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Norms in sinogram space and stability estimates for the Radon transform

Stefan Kindermann, Simon Hubmer

TL;DR

This work addresses the stability of Radon-transform-based reconstructions by introducing sinogram-space norms that connect to image-space norms. It defines Fourier-based Bessel-potential norms $H^{s,p}$ and associated sinogram norms $R^{s,p}_t$, $S^{s,p}_t$, and $R^{s,p}_t$, establishing norm-transfer results that extend classical Sobolev estimates to $L^p$ settings. A TV-inspired sinogram norm, $|\cdot|_{RTV}$, is developed and shown to be equivalent to the image-space total variation $|\cdot|_{TV}$ for indicator functions of convex sets in 2D, with Steiner symmetrization providing sharpening bounds, and conditional stability results derived for smooth nonnegative functions. These theoretical developments motivate a nonlinear backprojection framework based on proximal maps in sinogram space, applied to noisy and incomplete data, and demonstrated through numerical experiments with simulated and experimental Radon data. The proposed approach offers computationally efficient, edge-preserving regularization in Radon inversion, bridging variational regularization with nonlinear filtering in sinogram space and enabling robust reconstructions in challenging data regimes.

Abstract

We consider different norms for the Radon transform $Rf$ of a function $f$ and investigate under which conditions they can be estimated from above or below by some standard norms for $f$. We define Fourier-based norms for $Rf$ which can be related to Bessel-potential space norms for $f$. Furthermore, we define a variant of a total-variation norm for $Rf$ and provide conditions under which it is equivalent to the total-variation norm of $f$. To illustrate potential applications of these results, we propose a novel nonlinear backprojection method for inverting the Radon transform and present numerical results on simulated and experimental data.

Norms in sinogram space and stability estimates for the Radon transform

TL;DR

This work addresses the stability of Radon-transform-based reconstructions by introducing sinogram-space norms that connect to image-space norms. It defines Fourier-based Bessel-potential norms and associated sinogram norms , , and , establishing norm-transfer results that extend classical Sobolev estimates to settings. A TV-inspired sinogram norm, , is developed and shown to be equivalent to the image-space total variation for indicator functions of convex sets in 2D, with Steiner symmetrization providing sharpening bounds, and conditional stability results derived for smooth nonnegative functions. These theoretical developments motivate a nonlinear backprojection framework based on proximal maps in sinogram space, applied to noisy and incomplete data, and demonstrated through numerical experiments with simulated and experimental Radon data. The proposed approach offers computationally efficient, edge-preserving regularization in Radon inversion, bridging variational regularization with nonlinear filtering in sinogram space and enabling robust reconstructions in challenging data regimes.

Abstract

We consider different norms for the Radon transform of a function and investigate under which conditions they can be estimated from above or below by some standard norms for . We define Fourier-based norms for which can be related to Bessel-potential space norms for . Furthermore, we define a variant of a total-variation norm for and provide conditions under which it is equivalent to the total-variation norm of . To illustrate potential applications of these results, we propose a novel nonlinear backprojection method for inverting the Radon transform and present numerical results on simulated and experimental data.
Paper Structure (16 sections, 16 theorems, 129 equations, 5 figures)

This paper contains 16 sections, 16 theorems, 129 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bessel-potential space estimates
  3. Differential dimension
  4. Fourier-based norms and estimates
  5. Generalization
  6. Total variation in image and sinogram space
  7. Estimates and Properties
  8. Lower bound for convex sets in the plane
  9. Steiner Symmetrization in $\mathbb{R}^2$
  10. Conditional stability for smooth functions in $\mathbb{R}^2$
  11. Filtering and Regularization
  12. Proximal maps for $\|\cdot\|_{R_t^{s,p}}^p$ and results
  13. Proximal maps for $\text{\rm RTV}$ and results
  14. Noisy and incomplete data
  15. Conclusion
  16. ...and 1 more sections

Key Result

Theorem 1

For any $s\in\mathbb{R}$ and for $t \in (-n/2,n/2)$, the norms $\left\|f\right\|_{S^{s,2}_{t}({\mathbb{R}^n})}$ and $\left\|f\right\|_{S^{s,2}_0({\mathbb{R}^n})} = \left\|f\right\|_{H^{s,2}({\mathbb{R}^n})}$ are equivalent on the space of functions $f \in \mathcal{S}({\mathbb{R}^n})$ supported in a

Figures (5)

  • Figure 1: Comparison of the Sharafutdinov filter combined with filtered backprojection (iradon) for various choices of the regularization parameter and for different exact solutions (ordered row-wise). Left column: high regularization, middle column: medium regularization, right column: low regularization.
  • Figure 2: Comparison of the RTV filter combined with filtered backprojection (iradon) for various choices of the regularization parameter $\alpha$ and for different exact solutions (ordered row-wise). Left column: high regularization, middle column: medium regularization, right column: low regularization.
  • Figure 3: Illustration of the effect of an RTV-filter if $f$ is the indicator function of a disk. Left: Radon transform (dashed blue) and filtered Radon transform (red). Right: corresponding solutions in image space after applying $R^{-1}$.
  • Figure 4: Reconstruction using incomplete data: Columns: methods RTV, Sharafutdinov-filter, and CGNE inversion. Rows: Shepp-Logan phantom from Matlab, Walnut data from fipsFinnish, and THz tomography data from Foso22Hub22.
  • Figure 5: Comparision of the RTV filter with a variational TV regularization using a Pock-Chambolle method. Top row: RTV filter. Bottom row: TV-regularization. Left: full data. Center: Undersampled data with respect to $\sigma$. Right: Limited angle data: $\theta \in [0,90^\circ]$.

Theorems & Definitions (43)

  • Definition 1
  • Theorem 1: Shara
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • Theorem 3
  • Remark 1
  • Definition 2
  • Example 1
  • ...and 33 more