We present a deep-learning-based algorithm to jointly solve a reconstruction problem and a wavefront set extraction problem in tomographic imaging. The algorithm is based on a recently developed digital wavefront set extractor as well as the well-known microlocal canonical relation for the Radon transform. We use the wavefront set information about x-ray data to improve the reconstruction by requiring that the underlying neural networks simultaneously extract the correct ground truth wavefront set and ground truth image. As a necessary theoretical step, we identify the digital microlocal canonical relations for deep convolutional residual neural networks. We find strong numerical evidence for the effectiveness of this approach.

我们提出了一种基于深度学习的算法来联合解决断层成像中的重建问题和波前集提取问题。该算法基于最近开发的数字波前集提取器以及众所周知的 Radon 变换微局域规范关系。我们使用有关 X 射线数据的波前集信息，通过要求底层神经网络同时提取正确的地面实况波前集和地面实况图像来改进重建。作为必要的理论步骤，我们确定了深度卷积残差神经网络的数字微局部规范关系。我们为这种方法的有效性找到了强有力的数字证据。