Edge-Promoting Adaptive Bayesian Experimental Design for X-ray Imaging

Abstract

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . This work considers sequential edge-promoting Bayesian experimental design for (discretized) linear inverse problems, exemplified by X-ray tomography. The process of computing a total variation--type reconstruction of the absorption inside the imaged body via lagged diffusivity iteration is interpreted in the Bayesian framework. Assuming a Gaussian additive noise model, this leads to an approximate Gaussian posterior with a covariance structure that contains information on the location of edges in the posterior mean. The next projection geometry is then chosen through A-or D-optimal Bayesian design, which corresponds to minimizing the trace or the determinant of the updated posterior covariance matrix that accounts for the new projection. Two-and three-dimensional numerical examples based on simulated data demonstrate the functionality of the introduced approach.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . X-ray tomography, optimal projections, Bayesian experimental design, A-optimality, D-optimality, adaptivity, edge-promoting prior, lagged diffusivity \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs .

Bayesian design of measurements for magnetorelaxometry imaging *

“…However, if one proceeds sequentially, choosing the specifications for the next activation only after computing a maximum a posteriori (MAP) estimate for the MNP distribution via the lagged diffusivity iteration [39] based on the measurements from the previous activations, it is possible to interpret the MAP estimate as the mean of a Gaussian distribution whose covariance matrix is available as a side product of the iteration [5,8]. Basing the selection of the specifications for the next dipole activation on this covariance structure, one can devise a sequential Bayesian OED algorithm that adapts to the already collected data and has potential to produce edge-promoting experimental designs; see [19] for an application of this idea to sequential x-ray imaging. Let us also note that the sequential optimization approach can be extended to a non-parametric setting with convex priors (such as the smoothened TV) for which the posterior has good approximation properties by Gaussian distributions in the neighborhood of non-parametric MAP estimators [18].…”

“…The considered sequential approach to Bayesian OED has previously been investigated for choosing optimal projection geometries in x-ray imaging with a Gaussian prior in [7] and with a TV prior in [19]. However, these papers do not tackle simultaneous optimization of many projection geometries with a Gaussian prior due to computational restrictions.…”

“…Moreover, compared to x-ray tomography with a limited projection aperture, the measurement matrix for a single activation in MRXI is more ill-conditioned and provides information about the entire imaged object. Hence, the functionality of the sequential OED algorithm with the smoothened TV prior for MRXI is not obvious based solely on the material in [19].…”

“…The (simplified and discretized) measurement model of MRXI is described in section 2. Section 3 introduces the Bayesian framework for inverse problems and recalls the probabilistic interpretation of the lagged diffusivity iteration from [5,19]. In section 4, the optimality targets for A-and D-optimal designs are introduced, and their minimization is considered.…”

“…This work considers Bayesian optimal experimental design (OED) for choosing positions and orientations of activation coils in a simplified model for magnetorelaxometry imaging (MRXI). We implement both an offline algorithm for simultaneous optimization of multiple consecutive activations assuming a Gaussian prior for the imaged magnetic nanoparticle (MNP) distribution and an adaptive algorithm based on a total variation (TV) prior and the lagged diffusivity iteration as introduced for sequential x-ray imaging in [19].…”