Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction

Fridovich-Keil, Sara

doi:10.1109/tit.2026.3677729

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Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction

2026-05-01

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Fridovich-Keil, Sara *

Valdivia, Fabrizio

Gordon Wetzstein

Benjamin Recht

Soltanolkotabi, Mahdi

DOI：10.1109/tit.2026.3677729

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Abstract

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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.

Keywords:

Computed tomography

Vectors

Metals

Noise measurement

Three-dimensional displays

Compressed sensing

Weight measurement

Standards

Mathematical models

Image reconstruction

reconstruction algorithms