Aada Hakula

Aada Hakula, Pauliina Hirvi, Nuutti Hyvönen

Linear inverse problems are highly common in practical real-world applications from industry to medical imaging. The forward operator is often built on some approximations of the studied system. Handling inaccuracies in the forward operator in the context of inverse problems is a relatively unstudied problem. In this work, we assume that we have a set of candidate forward operator matrices and suggest principal component analysis (PCA) for modeling their variation from the mean. We combine the original linear problem with the included forward operator inaccuracy into a bilinear tensor inverse problem and present two optimization algorithms and Gibbs sampling for approximately solving the problem. As a real-world test case, we apply the algorithms to account for the inaccuracy that is present in the sensitivity profiles or Jacobian matrices in diffuse optical tomography when an atlas-based model of the head anatomy is used instead of the subject's own anatomical model in neonates over a wide range of gestational ages (29--44 weeks). We report visual and numerical improvements in the spatial localization and contrast-to-noise-ratio in reconstructions of simulated hemodynamic activity.

Aada Hakula, Pauliina Hirvi, Nuutti Hyvönen et al.

This paper presents a projection-based technique to mitigate the impact of modeling errors related to domain truncation, changes in the optode coupling coefficients, and misspecified optical parameters of different tissue types in diffuse optical tomography. The approach considers the primary Jacobian matrix of the forward map in the image reconstruction scheme, linking the primary unknown, i.e., the per-voxel absorption coefficient changes in the region of interest, to the optode measurements, as well as the nuisance Jacobians that do the same for the auxiliary unknown parameters of secondary interest. To mitigate mismodeled coupling coefficients or domain truncation, the method projects the linearized forward model defined by the primary Jacobian onto the orthogonal complement of the range of a nuisance Jacobian, or onto the orthogonal complement of the span of a number of first left singular vectors for the nuisance Jacobian that has been weighted to account for prior information on the measurement setup. In the case of a misspecified baseline optical parameter for some tissue type, the nullspace of the utilized orthogonal projection is defined to be the span of first left singular vectors for a (weighted) difference of two Jacobian matrices evaluated at two different levels for the considered tissue-wise optical parameter. The reconstruction is formed by applying Bayesian inversion with Gaussian prior and noise models to the projected linearized equation. We evaluate the method on simulated brain activity data obtained via Monte Carlo simulations of the radiative transfer equation in a voxelized head anatomy for a neonate with combined gestational and chronological age of 41.7 weeks.