A computational Bayesian inversion model is demonstrated. It is discretization invariant, describes prior information using function spaces with a wavelet basis and promotes reconstructions that are sparse in the wavelet transform domain. The method makes use of the Besov space prior with p = 1, q = 1 and s = 1, which is related to the total variation prior. Numerical evidence is presented in the context of a one-dimensional deconvolution task, suggesting that edgepreserving and noise-robust reconstructions can be achieved consistently at various resolutions.
Dental tomographic cone-beam x-ray imaging devices record truncated projections and reconstruct a region of interest (ROI) inside the head. Image reconstruction from the resulting local tomography data is an ill-posed inverse problem. A new Bayesian multiresolution method is proposed for local tomography reconstruction. The inverse problem is formulated in a well-posed statistical form where a prior model of the target tissues compensates for the incomplete x-ray projection data. Tissues are represented in a wavelet basis, and prior information is modeled in terms of a Besov norm penalty. The number of unknowns in the reconstruction problem is reduced by abandoning fine-scale wavelets outside the ROI. Compared to traditional voxel-based models, this multiresolution approach allows significant reduction of degrees of freedom without loss of accuracy inside the ROI, as shown by 2D examples using simulated and in vitro local tomography data.
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.
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