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Optimized filter functions for filtered back projection reconstructions

Matthias Beckmann, Judith Nickel

TL;DR

A formula is derived for the filter that minimizes the expected squared $\mathrm{L}^2$-norm of the difference between the FBP reconstruction, given infinite noisy measurement data, and the true target function.

Abstract

The method of filtered back projection (FBP) is a widely used reconstruction technique in X-ray computerized tomography (CT), which is particularly important in clinical diagnostics. To reduce scanning times and radiation doses in medical CT settings, enhancing the reconstruction quality of the FBP method is essential. To this end, this paper focuses on analytically optimizing the applied filter function. We derive a formula for the filter that minimizes the expected squared $\mathrm{L}^2$-norm of the difference between the FBP reconstruction, given infinite noisy measurement data, and the true target function. Additionally, we adapt our derived filter to the case of finitely many measurements. The resulting filter functions have a closed-form representation, do not require a training dataset, and, thus, provide an easy-to-implement, out-of-the-box solution. Our theoretical findings are supported by numerical experiments based on both simulated and real CT data.

Optimized filter functions for filtered back projection reconstructions

TL;DR

A formula is derived for the filter that minimizes the expected squared -norm of the difference between the FBP reconstruction, given infinite noisy measurement data, and the true target function.

Abstract

The method of filtered back projection (FBP) is a widely used reconstruction technique in X-ray computerized tomography (CT), which is particularly important in clinical diagnostics. To reduce scanning times and radiation doses in medical CT settings, enhancing the reconstruction quality of the FBP method is essential. To this end, this paper focuses on analytically optimizing the applied filter function. We derive a formula for the filter that minimizes the expected squared -norm of the difference between the FBP reconstruction, given infinite noisy measurement data, and the true target function. Additionally, we adapt our derived filter to the case of finitely many measurements. The resulting filter functions have a closed-form representation, do not require a training dataset, and, thus, provide an easy-to-implement, out-of-the-box solution. Our theoretical findings are supported by numerical experiments based on both simulated and real CT data.
Paper Structure (11 sections, 7 theorems, 117 equations, 9 figures, 1 table)

This paper contains 11 sections, 7 theorems, 117 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related results
  3. White noise and technical lemmata
  4. Optimized filter function
  5. Continuous setting
  6. Discrete setting
  7. Numerical Experiments
  8. Shepp-Logan phantom
  9. Modified Shepp-Logan phantom
  10. 2DeteCT
  11. Discussion and Outlook

Key Result

Lemma 3.3

\newlabelLem:Moments Approx Noise0 Let $H_a: \mathcal{S}(\mathbb R^n) \times \Omega \to \mathbb R$ be an approximate continuous white noise with support in $\Omega^\prime\subseteq \mathbb R^n$. Then, $H_a(\phi, \cdot): \Omega \to \mathbb R$ is a Gaussian random variable for all $\phi \in \mathcal{ for all $\psi, \phi\in \mathcal{S}(\mathbb R^n)$.

Figures (9)

  • Figure 1: Selection of classical low-pass filters used in our numerical experiments.
  • Figure 2: Phantoms used in our numerical experiments along with their sinograms (Radon data).
  • Figure 3: Noisy sinograms of the Shepp-Logan phantom with different noise levels.
  • Figure 4: Plots of the MSE and SSIM of FBP reconstructions for the Shepp-Logan phantom.
  • Figure 5: Reconstructions of Shepp-Logan phantom from noisy Radon data ($p_\mathrm{noise}=0.1$, $N_\varphi = 360$).
  • ...and 4 more figures

Theorems & Definitions (18)

  • Definition 3.1: Continuous white noise
  • Definition 3.2: Approximate continuous white noise
  • Lemma 3.3
  • Proof 1
  • Lemma 3.4
  • Proof 2
  • Lemma 3.5
  • Proof 3
  • Lemma 4.1
  • Proof 4
  • ...and 8 more