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Data-Driven Filter Design for Flexible and Noise-Robust Tomographic Imaging

Hamid Fathi, Alexander Skorikov, Tristan van Leeuwen

Abstract

While filtered back projection (FBP) is still the method of choice for fast tomographic reconstruction, its performance degrades noticeably in the presence of noise, incomplete sampling, or non-standard scan geometries. We propose a data-driven approach for learning FBP filters and projection weights directly from training data, with the goal of improving robustness without sacrificing computational efficiency. The resulting reconstructions adapt naturally to the noise level and acquisition geometry, while retaining the speed and simplicity of classical back-projection. The proposed method can be formulated as a regularized optimization problem for a linear inverse operator, which allows us to establish existence, uniqueness, and stability of the learned solution. From a spectral viewpoint, the learned filters act as data-adaptive gain functions that explicitly balance noise amplification and bias, in close analogy to a regularized pseudo-inverse. Experiments in both 2D and 3D show consistent improvements over conventional FBP and FDK in different case studies. Finally, we show that filters trained on synthetic laminography data generalize well to real-world measurements, delivering image quality comparable to advanced iterative methods without the high computational cost.

Data-Driven Filter Design for Flexible and Noise-Robust Tomographic Imaging

Abstract

While filtered back projection (FBP) is still the method of choice for fast tomographic reconstruction, its performance degrades noticeably in the presence of noise, incomplete sampling, or non-standard scan geometries. We propose a data-driven approach for learning FBP filters and projection weights directly from training data, with the goal of improving robustness without sacrificing computational efficiency. The resulting reconstructions adapt naturally to the noise level and acquisition geometry, while retaining the speed and simplicity of classical back-projection. The proposed method can be formulated as a regularized optimization problem for a linear inverse operator, which allows us to establish existence, uniqueness, and stability of the learned solution. From a spectral viewpoint, the learned filters act as data-adaptive gain functions that explicitly balance noise amplification and bias, in close analogy to a regularized pseudo-inverse. Experiments in both 2D and 3D show consistent improvements over conventional FBP and FDK in different case studies. Finally, we show that filters trained on synthetic laminography data generalize well to real-world measurements, delivering image quality comparable to advanced iterative methods without the high computational cost.
Paper Structure (28 sections, 48 equations, 10 figures, 4 tables)

This paper contains 28 sections, 48 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation
  4. The inverse problem and regularization
  5. Fast approximate inversion
  6. A Learned Inverse for tomographic reconstruction
  7. Learning a regularized pseudo-inverse
  8. General Assumptions
  9. Existence, uniqueness, and stability
  10. Interpretation as a regularized pseudo-inverse
  11. Reconstruction error
  12. Parameterizing the filter for tomographic reconstruction
  13. 2D Parallel beam
  14. 2D Circular fan-beam
  15. 2D Elliptical fan-beam
  16. ...and 13 more sections

Figures (10)

  • Figure 1: Sketch of acquisition geometry in a circular laminography setup.
  • Figure 2: Learned filters for different noisy datasets in a parallel beam geometry versus Ram-Lak filter shown in the frequency domain.
  • Figure 3: Learned fan-beam (A) filter and (B) weights for different noise levels (SNR 20/25/30 dB), compared to the Ram-Lak filter and the standard fan-beam weighting used in conventional FBP. The weights are normalized to the standard profile to remove the filter--weight scaling ambiguity; the filter is rescaled accordingly.
  • Figure 4: Qualitative comparison showing reconstructions of a 2D phantom for the ground truth, FBP, and the learned method, respectively.
  • Figure 5: Qualitative comparison of lateral and coronal slices in learned reconstruction (right column), respectively, against Ground Truth (left column) and conventional FDK (middle column).
  • ...and 5 more figures

Theorems & Definitions (2)

  • proof
  • proof