A reconstruction method for binary limited-data tomography using a dictionary-based sparse shape recovery

Binary tomography is concerned with reconstructing a binary image from a very small number or other limited CT projection data. This problem itself not only possesses several medical imaging applications but also can be considered a model of general inverse problems to recover the object shape from limited measured data. Several approaches such as the Mumford-Shah method and various level-set methods have been investigated, but most of them lead to a non-convex optimization due to the difficulty to handle the binary constraint. We propose a new method based on a convex optimization inspired by dictionary-based shape recovery. In the proposed method, the object boundary of the binary image is represented by a level set of linear combinations of basis vectors in the dictionary. Using the dictionary, the object boundary is reconstructed by finding weights of the linear combination that best match the measured data. We create the dictionary by using the Gaussian radial basis function (GRBF). More concretely, we use Gaussian functions as a basis function placed at sparse grid points to represent the parametric level-set function and provide more flexibility in the binary representation of the reconstructed image. The simulation results of CT image reconstruction from only four projection data demonstrate that the proposed method can recover the object boundary more accurately compared with other competitive methods. The significance of our approach is the formulation with a tractable convex program while keeping moderate mathematical rigorousness.

A very fast iterative algorithm for TV-regularized image reconstruction with applications to low-dose and few-view CT

This paper concerns iterative reconstruction for low-dose and few-view CT by minimizing a data-fidelity term regularized with the Total Variation (TV) penalty. We propose a very fast iterative algorithm to solve this problem. The algorithm derivation is outlined as follows. First, the original minimization problem is reformulated into the saddle point (primal-dual) problem by using the Lagrangian duality, to which we apply the first-order primal-dual iterative methods. Second, we precondition the iteration formula using the ramp flter of Filtered Backprojection (FBP) reconstruction algorithm in such a way that the problem solution is not altered. The resulting algorithm resembles the structure of so-called iterative FBP algorithm, and it converges to the exact minimizer of cost function very fast.

Proposal of fault-tolerant tomographic image reconstruction

This paper deals with tomographic image reconstruction under the situation where some of projection data bins are contaminated with abnormal data. Such situations occur in various instances of tomography. We propose a new reconstruction algorithm called the Fault-Tolerant reconstruction outlined as follows. The least-squares (L2-norm) error function ||Ax-b||_2^2 used in ordinary iterative reconstructions is sensitive to the existence of abnormal data. The proposed algorithm utilizes the L1-norm error function ||Ax-b||_1^1 instead of the L2-norm, and we develop a row-action-type iterative algorithm using the proximal splitting framework in convex optimization fields. We also propose an improved version of the L1-norm reconstruction called the L1-TV reconstruction, in which a weak Total Variation (TV) penalty is added to the cost function. Simulation results demonstrate that reconstructed images with the L2-norm were severely damaged by the effect of abnormal bins, whereas images with the L1-norm and L1-TV reconstructions were robust to the existence of abnormal bins.