# American Institute of Mathematical Sciences

February  2007, 1(1): 1-11. doi: 10.3934/ipi.2007.1.1

## Microlocal sequential regularization in imaging

 1 Case Western Reserve University, Department of Mathematics and Center for Modelling Integrated Metabolic Systems, 10900 Euclid Avenue, Cleveland, OH 44106, United States 2 Helsinki University of Technology, Department of Mathematics, P.O. Box 1100, FIN-02015 HUT, Finland

Received  September 2006 Published  January 2007

In this article, we consider imaging problems in which the data consist of noisy observations of the true image through a linear filter such as blurring, sparse sampling or tomographic projections. The image restoration problem is ill-posed and in order to obtain a meaningful result, the problem needs to be regularized or augmented by additional information. In this article, we consider Tikhonov regularization by a class of non-linear smoothness filters that are capable of detecting and restoring edges in the image. The regularization function is microlocal in the sense that it is sensitive to the location and the direction of the non-smoothness of the image. The implementation of the algorithm leads to a sequence of simple linear least squares problems, the penalty term being calculated as a direction-sensitive weighted finite difference approximation of the Laplacian. The algorithm is applied to two classical imaging problems, image zooming and limited angle tomography.
Citation: Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems & Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1
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