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Adaptive Weighted Total Variation boosted by learning techniques in few-view tomographic imaging

Elena Morotti, Davide Evangelista, Andrea Sebastiani, Elena Loli Piccolomini

TL;DR

This work tackles few-view computed tomography reconstruction under unknown noise levels by introducing a spatially adaptive weighted total variation regularization. The weights are computed from a fast intermediate image produced by a neural reconstructor, and they are fixed from the outset to preserve a rigorous variational framework; the overall objective is $\min_{\boldsymbol{x}\in\mathcal{X}} \|\boldsymbol{K}\boldsymbol{x}-\boldsymbol{y}^\delta\|_2^2 + \lambda TV_{\boldsymbol{w}}(\boldsymbol{x})$ with $TV_{\boldsymbol{w}}(\boldsymbol{x}) = \sum_i w_i \sqrt{(D_h\boldsymbol{x})_i^2+(D_v\boldsymbol{x})_i^2}$. The paper proves existence, uniqueness, and stability (noise and reconstructor) for the $\Psi$-$W\ell_1$ formulation and demonstrates through synthetic COULE data and Mayo real CT data that the approach outperforms global TV and iterative reweighting baselines, achieving high-fidelity reconstructions from very sparse views. It also shows that gradient-focused training of the weight network improves edge and texture preservation, while end-to-end networks alone remain less stable under unseen noise. The method has practical implications for reducing radiation dose and acquisition time in CT, by enabling reliable reconstructions from limited-angle measurements. Theoretical results and numerical gains together underpin a robust, interpretable framework combining learning with variational regularization.

Abstract

This study presents the development of a spatially adaptive weighting strategy for Total Variation regularization, aimed at addressing under-determined linear inverse problems. The method leverages the rapid computation of an accurate approximation of the true image (or its gradient magnitude) through a neural network. Our approach operates without requiring prior knowledge of the noise intensity in the data and avoids the iterative recomputation of weights. Additionally, the paper includes a theoretical analysis of the proposed method, establishing its validity as a regularization approach. This framework integrates advanced neural network capabilities within a regularization context, thereby making the results of the networks interpretable. The results are promising as they enable high-quality reconstructions from limited-view tomographic measurements.

Adaptive Weighted Total Variation boosted by learning techniques in few-view tomographic imaging

TL;DR

This work tackles few-view computed tomography reconstruction under unknown noise levels by introducing a spatially adaptive weighted total variation regularization. The weights are computed from a fast intermediate image produced by a neural reconstructor, and they are fixed from the outset to preserve a rigorous variational framework; the overall objective is with . The paper proves existence, uniqueness, and stability (noise and reconstructor) for the - formulation and demonstrates through synthetic COULE data and Mayo real CT data that the approach outperforms global TV and iterative reweighting baselines, achieving high-fidelity reconstructions from very sparse views. It also shows that gradient-focused training of the weight network improves edge and texture preservation, while end-to-end networks alone remain less stable under unseen noise. The method has practical implications for reducing radiation dose and acquisition time in CT, by enabling reliable reconstructions from limited-angle measurements. Theoretical results and numerical gains together underpin a robust, interpretable framework combining learning with variational regularization.

Abstract

This study presents the development of a spatially adaptive weighting strategy for Total Variation regularization, aimed at addressing under-determined linear inverse problems. The method leverages the rapid computation of an accurate approximation of the true image (or its gradient magnitude) through a neural network. Our approach operates without requiring prior knowledge of the noise intensity in the data and avoids the iterative recomputation of weights. Additionally, the paper includes a theoretical analysis of the proposed method, establishing its validity as a regularization approach. This framework integrates advanced neural network capabilities within a regularization context, thereby making the results of the networks interpretable. The results are promising as they enable high-quality reconstructions from limited-view tomographic measurements.
Paper Structure (19 sections, 17 theorems, 61 equations, 8 figures, 3 tables)

This paper contains 19 sections, 17 theorems, 61 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Structure of the paper.
  4. The proposed adaptive weighted TV model
  5. Experiments on a synthetic image
  6. Weights Computation via a Neural Network
  7. On the well-posedness of the proposed regularization method
  8. Numerical Results
  9. Experiments on COULE synthetic dataset
  10. Experiments on Mayo real medical image dataset
  11. Conclusions
  12. Proof of well-posedness of Psi-WL1 scheme
  13. Proof of Lemma \ref{['lemma:coercivity']}
  14. Proof of Lemma \ref{['lemma:TV_gradient']}
  15. Proof of Lemma \ref{['lemma:J_properties_on_M']}
  16. ...and 4 more sections

Key Result

Proposition 1

For any $\eta > 0$, any $p\in(0,1)$, and any $\tilde{\boldsymbol{x}} \in \mathbb{R}^n$, $\left(w(\tilde{\boldsymbol{x}})\right)_i \in (0, 1]$, $\forall i = 1, \dots, n$. Moreover, $\left(\boldsymbol{w}(\tilde{\boldsymbol{x}})\right)_i = 1$ on a pixel $i \in 1, \dots, n$ if and only if $(| \boldsymbo

Figures (8)

  • Figure 1: A plot of $\left(\boldsymbol{w}(\tilde{\boldsymbol{x}})\right)_i$ for different values of $\eta$, over $(| \boldsymbol{D} \tilde{\boldsymbol{x}} |)_i$, for $p=0.3$.
  • Figure 2: Workflow of the considered scheme, where the reconstructor $\Psi$ is used to achieve a useful image $\tilde{\boldsymbol{x}} = \Psi(\boldsymbol{y}^\delta)$ for adaptive $\ell_1$ regularization.
  • Figure 3: Results of the experiment on the synthetic image with higher noise ($\nu=0.02$). In the first row, from left to right: the entire ground truth image with a red square depicting the crop of interest, cropped zooms on the reconstructions by GT-$W\ell_1$, FBP-$W\ell_1$, TV-$W\ell_1$ and by the global TV model. In the second row: plot of the Relative Error over the iterations.
  • Figure 4: The reconstructor $\Psi$ when it is constituted by two steps: a Filtered Back Projection and a neural network.
  • Figure 5: Images used as ground truth in the numerical experiments with some zoom-ins remarking regions of interest. On the left, the $\boldsymbol{x}^{GT}$ coming from the test subset of COULE; on the right, the one from the Mayo Clinic dataset.
  • ...and 3 more figures

Theorems & Definitions (40)

  • Proposition 1
  • proof
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1: Existence
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 30 more