PENALIZED LEAST-SQUARES IMAGE RECONSTRUCTION FOR BOREHOLE TOMOGRAPHY

The Algebraic Reconstruction Technique (ART), which is based on the Kaczmarz's method, is very popular in many tomographic applications of image reconstruction. However, this algorithm gives satisfactory approximations to the exact solution only for consistent data. For inconsistent problems (which is a real case in practice), the reconstruction is speckled with noise, and the convergence is not asymptotical. In a previous paper we made a systematic analysis, both theoretical and experimental, for this case by using an extension of the classical Kaczmarz's algorithm. But, although the results were much better and very promising comparing them with some classical and widely used projection algorithms, another dicult y still could not be eliminate, namely the exact solution of the discrete problem is not always enough accurate. This aspect is of course related to the mathematical modeling of the real problem (image), which gives us only a classical least-squares formulation. In this paper, we considered a penalized least-squares objective. The penalty term is dened with the Gibbs prior that incorporates nearest neighbor interactions among adjacent pixels to enforce an overall smoothness in the image. Then, we derived a version of the above mentioned extended Kaczmarz algorithm for this new penalized least-squares problem. Our synthetic experiments showed that this algorithm has not only a good noisy performance, but it also moderates parasite eects of the limited angular ray-coverage in borehole tomography. The eects are visible in the form of vertical smearings from inhomogeneous features in the image.