A Mumford-Shah-Like Method for Limited Data Tomography with an Application to Electron Tomography

In this article the Mumford-Shah-like method of [R. Ramlau and W. Ring, J. Comput. Phys., 221 (2007), pp. 539-557] for complete tomographic data is generalized and applied to limited angle and region of interest tomography data. With the Mumford-Shah-like method, one reconstructs a piecewise constant function and simultaneously a segmentation from its (complete) Radon transform data. For limited data, the ability of the Mumford-Shah-like method to find a segmentation, and by that the singularity set of a function, is exploited. The method is applied to generated data from a torso phantom. The results demonstrate the performance of the method in reconstructing the singularity set, the density distribution itself for limited angle data, and also some quantitative information about the density distribution for region of interest data. As a second example limited angle region of interest tomography is considered as a simplified model for electron tomography (ET). For this problem we combine Lambda tomography and the Mumford-Shah-like method. The combined method is applied to simulated ET data.