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.

翻译：在计算机断层扫描（CT）中，前向模型由线性Radon变换和基于比尔-朗伯定律的光衰减指数非线性组成。传统重建通常先对非线性进行预处理以反转其效应，再求解凸逆问题。然而，这种为使用Radon变换所需的非线性测量预处理在高密度材料（如金属）附近存在病态条件，导致CT重建方法在近高密度区域数值敏感且易产生伪影。本文研究一种通过非线性前向模型直接从原始测量中重建信号的技术。尽管该优化问题是非凸的，但我们证明梯度下降能以几何速率收敛到全局最优解，从而在近乎最少的随机测量次数下完美重建原始信号。我们还在测量数远小于信号维度的欠定条件下证明了类似结论——通过对优化变量施加约束来利用信号的先验结构信息。通过锥束CT在合成与真实三维体数据上的实验，我们展示了直接非线性CT重建的优越性。实验表明，与商业重建算法相比，该方法可减少含金属牙冠人类头骨CT图像中的金属伪影。