Esa Niemi

Heikki Haario, Aki Kallonen, Marko Laine et al.

A two-dimensional tomographic problem is studied. The target is assumed to be a homogeneous object bounded by a smooth curve. A Non Uniform Rational Basis Splines (NURBS) curve is used as computational representation of the boundary. This approach conveniently provides the result in a format readily compatible with computer-aided design (CAD) software. However, the linear tomography task becomes a nonlinear inverse problem due to the NURBS-based parameterization. Therefore, Bayesian inversion with Markov chain Monte Carlo (MCMC) sampling is used for calculating an estimate of the NURBS control points. The reconstruction method is tested with both simulated data and measured X-ray projection data. The proposed method recovers the shape and the attenuation coefficient significantly better than the baseline algorithm (optimally thresholded total variation regularization), but at the cost of heavier computation.

Kati Niinimäki, Matti Lassas, Keijo Hämäläinen et al.

A computational method is introduced for choosing the regularization parameter for total variation (TV) regularization. The approach is based on computing reconstructions at a few different resolutions and various values of regularization parameter. The chosen parameter is the smallest one resulting in approximately discretization-invariant TV norms of the reconstructions. The method is tested with X-ray tomography data measured from a walnut and compared to the S-curve method. The proposed method seems to automatically adapt to the desired resolution and noise level, and it yields useful results in the tests. The results are comparable to those of the S-curve method; however, the S-curve method needs a priori information about the sparsity of the unknown, while the proposed method does not need any a priori information (apart from the choice of a desired resolution). Mathematical analysis is presented for (partial) understanding of the properties of the proposed parameter choice method. It is rigorously proven that the TV norms of the reconstructions converge with any choice of regularization parameter.