A fast iterative P-thresholding algorithm for sparse reconstruction of electrical tomography

The image reconstruction of Electrical Tomography (ET) is an ill-posed nonlinear inverse problem. Regularization methods are introduced to treat this ill-posedness. Considering the sparsity property of the internal medium distribution of the electrical field, the conventional L2 regularization penalty term is replaced by sparsity constraints, and the soft or hard thresholding function is adopted to provide the solution to the corresponding penalized least-squares problem. Both of them are the proximal mappings of L1 or L0 penalty function. It is important to design the objective function with sparsity constraint flexibly. So a new generalized p-thresholding function, which can be seen as a mapping of a wide class of penalty functions with sparsity constraints, is proposed to solve the sparse regularization of ET. The new sparse objective function can be designed by flexibly changing the value of p. The proposed fast iterative p-thresholding algorithm (FIPTA) provides a more accurate result with less iterations than traditional methods using soft thresholding function. FIPTA can also preserve the computational simplicity by avoiding the directly time consuming large-scale matrix calculations and improve the calculation speed. This method is discussed using simulated data and further tested with the practical ERT system, both theoretically and practically.