods often offer better imaging quality and allow for reconstructions with lower imaging dose than classical methods in computed tomography. However, the computational speed is a major concern for these iterative methods, for which the x-ray transform and its adjoint are two most time-consuming components. The speed issue becomes even notable for the 3D imaging such as cone beam scans or helical scans, since the x-ray transform and its adjoint are frequently computed as there is usually not enough computer memory to save the corresponding system matrix. The purpose of this paper is to optimize the algorithm for computing the x-ray transform and its adjoint, and their parallel computation. Methods: The fast and highly parallelizable algorithms for the x-ray transform and its adjoint are proposed for the infinitely narrow beam in both 2D and 3D. The extension of these fast algorithms to the finite-size beam is proposed in 2D and discussed in 3D. Results: The CPU and GPU codes are available at https://sites.google.com/site/fastxraytransform. The proposed algorithm is faster than Siddon's algorithm for computing the x-ray transform. In particular, the improvement for the parallel computation can be an order of magnitude. Conclusions:The authors have proposed fast and highly parallelizable algorithms for the x-ray transform and its adjoint, which are extendable for the finite-size beam. The proposed algorithms are suitable for parallel computing in the sense that the computational cost per parallel thread is O(1).