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Revisiting $Ψ$DONet: microlocally inspired filters for incomplete-data tomographic reconstructions

Tatiana A. Bubba, Luca Ratti, Andrea Sebastiani

TL;DR

This work addresses incomplete-data tomography, where limited-angle or sparse-angle measurements hamper accurate edge recovery and induce streak artifacts. It provides a microlocal interpretation of ΨDONet and introduces a novel continuous, wavelet-based formulation that clarifies how the learned correction acts as a pseudodifferential operator to regularize singularities. Three geometry-aware filter variants (bow, x, spa) are proposed to align the learned corrections with visible singular directions, significantly reducing learnable parameters while preserving or improving reconstruction quality on limited- and sparse-angle data. Numerical results on synthetic ellipse data demonstrate competitive PSNR/SSIM and notable artifact suppression, highlighting practical improvements for CT in challenging incomplete-data scenarios.

Abstract

In this paper, we revisit a supervised learning approach based on unrolling, known as $Ψ$DONet, by providing a deeper microlocal interpretation for its theoretical analysis, and extending its study to the case of sparse-angle tomography. Furthermore, we refine the implementation of the original $Ψ$DONet considering special filters whose structure is specifically inspired by the streak artifact singularities characterizing tomographic reconstructions from incomplete data. This allows to considerably lower the number of (learnable) parameters while preserving (or even slightly improving) the same quality for the reconstructions from limited-angle data and providing a proof-of-concept for the case of sparse-angle tomographic data.

Revisiting $Ψ$DONet: microlocally inspired filters for incomplete-data tomographic reconstructions

TL;DR

This work addresses incomplete-data tomography, where limited-angle or sparse-angle measurements hamper accurate edge recovery and induce streak artifacts. It provides a microlocal interpretation of ΨDONet and introduces a novel continuous, wavelet-based formulation that clarifies how the learned correction acts as a pseudodifferential operator to regularize singularities. Three geometry-aware filter variants (bow, x, spa) are proposed to align the learned corrections with visible singular directions, significantly reducing learnable parameters while preserving or improving reconstruction quality on limited- and sparse-angle data. Numerical results on synthetic ellipse data demonstrate competitive PSNR/SSIM and notable artifact suppression, highlighting practical improvements for CT in challenging incomplete-data scenarios.

Abstract

In this paper, we revisit a supervised learning approach based on unrolling, known as DONet, by providing a deeper microlocal interpretation for its theoretical analysis, and extending its study to the case of sparse-angle tomography. Furthermore, we refine the implementation of the original DONet considering special filters whose structure is specifically inspired by the streak artifact singularities characterizing tomographic reconstructions from incomplete data. This allows to considerably lower the number of (learnable) parameters while preserving (or even slightly improving) the same quality for the reconstructions from limited-angle data and providing a proof-of-concept for the case of sparse-angle tomographic data.

Paper Structure

This paper contains 16 sections, 4 theorems, 38 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Some microlocal analysis concepts
  4. Microlocal analysis for tomographic imaging
  5. A pseudodifferential correction via $\Psi$DONet
  6. Sparsity-promoting regularization in tomography
  7. $\Psi$DONet in a nutshell
  8. $\Psi$DONet: revisiting theory and practice
  9. A novel continuous formulation of $\Psi$DONet
  10. On introducing edges via soft-thresholding
  11. On preventing streak artifacts via kernel smoothing
  12. Optimizing $\Psi$DONet with special filters
  13. Numerical results
  14. Limited-angle tomography
  15. Sparse-angle tomography
  16. ...and 1 more sections

Key Result

Proposition 1

Any $\mathcal{A} \in \Psi^m$ has the pseudolocal property and the microlocal property

Figures (6)

  • Figure 1: Tomographic reconstructions (using FBP) of a cucumber slice from: (b) full data; (c) sparse-angle data; and (d) limited-angle data. Discontinuities along lines are evident when only limited data are are available.
  • Figure 2: Different filter geometries for $\Psi$DONet. (a) $\Psi$DONet filters initially proposed in bubba2021deep. (b) $\Psi$DONet-bow filters for limited-angle tomography. (c) $\Psi$DONet-x filters for limited-angle tomography. (d) $\Psi$DONet-sparse filters for sparse-angle tomography.
  • Figure 3: Visualization of the reconstructions of two images from the test set with $A=[-\pi/3,\pi/3]$.
  • Figure 4: Visualization of the reconstructions of two images from the test set with $A=[-\pi/6,\pi/6]$.
  • Figure 5: Visualization of the reconstructions of two images with 12 angles.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Definition 1: singular support
  • Definition 2: wavefront set
  • Definition 3: Pseudodifferential operators
  • Proposition 1: Pseudolocal and microlocal property of $\Psi$DOs
  • Theorem 2: quinto2017artifactsquinto1993singularities
  • Proposition 3: $\operatorname{sing\ supp}$ version of andrade2022deep
  • Corollary 4