Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction
Sara Fridovich-Keil Fabrizio Valdivia Gordon Wetzstein Benjamin Recht Mahdi Soltanolkotabi
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 as a preprocessing step and then solving a
convex inverse problem. However, this nonlinear measurement preprocessing
required to use the Radon transform 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. We show that this approach reduces metal
artifacts compared to a commercial reconstruction of a human skull with metal
dental crowns.