  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.
