Spectral Computed Tomography (CT) is an emerging technology that enables us to estimate the concentration of basis materials within a scanned object by exploiting different photon energy spectra. In this work, we aim at efficiently solving a model-based maximum-a-posterior problem to reconstruct multi-materials images with application to spectral CT. In particular, we propose to solve a regularized optimization problem based on a plug-in image-denoising function using a randomized second order method. By approximating the Newton step using a sketching of the Hessian of the likelihood function, it is possible to reduce the complexity while retaining the complex prior structure given by the data-driven regularizer. We exploit a non-uniform block sub-sampling of the Hessian with inexact but efficient conjugate gradient updates that require only Jacobian-vector products for denoising term. Finally, we show numerical and experimental results for spectral CT materials decomposition.

This article is part of the theme issue ‘Synergistic tomographic image reconstruction: part 1’.

光谱计算机断层扫描（CT）是一项新兴技术，使我们能够通过利用不同的光子能谱来估计扫描对象中基础材料的浓度。在这项工作中，我们旨在有效解决基于模型的最大后验问题，以将其应用于光谱CT来重建多材料图像。特别地，我们提出使用随机二阶方法基于插件图像去噪函数来解决正则化优化问题。通过使用似然函数的Hessian草图来逼近牛顿步，可以降低复杂度，同时保留由数据驱动的正则化器给出的复杂先验结构。我们利用不精确但有效的共轭梯度更新方法对Hessian进行非均匀块子采样，该方法仅需要Jacobian向量乘积来对项进行降噪。最后，我们给出了光谱CT材料分解的数值和实验结果。

本文是主题主题“协同层析图像重建：第1部分”的一部分。