Nonlinear tomographic reconstruction via nonsmooth optimization

TL;DR

This work addresses nonlinear tomographic reconstruction in CT when measurements follow a nonlinear exponential model, showing that a nonsmooth $\ell_1$-loss formulation paired with a scaled Polyak subgradient method yields geometric convergence with polynomial sample complexity under Gaussian designs. The key theoretical contributions are Lipschitzity and sharp-growth regularity, plus an aiming inequality that yields a modest condition number $\kappa = O(\|x_\star\|^2)$ for $m \gtrsim d\|x_\star\|^4$, significantly improving over prior smooth-loss approaches. The paper also introduces an adaptive variant AdPolyakSGM that avoids knowing $\|x_\star\|$, with competitive empirical performance across Gaussian and RWHT designs and CT-like imaging, including TV-regularized reconstructions. These results imply practical gains in sample efficiency and speed for nonlinear tomographic reconstruction, offering a robust optimization pathway for CT with high dynamic range and metal artifacts.

Abstract

We study iterative signal reconstruction in computed tomography (CT), wherein measurements are produced by a linear transformation of the unknown signal followed by an exponential nonlinear map. Approaches based on pre-processing the data with a log transform and then solving the resulting linear inverse problem are tempting since they are amenable to convex optimization methods; however, such methods perform poorly when the underlying image has high dynamic range, as in X-ray imaging of tissue with embedded metal. We show that a suitably initialized subgradient method applied to a natural nonsmooth, nonconvex loss function produces iterates that converge to the unknown signal of interest at a geometric rate under the statistical model proposed by Fridovich-Keil et al. (arXiv:2310.03956). Our recovery program enjoys improved conditioning compared to the formulation proposed by the latter work, enabling faster iterative reconstruction from substantially fewer samples.

Figures (8)

• Figure 1: Empirical probability of recovering $x_{\star}$ for various signal scales and oversampling factors over synthetic problems with $d = 128$ and Gaussian measurements. The empirical probability for each configuration is calculated over $25$ independently generated problem instances. Lighter tiles indicate higher probability of recovery. We compare \ref{['alg:polyaksgm']} with $\eta = 1$ against gradient descent with optimized stepsize $\eta$ using a logarithmically spaced grid.
• Figure 2: Comparison of PolyakSGM (\ref{['alg:polyaksgm']}) using $\eta = 1$ with gradient descent \ref{['eq:gd-baseline']} using tuned stepsize $\eta$ for synthetic problems with Gaussian and randomized Walsh-Hadamard measurements. The iterates produced by \ref{['alg:polyaksgm']} converge to $x_{\star}$ at moderately faster rates.
• Figure 3: Comparison between \ref{['alg:polyaksgm']} and \ref{['eq:gd-baseline']} using theoretically prescribed stepsizes for synthetic problem instances. In all cases, \ref{['alg:polyaksgm']} converges faster than gradient descent.
• Figure 4: Number of iterations to achieve suboptimality of $\varepsilon = 10^{-5}$ for \ref{['alg:polyaksgm']} and gradient descent for different stepsizes $\eta$. Reporting median over 10 independent runs (solid line) $\pm$ 3 standard deviations (shaded area); see \ref{['sec:maxiter-comparison']} for description.
• Figure 5: Comparison between \ref{['alg:polyaksgm']} and \ref{['alg:adpolyak']} for synthetic problems with Gaussian and randomized Walsh-Hadamard measurements.

Theorems & Definitions (60)

• Theorem 1.1: Informal; adapted from FVW+23
• Remark 1
• Lemma 2.1
• Corollary 2.1
• Lemma 2.2
• Proposition 2.1: Lipschitz continuity
• Corollary 2.2
• Lemma 2.3: Solvability lemma
• Remark 2
• Lemma 2.4: Subdifferential inner product