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Accurate, provable, and fast nonlinear tomographic reconstruction: A variational inequality approach

Mengqi Lou, Kabir Aladin Verchand, Sara Fridovich-Keil, Ashwin Pananjady

TL;DR

This work tackles nonlinear CT reconstruction under a forward model with exponential attenuation and polychromatic X-ray sources by introducing EXACT, an extragradient-based algorithm that finds a fixed point of a monotone variational inequality under a convex constraint. The authors establish rigorous convergence guarantees for EXACT in realistic nonnegative-measurement CT settings and in Gaussian measurement scenarios, deriving explicit sample and iteration complexity trade-offs. Theoretical results are complemented by photon-counting CT and Gaussian-model experiments, showing that EXACT can achieve comparable reconstruction quality with substantially reduced X-ray dose and faster computation relative to competitive baselines. Overall, the paper provides provable, efficient recovery guarantees for nonlinear CT with practical noise models and measurement designs, offering guidance for measurement budgeting and algorithmic design in spectral CT applications.

Abstract

We consider the problem of signal reconstruction for computed tomography (CT) under a nonlinear forward model that accounts for exponential signal attenuation, a polychromatic X-ray source, general measurement noise (e.g. Poisson shot noise), and observations acquired over multiple wavelength windows. We develop a simple iterative algorithm for single-material reconstruction, which we call EXACT (EXtragradient Algorithm for Computed Tomography), based on formulating our estimate as the fixed point of a monotone variational inequality. We prove guarantees on the statistical and computational performance of EXACT under practical assumptions on the measurement process. We also consider a recently introduced variant of this model with Gaussian measurements, and present sample and iteration complexity bounds for EXACT that improve upon those of existing algorithms. We apply our EXACT algorithm to a CT phantom image recovery task and show that it often requires fewer X-ray projection exposures, lower source intensity, and less computation time to achieve similar reconstruction quality to existing methods.

Accurate, provable, and fast nonlinear tomographic reconstruction: A variational inequality approach

TL;DR

This work tackles nonlinear CT reconstruction under a forward model with exponential attenuation and polychromatic X-ray sources by introducing EXACT, an extragradient-based algorithm that finds a fixed point of a monotone variational inequality under a convex constraint. The authors establish rigorous convergence guarantees for EXACT in realistic nonnegative-measurement CT settings and in Gaussian measurement scenarios, deriving explicit sample and iteration complexity trade-offs. Theoretical results are complemented by photon-counting CT and Gaussian-model experiments, showing that EXACT can achieve comparable reconstruction quality with substantially reduced X-ray dose and faster computation relative to competitive baselines. Overall, the paper provides provable, efficient recovery guarantees for nonlinear CT with practical noise models and measurement designs, offering guidance for measurement budgeting and algorithmic design in spectral CT applications.

Abstract

We consider the problem of signal reconstruction for computed tomography (CT) under a nonlinear forward model that accounts for exponential signal attenuation, a polychromatic X-ray source, general measurement noise (e.g. Poisson shot noise), and observations acquired over multiple wavelength windows. We develop a simple iterative algorithm for single-material reconstruction, which we call EXACT (EXtragradient Algorithm for Computed Tomography), based on formulating our estimate as the fixed point of a monotone variational inequality. We prove guarantees on the statistical and computational performance of EXACT under practical assumptions on the measurement process. We also consider a recently introduced variant of this model with Gaussian measurements, and present sample and iteration complexity bounds for EXACT that improve upon those of existing algorithms. We apply our EXACT algorithm to a CT phantom image recovery task and show that it often requires fewer X-ray projection exposures, lower source intensity, and less computation time to achieve similar reconstruction quality to existing methods.

Paper Structure

This paper contains 42 sections, 8 theorems, 208 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction and methodology
  2. Formal model and approach
  3. Additional notation
  4. Contributions and comparison to related work
  5. Theoretical results
  6. Positive measurements and positive signal
  7. Gaussian measurements
  8. Experimental results
  9. Photon-counting CT simulation
  10. Gaussian model
  11. Proofs
  12. Proof of Theorem \ref{['general-thm']}
  13. Proof of Theorem \ref{['thm-positive-measurements']}
  14. Strong pseudo-monotonicity.
  15. Lipschitz constant.
  16. ...and 27 more sections

Key Result

Theorem 1

Suppose that the data $\{(a_i, y_i)\}_{i=1}^{n}$ satisfy the single-index relationship model, Assumption assump-X holds, and that the operator $F$ in Eq. operator is both $L$-Lipschitz lipschitzness as well as $\nu$-strongly pseudo-monotone strong-monotone. Then the iterates $\{x_{t}\}$ generated ac

Figures (6)

  • Figure 1: Spectral density of the simulated X-ray source (left), detector sensitivity for 3 simulated windows (middle), and target phantom made of simulated PMMA with 4 regions of interest (right, colorbar in Hounsfield Units).
  • Figure 2: RMSE (left) and runtime (right), varying the number of projection exposures while the X-ray source outputs on average $10^6$ photons per detector cell. Until the number of projection exposures drops below 10, at which point all algorithms fail, our proposed Extragradient method EXACT (defined in \ref{['extragrad-method']}) converges fastest and to lowest error.
  • Figure 3: Full reconstructions (in Hounsfield Units, HU) and zoom-ins (bottom row) after convergence of each algorithm using 10 projection exposures and an average source intensity of $10^6$ photons per detector cell, a total of 5$\times$ reduction in X-ray dose compared to the 50-projection exposure default setting in which all algorithms perform well. The EXACT ("Extragradient") and MSE GD reconstructions are visually faithful to the target in \ref{['fig:simulation-setup']}, the ADMM reconstruction is perceptually accurate but exhibits less uniformity in the constant regions, and the PolyakSGM reconstruction exhibits more severe artifacts that obscure part of the lowest-contrast ROI.
  • Figure 4: RMSE (left) and runtime (middle, right), varying the source intensity while the number of projection exposures is fixed at 10, the smallest value for which recovery is possible in \ref{['fig:1e6photons-varymeasurements']}. All algorithms suffer higher reconstruction error as source intensity and thus SNR is decreased. The rightmost plot is a zoom-in of the center plot to illustrate the runtimes of the three fastest algorithms.
  • Figure 5: Full reconstructions (in Hounsfield Units, HU) and zoom-ins (bottom row) after convergence of each algorithm using 10 projection exposures and an average source intensity of $10^5$ photons per detector cell, a total of 50$\times$ reduction in X-ray dose compared to the 50-projection exposure, $10^6$ photons default setting in which all algorithms perform well. The EXACT ("Extragradient") reconstruction exhibits minor artifacts at the edges of several ROIs, the MSE GD and ADMM reconstructions exhibit artifacts that partially obscure the lowest-contrast ROI, and the PolyakSGM reconstruction exhibits more severe artifacts that obscure several ROIs.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4