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The method of the approximate inverse for limited-angle CT

Bernadette Hahn, Gael Rigaud, Richard Schmähl

TL;DR

This work tackles limited-angle CT, where incomplete angular data cause streaks and missing details. It introduces a model-driven reconstruction framework based on the method of the approximate inverse to build reconstruction kernels (LARK) that operate directly on measured data, augmented by a spectral filter and mollification to stabilize ill-posedness. To cope with real, semi-discrete data, the authors develop a constrained variant (CLARK) that couples the kernel with a denoising/regularization step, and provide error analysis for semi-discrete settings alongside practical discretization strategies. The approach is validated on synthetic phantoms and real measurements, showing improved feature recovery and reduced artefacts compared to FBP and TV, while highlighting dependence on missing angle extent and noise. Overall, LARK/CLARK offer a principled, interpretable starting point for reconstruction in severely ill-posed limited-angle CT and may serve as a foundation for learning-based enhancements.

Abstract

Limited-angle computerized tomography stands for one of the most difficult challenges in imaging. Although it opens the way to faster data acquisition in industry and less dangerous scans in medicine, standard approaches, such as the filtered backprojection (FBP) algorithm or the widely used total-variation functional, often produce various artefacts that hinder the diagnosis. With the rise of deep learning, many modern techniques have proven themselves successful in removing such artefacts but at the cost of large datasets. In this paper, we propose a new model-driven approach based on the method of the approximate inverse, which could serve as new starting point for learning strategies in the future. In contrast to FBP-type approaches, our reconstruction step consists in evaluating linear functionals on the measured data using reconstruction kernels that are precomputed as solution of an auxiliary problem. With this problem being uniquely solvable, the derived limited-angle reconstruction kernel (LARK) is able to fully reconstruct the object without the well-known streak artefacts, even for large limited angles. However, it inherits severe ill-conditioning which leads to a different kind of artefacts arising from the singular functions of the limited-angle Radon transform. The problem becomes particularly challenging when working on semi-discrete (real or analytical) measurements. We develop a general regularization strategy, named constrained limited-angle reconstruction kernel (CLARK), by combining spectral filter, the method of the approximate inverse and custom edge-preserving denoising in order to stabilize the whole process. We further derive and interpret error estimates for the application on real, i.e. semi-discrete, data and we validate our approach on synthetic and real data.

The method of the approximate inverse for limited-angle CT

TL;DR

This work tackles limited-angle CT, where incomplete angular data cause streaks and missing details. It introduces a model-driven reconstruction framework based on the method of the approximate inverse to build reconstruction kernels (LARK) that operate directly on measured data, augmented by a spectral filter and mollification to stabilize ill-posedness. To cope with real, semi-discrete data, the authors develop a constrained variant (CLARK) that couples the kernel with a denoising/regularization step, and provide error analysis for semi-discrete settings alongside practical discretization strategies. The approach is validated on synthetic phantoms and real measurements, showing improved feature recovery and reduced artefacts compared to FBP and TV, while highlighting dependence on missing angle extent and noise. Overall, LARK/CLARK offer a principled, interpretable starting point for reconstruction in severely ill-posed limited-angle CT and may serve as a foundation for learning-based enhancements.

Abstract

Limited-angle computerized tomography stands for one of the most difficult challenges in imaging. Although it opens the way to faster data acquisition in industry and less dangerous scans in medicine, standard approaches, such as the filtered backprojection (FBP) algorithm or the widely used total-variation functional, often produce various artefacts that hinder the diagnosis. With the rise of deep learning, many modern techniques have proven themselves successful in removing such artefacts but at the cost of large datasets. In this paper, we propose a new model-driven approach based on the method of the approximate inverse, which could serve as new starting point for learning strategies in the future. In contrast to FBP-type approaches, our reconstruction step consists in evaluating linear functionals on the measured data using reconstruction kernels that are precomputed as solution of an auxiliary problem. With this problem being uniquely solvable, the derived limited-angle reconstruction kernel (LARK) is able to fully reconstruct the object without the well-known streak artefacts, even for large limited angles. However, it inherits severe ill-conditioning which leads to a different kind of artefacts arising from the singular functions of the limited-angle Radon transform. The problem becomes particularly challenging when working on semi-discrete (real or analytical) measurements. We develop a general regularization strategy, named constrained limited-angle reconstruction kernel (CLARK), by combining spectral filter, the method of the approximate inverse and custom edge-preserving denoising in order to stabilize the whole process. We further derive and interpret error estimates for the application on real, i.e. semi-discrete, data and we validate our approach on synthetic and real data.

Paper Structure

This paper contains 17 sections, 11 theorems, 63 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. The continuous limited-angle Radon transform
  3. Mathematical background
  4. The method of the approximate inverse for $\mathcal{R}_\phi$
  5. Handling data errors under severe ill-posedness
  6. Towards the use on real data
  7. Computation of the discrete reconstruction kernel
  8. Ideal case: Exact interpolation in the pixel basis
  9. Numerical results
  10. The real-data case: a representation issue
  11. An introduction to interpolation with radial basis functions
  12. Reconstruction error for dominated functions
  13. Smoothing data to enforce dominance
  14. Numerical results
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 1

Let $C \subseteq S^1$ be a set of directions such that no non-trivial homogeneous polynomial vanishes on $C$. If $f\in C^\infty(\mathbb{R}^2)$ is compactly supported and $\mathcal{R}_\Phi f(\cdot,\theta) = 0$ for $\theta \in C$, then $f=0$.

Figures (12)

  • Figure 1: Illustration of standard limited-angle reconstructions for the Shepp-Logan phantom on a grid 201x201. For a better display in the diagonal of the limited-angle artefacts, the phantom is rotated of $45^\circ$ regarding the CT-scan. (a) Noisy data $g^\delta$ with in red the cut-off area here for $\Phi=40^\circ$. (b-c) FBP reconstruction using the Shepp-Logan-filter for $\Phi=0^\circ$ and $\Phi=40^\circ$, respectively.
  • Figure 2: Simulated on a 121x121 grid for $\Phi=30^\circ$. (a) Singular values of $\mathcal{R}_\Phi$ on a log-scale with two target singular values depicted as a red disk. (b-c) and resp. (d-e) Singular function $v_{ml}(\mathrm{x})$ and the magnitude of its 2D-Fourier transform for the first target singular value and the second one respectively. (c) and (e) highlight in red (dashed line) the missing cone.
  • Figure 3: Visualization of the different mollifiers for the central pixel on a $N\times N$ grid with $N=121$ and $\Phi=30^\circ$ for: (a) $E^\gamma$, (b) $E^{\gamma,\tau}$, (c) $\arctan \left(N \cdot E^{\gamma,\tau}\right)$ (for a better visualization) where $\gamma = 1/N$ and $\tau = 5\sigma_n$.
  • Figure 4: Evolution of the reconstruction kernel LARK at reconstruction point $\mathrm{z}=0$ as $\Phi$ increases.
  • Figure 5: Evolution of the singular values of $A_\Phi$ for the pixel grid and for different $\Phi$.
  • ...and 7 more figures

Theorems & Definitions (32)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark
  • Remark
  • Definition 3
  • Lemma 4
  • proof
  • Definition 5
  • ...and 22 more