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

Abstract

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.