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Learned RESESOP for solving inverse problems with inexact forward operator

Mathias S. Feinler, Bernadette N. Hahn

TL;DR

This work proposes a learned version of ReSeSOp which allows to approximate inexactness levels on the fly and generalizes established unrolled iterative reconstruction schemes to inexact forward operators and is particularly tailored to the structure of dynamic problems.

Abstract

When solving inverse problems, one has to deal with numerous potential sources of model inexactnesses, like object motion, calibration errors, or simplified data models. Regularized Sequential Subspace Optimization (ReSeSOp) allows to compensate for such inaccuracies within the reconstruction step by employing consecutive projections onto suitably defined subspaces. However, this approach relies on a priori estimates for the model inexactness levels which are typically unknown. In dynamic imaging applications, where inaccuracies arise from the unpredictable dynamics of the object, these estimates are particularly challenging to determine in advance. To overcome this limitation, we propose a learned version of ReSeSOp which allows to approximate inexactness levels on the fly. The proposed framework generalizes established unrolled iterative reconstruction schemes to inexact forward operators and is particularly tailored to the structure of dynamic problems. We also present a comprehensive mathematical analysis regarding the effect of dependencies within the forward problem, clarifying when and why dividing the overall problem into subproblems is essential. The proposed method is evaluated on various examples from dynamic imaging, including datasets from a rheological CT experiment, brain MRI, and real-time cardiac MRI. The respective results emphasize improvements in reconstruction quality while ensuring adequate data consistency.

Learned RESESOP for solving inverse problems with inexact forward operator

TL;DR

This work proposes a learned version of ReSeSOp which allows to approximate inexactness levels on the fly and generalizes established unrolled iterative reconstruction schemes to inexact forward operators and is particularly tailored to the structure of dynamic problems.

Abstract

When solving inverse problems, one has to deal with numerous potential sources of model inexactnesses, like object motion, calibration errors, or simplified data models. Regularized Sequential Subspace Optimization (ReSeSOp) allows to compensate for such inaccuracies within the reconstruction step by employing consecutive projections onto suitably defined subspaces. However, this approach relies on a priori estimates for the model inexactness levels which are typically unknown. In dynamic imaging applications, where inaccuracies arise from the unpredictable dynamics of the object, these estimates are particularly challenging to determine in advance. To overcome this limitation, we propose a learned version of ReSeSOp which allows to approximate inexactness levels on the fly. The proposed framework generalizes established unrolled iterative reconstruction schemes to inexact forward operators and is particularly tailored to the structure of dynamic problems. We also present a comprehensive mathematical analysis regarding the effect of dependencies within the forward problem, clarifying when and why dividing the overall problem into subproblems is essential. The proposed method is evaluated on various examples from dynamic imaging, including datasets from a rheological CT experiment, brain MRI, and real-time cardiac MRI. The respective results emphasize improvements in reconstruction quality while ensuring adequate data consistency.

Paper Structure

This paper contains 26 sections, 4 theorems, 58 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Interpreting the inexactness as noise
  3. Explicit estimation strategies to account for the inexactness
  4. Adapting data consistency terms
  5. Regularized sequential subspace optimization (ReSeSOp)
  6. Outline of the paper
  7. Our solution framework using regularized sequential subspace optimization
  8. The basic principle of ReSeSOp
  9. The case of exact data and exact operator
  10. ReSeSOp for noisy data and inexact operator
  11. Application in dynamic imaging and accounting for local model inexactnesses
  12. Learned ReSeSOp
  13. ReSeSOp with an additional regularizer
  14. Learned ReSeSOp algorithm
  15. Training
  16. ...and 11 more sections

Key Result

Lemma 3.1

If all suboperators are pairwise orthogonal, we can treat subproblems individually.

Figures (7)

  • Figure 1: $\text{NN}_\theta$ with parameters $\theta$ for images of resolution $N\times N=192 \times 192$. The outputs are the regularization term $R^{(k)}$ and stepsizes $\{\kappa_i^{(k)}\}_i$. The inputs are the iterate $s^{(k)}$, old stepsizes $\{\kappa_i^{(k-1)}\}_i$, current inexactness levels $\{\mathcal{E}_i^{(k)} = \|\mathcal{A}^\eta_i s^{(k)} - y_i\|\}_i$ and search directions $\{u_i^{(k)}\}_i$.
  • Figure 2: Comparison of rheological Flow CT case. Resolution: $288 \times 288$, $N^\mathrm{dir} = 16$, 192 angles distributed over 180 degrees and 406 receiver points.
  • Figure 3: Visualization of one representative set of parameters for uniformly and non-uniformly distributed deformation. The deformation at $i=8$ is designed as the reference configuration. Consequently, we only visualize $\alpha_i - \alpha_8$, and analogously for other parameters.
  • Figure 4: Comparison on dynamic case with CARTESIAN trajectory of learned ReSeSOp, CG and learned primal to the ground truth. Resolution : $384\times384$, $N^\mathrm{dir}=15, N^\mathrm{coils} = 15$. non-uniformly distributed deformations
  • Figure 5: Comparison on dynamic case with RADIAL trajectory of learned ReSeSOp, CG and learned primal to the ground truth. Resolution : $384\times384$, $N^\mathrm{dir}=15, N^\mathrm{coils} = 15$. uniformly distributed deformations
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Corollary 3.3.1
  • Remark