Feature reconstruction in inverse problems

A strategy for the derivation of fast, accurate and stable algorithms for combining the reconstruction and the feature extraction step for solving linear ill-posed problems in just one method is presented. The precomputation of special reconstruction kernels with optimized parameters for the combination of the two tasks allows for fast implementations and better results than separate realizations. The concept of order optimality is generalized to the solution of feature reconstruction and to Banach spaces in order to find criteria for the selection of suitable mollifiers. Results from real data in different tomographic modalities and scanning geometries are presented with the direct calculation of derivatives, as in Canny edge detectors, and the Laplacian of the solution used in many segmentation algorithms. The method works also when the searched-for solution is not smooth or when the data are very noisy. This shows the versatility of the approach.