A Hierarchical Bayesian Inversion Method for Electromagnetic Imaging of Inhomogeneous Objects With Piecewise Homogeneities

Electromagnetic imaging of inhomogeneous objects is a nonlinear inverse scattering problem which can be effectively handled by means of linear inversion methods or nonlinear optimization methods. In addition to being nonlinear, inverse scattering problems are also ill-posed which can be overcome through a regularization scheme by taking into account some kind of a priori information. However, it is not straightforward to incorporate such inhomogeneity prior information into conventional inversion methods. In this work, we propose an innovative method to solve the problem of imaging inhomogeneous objects with piecewise homogeneity. Under Born approximation (BA), this problem is cast into a linear inversion one which is then addressed by means of a hierarchical Bayesian inversion method. To account for the piecewise homogeneity of the image, a Gaussian mixture model (GMM) is developed to enforce the feature. Specifically, a hidden label variable which is associated with each pixel is introduced for the class of materials and the contrast of each pixel is modeled as a mixture of independent Gaussian priors. Then, the proposed hierarchical Bayesian inversion method incorporates the GMM to the variational Bayesian inference framework to approximate the posterior distribution of all the unknowns (the unknown contrast, hidden label variable, and other model parameters). In addition, the damped generalized approximate message passing (GAMP) is utilized to circumvent the matrix inversion, making the overall process very computationally efficient. Extensive numerical examples will be shown to demonstrate the effectiveness of the proposed method.