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.

在本文中，我们研究了一种技术，通过非线性正演模型直接从原始测量数据中重建信号。尽管这种优化是非凸的，但我们证明了梯度下降算法可以以几何速度收敛到全局最优解，使用近乎最小数量的随机测量完美重建基础信号。我们还通过对优化变量施加先验结构信息的约束，在信号的维度明显小于测量数量的欠定设置中证明了类似的结果。我们通过合成和真实三维体积上的锥束 CT 实验展示了直接非线性 CT 重建的好处，并证明与经商业重建的带金属牙冠人类头骨相比，这种方法能减少金属伪影。