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Data-Driven Filter Design in FBP: Transforming CT Reconstruction with Trainable Fourier Series

Yipeng Sun, Linda-Sophie Schneider, Fuxin Fan, Mareike Thies, Mingxuan Gu, Siyuan Mei, Yuzhong Zhou, Siming Bayer, Andreas Maier

TL;DR

A Fourier series-based trainable filter for computed tomography (CT) reconstruction within the filtered backprojection (FBP) framework overcomes the limitation in noise reduction by optimizing Fourier series coefficients to construct the filter, maintaining computational efficiency with minimal increment for the trainable parameters compared to other deep learning frameworks.

Abstract

In this study, we introduce a Fourier series-based trainable filter for computed tomography (CT) reconstruction within the filtered backprojection (FBP) framework. This method overcomes the limitation in noise reduction by optimizing Fourier series coefficients to construct the filter, maintaining computational efficiency with minimal increment for the trainable parameters compared to other deep learning frameworks. Additionally, we propose Gaussian edge-enhanced (GEE) loss function that prioritizes the $L_1$ norm of high-frequency magnitudes, effectively countering the blurring problems prevalent in mean squared error (MSE) approaches. The model's foundation in the FBP algorithm ensures excellent interpretability, as it relies on a data-driven filter with all other parameters derived through rigorous mathematical procedures. Designed as a plug-and-play solution, our Fourier series-based filter can be easily integrated into existing CT reconstruction models, making it an adaptable tool for a wide range of practical applications. Code and data are available at https://github.com/sypsyp97/Trainable-Fourier-Series.

Data-Driven Filter Design in FBP: Transforming CT Reconstruction with Trainable Fourier Series

TL;DR

A Fourier series-based trainable filter for computed tomography (CT) reconstruction within the filtered backprojection (FBP) framework overcomes the limitation in noise reduction by optimizing Fourier series coefficients to construct the filter, maintaining computational efficiency with minimal increment for the trainable parameters compared to other deep learning frameworks.

Abstract

In this study, we introduce a Fourier series-based trainable filter for computed tomography (CT) reconstruction within the filtered backprojection (FBP) framework. This method overcomes the limitation in noise reduction by optimizing Fourier series coefficients to construct the filter, maintaining computational efficiency with minimal increment for the trainable parameters compared to other deep learning frameworks. Additionally, we propose Gaussian edge-enhanced (GEE) loss function that prioritizes the norm of high-frequency magnitudes, effectively countering the blurring problems prevalent in mean squared error (MSE) approaches. The model's foundation in the FBP algorithm ensures excellent interpretability, as it relies on a data-driven filter with all other parameters derived through rigorous mathematical procedures. Designed as a plug-and-play solution, our Fourier series-based filter can be easily integrated into existing CT reconstruction models, making it an adaptable tool for a wide range of practical applications. Code and data are available at https://github.com/sypsyp97/Trainable-Fourier-Series.
Paper Structure (13 sections, 13 equations, 3 figures, 1 table)

This paper contains 13 sections, 13 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method & Materials
  3. Filtered Backprojection
  4. Projection Filtering
  5. Backprojection
  6. Trainable Fourier Series Filter
  7. Sinogram Filtering
  8. Deep Learning Computed Tomography
  9. Fourier Series Filter Construction
  10. Gradient Variance Loss
  11. Gaussian Edge-Enhanced Loss
  12. Experiments and Results
  13. Conclusion and Discussion

Figures (3)

  • Figure 1: Deep learning computed tomography: Reconstruction network for ${I}_{rec} = {ReLU}(A^{-1}F^{-1}HFP)$ from sinogram $P$ to image ${I}_{rec}$.
  • Figure 2: Comparative analysis of reconstruction results: When compared to MSE, our composite loss function demonstrates superior detail retention and image sharpness. In comparison to the traditional FBP method, our approach excels in high-frequency noise suppression and provides clearer structural definition.
  • Figure 3: Comparison of reconstruction in Laplace map: In contrast to supervised training using MSE, our approach demonstrates superior edge sharpness. When compared to the conventional FBP method, our approach showcases a more distinct edge structure and reduced high-frequency noise.