Multigrid Regularized Image Reconstruction for Limited-Data Tomography

Abstract

Limited-data tomography, to which electromagnetic geotomography belongs, is analyzed in this paper. In this technique, a discrete forward projection model may be expressed by a rank-deficient system of linear equations whose the nullspace is non-trivial. This means that some image components may fall into the nullspace, and hence the minimal-norm least-square solution, to which many image reconstructions methods converge, may be different from the true one. The Algebraic Reconstruction Technique (ART), Simultaneous Iterative Reconstruction Technique (SIRT), or Conjugate Gradients Least Squares (CGLS) are examples of such methods. In this paper, we deal with the question of how to partially recover the missing image components. First, we analyze the advantages of using the iterative Tikhonov regularization and the Maximum A Posteriori (MAP) algorithm with Gibbs prior. Then, we conclude that the missing (nullspace) image components can be partially recovered if the MAP algorithm is implemented through a multigrid technique. The results, which are presented for synthetic noise-free and noisy data, demonstrate the validity of our assumption. The problem of estimating the regularization and scaling parameters in the MAP algorithm is also addressed.

Two-Dimensional Tomographic Simultaneous Multi-Species Visualization—Part I: Experimental Methodology and Application to Laminar and Turbulent Flames

“…In statistics, Tikhonov regularization and compressive sensing are also known as ridge and lasso regression, respectively. Since the classical L-curve method is not recommended for determining the optimal regularization parameters for the specified regularization scheme used here [64,72], the optimal parameters were determined by trial-and-error and phantom studies. Equation (3) was solved by a steepest descent solver until the scalar objective function values of two successive iteration steps fulfill the convergence criterion:…”

Two-Dimensional Tomographic Simultaneous Multispecies Visualization—Part II: Reconstruction Accuracy

“…In statistics and machine learning, Tikhonov regularization and compressive sensing are also known as ridge and lasso regression, respectively. Since the classical L-curve method is not recommended for determining the optimal regularization parameters for the specified regularization scheme used here [40,59], the optimal parameters were determined by trial-and-error and phantom studies. Basically, the reconstruction was tested with different parameters and a combination was chosen in which the distribution is not excessively smeared by the regularization.…”