Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction

TL;DR

This paper proves that gradient descent can reliably solve nonlinear tomographic reconstruction problems directly from raw measurements, leading to improved image quality and reduced artifacts in CT scans, especially near high-density materials.

Contribution

It provides the first theoretical guarantees for gradient descent convergence in nonlinear CT reconstruction, even with fewer measurements and structural priors.

Findings

Gradient descent converges to the global optimum at a geometric rate.

Direct nonlinear reconstruction reduces metal artifacts in CT images.

Logarithmic preprocessing alone can cause artifacts without other factors.

Abstract

In computed tomography (CT), the forward model consists of a linear Radon transform followed by an exponential nonlinearity based on the attenuation of light according to the Beer-Lambert Law. Conventional reconstruction often involves inverting this nonlinearity and then solving a linear inverse problem. However, this nonlinear measurement preprocessing is poorly conditioned in the vicinity of high-density materials, such as metal. This preprocessing makes CT reconstruction methods numerically sensitive and susceptible to artifacts near high-density regions. In this paper, we study a technique where the signal is directly reconstructed from raw measurements through the nonlinear forward model. Though this optimization is nonconvex, we show that gradient descent provably converges to the global optimum at a geometric rate, perfectly reconstructing the underlying signal with a near minimal number of random measurements. We also prove similar results in the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal. This is achieved by enforcing prior structural information about the signal through constraints on the optimization variables. We illustrate the benefits of direct nonlinear CT reconstruction with cone-beam CT experiments on synthetic and real 3D volumes, in which metal artifacts are reduced compared to standard linear reconstruction methods. Our experiments also demonstrate that logarithmic preprocessing alone is sufficient to produce metal artifacts, even in the absence of other causes such as beam hardening.