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Spotlight inversion by orthogonal projections

Daniela Calvetti, Nuutti Hyvönen, Ville Kolehmainen, Erkki Somersalo

TL;DR

Spotlight inversion employs orthogonal projections to suppress nuisance parameters in linear inverse problems, enabling ROI-focused recovery by projecting onto the orthogonal complement of the nuisance subspace and solving a reduced system $b'={\mathsf A}_1'x_1+\varepsilon'$. In noiseless, square systems this yields exact recovery of the targeted components, while in ill-posed problems it integrates with standard regularization or Gaussian priors, with a Bayesian interpretation clarified through marginalization versus conditioning and a uninformative-prior limit. The paper develops truncation schemes for ill-determined nuisance ranks, including a clutter-to-noise ratio criterion and randomized SVD strategies, and demonstrates the approach on two computed examples: PDE-model reduction and local tomography ROI reconstruction. The method provides a computationally efficient, linear-algebraic alternative to Bayesian marginalization for nuisance suppression, with practical impact in high-dimensional imaging and inverse problems, and it offers avenues for extension to nonlinear problems and connections to classical regression theory.

Abstract

Many computational problems involve solving a linear system of equations, although only a subset of the entries of the solution are needed. In inverse problems, where the goal is to estimate unknown parameters from indirect noisy observations, it is not uncommon that the forward model linking the observed variables to the unknowns depends on variables that are not of primary interest, often referred to as nuisance parameters. In this article, we consider linear problems, and propose a novel projection technique to eliminate, or at least mitigate, the contribution of the nuisance parameters in the model. We refer to this approach as spotlight inversion, as it allows to focus on only the portion of primary interest of the unknown parameter vector, leaving the uninteresting part in the shadow. The viability of the approach is illustrated with two computed examples, one where it works as model reduction for a finite element approximation of an elliptic PDE, the other amounting to local fanbeam X-ray tomography, spotlighting the region of interest that is part of the full target.

Spotlight inversion by orthogonal projections

TL;DR

Spotlight inversion employs orthogonal projections to suppress nuisance parameters in linear inverse problems, enabling ROI-focused recovery by projecting onto the orthogonal complement of the nuisance subspace and solving a reduced system . In noiseless, square systems this yields exact recovery of the targeted components, while in ill-posed problems it integrates with standard regularization or Gaussian priors, with a Bayesian interpretation clarified through marginalization versus conditioning and a uninformative-prior limit. The paper develops truncation schemes for ill-determined nuisance ranks, including a clutter-to-noise ratio criterion and randomized SVD strategies, and demonstrates the approach on two computed examples: PDE-model reduction and local tomography ROI reconstruction. The method provides a computationally efficient, linear-algebraic alternative to Bayesian marginalization for nuisance suppression, with practical impact in high-dimensional imaging and inverse problems, and it offers avenues for extension to nonlinear problems and connections to classical regression theory.

Abstract

Many computational problems involve solving a linear system of equations, although only a subset of the entries of the solution are needed. In inverse problems, where the goal is to estimate unknown parameters from indirect noisy observations, it is not uncommon that the forward model linking the observed variables to the unknowns depends on variables that are not of primary interest, often referred to as nuisance parameters. In this article, we consider linear problems, and propose a novel projection technique to eliminate, or at least mitigate, the contribution of the nuisance parameters in the model. We refer to this approach as spotlight inversion, as it allows to focus on only the portion of primary interest of the unknown parameter vector, leaving the uninteresting part in the shadow. The viability of the approach is illustrated with two computed examples, one where it works as model reduction for a finite element approximation of an elliptic PDE, the other amounting to local fanbeam X-ray tomography, spotlighting the region of interest that is part of the full target.

Paper Structure

This paper contains 11 sections, 4 theorems, 108 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Spotlight projections for Model Reduction in Noiseless Case
  3. Spotlight projections for inverse problems
  4. Marginalization and error modeling
  5. Projection versus marginalization
  6. Partial projection, ill-determined rank and underdetermined problems
  7. Effect of Spotlighting on the Signal-to-Noise Ratio
  8. Computed examples
  9. Forward model of the electrical impedance tomography
  10. Local tomography
  11. Conclusions and outlook

Key Result

Theorem 2.1

If ${\mathsf A}\in {\mathbb R}^{n\times n}$ is invertible, and ${\mathsf A} x = b$, then $x_1\in{\mathbb R}^k$ can be uniquely recovered by solving the projected least squares problem

Figures (8)

  • Figure 1: Two FEM meshes with boundary refinement. The left one is used for data generation, the right one for solving the inverse problem. The 32 electrodes are indicated by the red line segments around the domain.
  • Figure 2: Interior potential in $\Omega\setminus\Omega_c$ computed by using the full model (left) and the spotlight projections (middle). The difference, shown on the right, is negligible.
  • Figure 3: Electrode voltages computed by the spotlight inversion (black dots), and the reference arising from the full model (red squares).
  • Figure 4: Left: The MAP estimate from local tomography data using the full domain model $b={\mathsf A} x$. Right: Spotlight part from the MAP estimate on the left. We denote the spotlight part as $x^{\ast}_1$ and use it as ground truth reference for the estimates computed using the spotlight model ${\mathsf A}_1 x_1$.
  • Figure 5: Clutter-to-noise ratio $R_r$ for the local tomography experiment. Horizontal axis is the number $r$ of svd vectors (i.e., number of columns in $U_r$). The point where $R_r =1$ is at $r=587$, denoted by the red dashed line.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3