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A bilinear inverse problem with forward operator inaccuracy applied to neonatal atlas-based diffuse optical tomography

Aada Hakula, Pauliina Hirvi, Nuutti Hyvönen

Abstract

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.

A bilinear inverse problem with forward operator inaccuracy applied to neonatal atlas-based diffuse optical tomography

Abstract

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.
Paper Structure (34 sections, 3 theorems, 62 equations, 8 figures, 2 tables)

This paper contains 34 sections, 3 theorems, 62 equations, 8 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Bilinear Inverse Problem
  3. Application to Diffuse Optical Tomography
  4. Outline
  5. Bilinear Finite-Dimensional Inverse Problem
  6. Modeling Forward Operator Inaccuracy
  7. Notation and Deterministic Formulation
  8. Bayesian Formulation
  9. On computing a maximum MAP estimate
  10. On Properties of the Tikhonov Functional
  11. Block Coordinate Descent
  12. Gauss--Newton Algorithm
  13. Conditional Mean Estimate
  14. Gibbs Sampler
  15. Linearized Diffuse Optical Tomography
  16. ...and 19 more sections

Key Result

Proposition 2.2

\newlabelprop:marg_post0 Assume the measurement model eq:Btensor_inv_prob and that $E$, $Y$ and $X$ are a priori mutually independent with the distributions eq:priors. Then, the posterior density of $X$ given $B = b$ is where the omitted constant does not depend on $x$ and are, respectively, the covariance matrices of the Gaussian distributions for $Y$ given $B=b$ and $X = x$ and for $B$ given

Figures (8)

  • Figure 1: Visualization of $\Phi(x,y)$ from \ref{['eq:simple_Tikhonov']}. Left: $\beta = 0.1$, the diamond corresponds to a saddle point and the dots mark the local minimizers. Right: $\beta = 1$ and the dot marks the global minimizer.
  • Figure 1: a) The 15 sources (red crosses), 21 detectors (blue rings) and seven perturbation sites (yellow circles; 1--7) on the reference neonate. Nz = nasion, Cz = vertex, Iz = inion, LPA = left pre-auricular point. b) Perturbations at location 2. GM = gray matter, WM = white matter, S&S = scalp and skull, CSF = cerebrospinal fluid.
  • Figure 1: The median of registration errors for the 214 other neonates in the atlas considering one registered neonate head as the target (horizontal axis, ordered by age). The error for each neonate pair is the mean distance between the registered surface meshes under the measurement probe. The actual common target (red square) in the registration was number 182. Blue circles refer to the selections in Table \ref{['table:cases']}.
  • Figure 2: Statistics for the relative error of the PC representation for the rows of the Jacobians in the 2-mm resolution with the first 10 row-wise PCs. (a) log-amplitude. (b) phase. The squares depict the mean relative errors and the vertical lines their standard deviations for rows in the SDSs ranges indicated on the horizontal axis.
  • Figure 3: First row: Case 3. Second row: Case 4. Reconstructions averaged over axial slices covering the perturbed region, computed with the true Jacobian of the target neonate (a, d), the mean Jacobian over the other 214 neonates registered to the target head shape (b, e), and by improving the rows of the mean Jacobian along their first 10 PCs with the G--N algorithm (c, f). The black lines mark the target perturbations with contrast of 6 m$^{-1}$ or 8 m$^{-1}$ (30.6 $\mu$M or 40.8 $\mu$M at 798 nm, respectively).
  • ...and 3 more figures

Theorems & Definitions (10)

  • Remark 2.1
  • Proposition 2.2
  • Proof 1
  • Remark 2.3
  • Proposition 3.1
  • Proof 2
  • Example 1
  • Theorem 3.2
  • Proof 3
  • Remark 5.1