New Level-Set-Based Shape Recovery Method and its application to sparse-view shape tomography

The recovery of shapes from a few numbers of their projections is very important in Computed tomography. In this paper, we propose a novel scheme based on a collocation set of Gaussian functions to represent any object by using a limited number of projections. This approach provides a continuous representation of both the implicit function and its zero level set. We show that the appropriate choice of a basis function to represent the parametric level-set leads to an optimization problem with a modest number of parameters, which exceeds many difficulties with traditional level set methods, such as regularization, re-initialization, and use of signed distance function. For the purposes of this paper, we used a dictionary of Gaussian function to provide flexibility in the representation of shapes with few terms as a basis function located at lattice points to parameterize the level set function. We propose a convex program to recover the dictionary coefficients successfully so it works stably with only four projections by overcoming the issue of local-minimum of the cost function. Finally, the performance of the proposed approach in three examples of inverse problems shows that our method compares favorably to Sparse Shape Composition (SSC), Total Variation, and Dual Problem.