AnLq–Lpoptimization framework for image reconstruction of electrical resistance tomography

Image reconstruction in electrical resistance tomography (ERT) is an ill-posed and nonlinear problem, which is easily affected by measurement noise. The regularization method with L 2 constraint term or L 1 constraint term is often used to solve the inverse problem of ERT. It shows that the reconstruction method with L 2 regularization puts smoothness to obtain stability in the image reconstruction process, which is blurry at the interface of different conductivities. The regularization method with L 1 norm is powerful at dealing with the over-smoothing effects, which is beneficial in obtaining a sharp transaction in conductivity distribution. To find the reason for these effects, an L q – L p optimization framework (1 ⩽ q ⩽ 2, 1 ⩽ p ⩽ 2) for the image reconstruction of ERT is presented in this paper. The L q – L p optimization framework is solved based on an approximation handling with Gauss–Newton iteration algorithm. The optimization framework is tested for image reconstruction of ERT with different models and the effects of the L p regularization term on the quality of the reconstructed images are discussed with both simulation and experiment. By comparing the reconstructed results with different p in the regularization term, it is found that a large penalty is implemented on small data in the solution when p is small and a lesser penalty is implemented on small data in the solution when p is larger. It also makes the reconstructed images smoother and more easily affected by noise when p is larger.