Non-convex ℓ_p regularization for sparse reconstruction of electrical impedance tomography

Abstract

This work is to investigate the image reconstruction of electrical impedance tomography from the electrical measurements made on an object's surface. An ${\ell }_p$-norm (0<p<1) sparsity-promoting regularization is considered to deal with the fully non-linear electrical impedance tomography problem, and a novel type of smoothing gradient-type iteration scheme is introduced. To avoid the difficulty in calculating its gradient in the optimization process, a smoothing Huber potential function is utilized to approximate the ${\ell }_p$-norm penalty. We then propose the smoothing algorithm in the general frame and establish that any accumulation point of the generated iteration sequence is a first-order stationary point of the original problem. Furthermore, one iteration scheme based on the homotopy perturbation technology is derived to find the minimizers of the Huberized approximated objective function. Numerical experiments show that non-convex ${\ell }_p$-norm sparsity-promoting regularization improves the spatial resolution and is more robust with respect to noise, in comparison with ${\ell }_1$-norm regularization.

Keywords:

• Electrical impedance tomography
• image reconstruction
• regularization methods
• ${\ell }_p$ sparsity minimization

2010 Mathematics Subject Classifications:

• 65N21
• 65R32
• 65J20

Acknowledgements

The author thanks the four anonymous referees for useful comments and suggestions which improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Heilongjiang University Scientific Staring Research Foundation (No. RCCX201719), partly supported by Heilongjiang Provincial Key Laboratory for Complex Systems Theory and Computation (No. 201907), and partly supported by the Fundamental Research Funds for the Universities of Heilongjiang Province-Heilongjiang University Special Fund Project (No. RCYJTD201804).