**Multigrid Regularized Image Reconstruction for Limited-Data Tomography**

Institute of Telecommunications, Teleinformatics and Acoustics

Wroclaw University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wroclaw, Poland

e-mail: Rafal.Zdunek@pwr.wroc.pl

### Received:

Rec. January 11, 2007

### DOI: 10.12921/cmst.2007.13.01.67-77

### OAI: oai:lib.psnc.pl:633

### 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.

### Key words:

electromagnetic geotomography, hyperparameter estimation, limited-data tomography, multigrid image reconstruction

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