T. Hohage

D. Fournier, L. Gizon, M. Holzke et al.

A major goal of helioseismology is the three-dimensional reconstruction of the three velocity components of convective flows in the solar interior from sets of wave travel-time measurements. For small amplitude flows, the forward problem is described in good approximation by a large system of convolution equations. The input observations are highly noisy random vectors with a known dense covariance matrix. This leads to a large statistical linear inverse problem. Whereas for deterministic linear inverse problems several computationally efficient minimax optimal regularization methods exist, only one minimax-optimal linear estimator exists for statistical linear inverse problems: the Pinsker estimator. However, it is often computationally inefficient because it requires a singular value decomposition of the forward operator or it is not applicable because of an unknown noise covariance matrix, so it is rarely used for real-world problems. These limitations do not apply in helioseismology. We present a simplified proof of the optimality properties of the Pinsker estimator and show that it yields significantly better reconstructions than traditional inversion methods used in helioseismology, i.e.\ Regularized Least Squares (Tikhonov regularization) and SOLA (approximate inverse) methods. Moreover, we discuss the incorporation of the mass conservation constraint in the Pinsker scheme using staggered grids. With this improvement we can reconstruct not only horizontal, but also vertical velocity components that are much smaller in amplitude.

T. Hohage, C. Rügge

Processing of Diffusion MRI data obtained from High Angular Resolution measurements consists of a series of steps, starting with the estimation of an orientation distribution function (ODF), which is then used as input for e.g. tractography algorithms. It is important that ODF reconstruction methods yield accurate, coherent ODFs, in particular for low SNR or coarsely sampled data sets. As the diffusion process is modelled independently in each voxel, reconstructions are often carried out for each voxel separately, disregarding the observation that neighboring voxels are often quite similar if they belong to the same fiber structure. There are surprisingly few approaches that make use of this kind of spatial regularity to improve coherence and stability of the reconstruction. In this work, we focus on a variation of a method proposed by Reisert and Kiselev based on the concept of fiber continuity. The method has already been shown to yield good numerical results, but has not yet been analyzed theoretically. Under suitable smoothness assumptions, we apply results on constrained Tikhonov-type regularization with approximate operator to show convergence of reconstructions from discrete, noisy data for linear forward models. Further, we numerically illustrate the performance of the method on phantom and in-vivo data.