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Data-driven Morozov regularization of inverse problems

Markus Haltmeier, Richard Kowar, Markus Tiefenthaler

TL;DR

This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization, and proposes a refined training strategy that improves adaptation to ill-posed problems compared to NETT’s original strategy, which primarily focuses on addressing non-uniqueness.

Abstract

The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT's original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer.

Data-driven Morozov regularization of inverse problems

TL;DR

This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization, and proposes a refined training strategy that improves adaptation to ill-posed problems compared to NETT’s original strategy, which primarily focuses on addressing non-uniqueness.

Abstract

The solution of inverse problems is crucial in various fields such as medicine, biology, and engineering, where one seeks to find a solution from noisy observations. These problems often exhibit non-uniqueness and ill-posedness, resulting in instability under noise with standard methods. To address this, regularization techniques have been developed to balance data fitting and prior information. Recently, data-driven variational regularization methods have emerged, mainly analyzed within the framework of Tikhonov regularization, termed Network Tikhonov (NETT). This paper introduces Morozov regularization combined with a learned regularizer, termed DD-Morozov regularization. Our approach employs neural networks to define non-convex regularizers tailored to training data, enabling a convergence analysis in the non-convex context with noise-dependent regularizers. We also propose a refined training strategy that improves adaptation to ill-posed problems compared to NETT's original strategy, which primarily focuses on addressing non-uniqueness. We present numerical results for attenuation correction in photoacoustic tomography, comparing DD-Morozov regularization with NETT using the same trained regularizer, both with and without an additional total variation regularizer.
Paper Structure (26 sections, 4 theorems, 16 equations, 4 figures, 2 tables)

This paper contains 26 sections, 4 theorems, 16 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Morozov regularization
  3. Neural network regularizers
  4. Outline
  5. Theory
  6. Convergence analysis
  7. Training strategy
  8. Application
  9. Attenuation correction in PAT using the NSW model
  10. PAT in attenuating media:
  11. The NSW model:
  12. Attenuation correction:
  13. Note on the ill-posedness:
  14. Implementation details
  15. Forward operator:
  16. ...and 11 more sections

Key Result

Lemma 2.2

The regularizers $\mathcal{R}, \mathcal{R}_\delta$ are coercive and weakly sequentially lower semicontinuous. Further, the feasible set $\{x \in \mathbb X \mid \lVert\mathbf A x- y_\delta\rVert \leq \delta \}$ is weakly closed and non-empty for all $\delta >0$ and all data $y_\delta$ with $\lVert\m

Figures (4)

  • Figure 1: Left: Singular values of the $\mathbf A \in\mathds{R}^{d \times d}$ modeling dissipation on a logarithmic scale. Middle: Test signal $x \in \mathds{R}^d$. Right: Exact data $\mathbf A x$ for input from the middle picture. The horizontal axis in the middle and right images represents time.
  • Figure 2: Top left: Noisy data $y_\delta$. Top right: BP reconstruction. Bottom left: SVD reconstruction. Bottom right: DD-Morozov regularization .
  • Figure 3: Comparison of several methods using either the network output (top) or the zero signal as initial guess (bottom). The columns from left to right show Tikhonov regularization with TV as the regularizer, the NETT, DD-Morozov without TV, and DD-Morozov with TV as an additional regularizer. For the network as initial guess, the truncated SVD reconstruction applied to noisy data has been used as input for the neural network that has been trained for the regularizer.
  • Figure 4: Mean value and standard deviation of the $\ell^2$ reconstruction error as a function of the noise level $\delta$.

Theorems & Definitions (9)

  • Lemma 2.2
  • proof
  • Lemma 2.3: Existence
  • proof
  • Theorem 2.4: Weak convergence
  • proof
  • Theorem 2.6: Stong convergence
  • proof
  • Remark 2.7: Choice of the perturbations