Task adapted reconstruction for inverse problems

The paper considers the problem of performing a post-processing task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and post-processing as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the post-processing task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any post-processing that can be encoded as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation.

本文考虑在一个模型参数上执行后处理任务的问题，该模型参数在一个不适定的逆问题中只能通过含噪数据间接观测到。一个关键方面是将重建和后处理的步骤在统计估计问题中形式化为适当的估计量（非随机决策规则）。实现过程利用（深度）神经网络为这两个步骤的估计量族提供可微参数化。这些网络被组合起来，并针对合适的有监督训练数据进行联合训练，以最小化一个联合可微损失函数，从而产生一种端到端的适应任务的重建方法。所提出的框架是通用的，但具有适应性，具有即插即用的结构，可用于调整逆问题和手头的后处理任务。更准确地说，与逆问题相关的数据模型（正向算子和噪声的统计模型）是可交换的，例如，通过使用由一种学习迭代方法给出的神经网络架构。此外，可以使用任何可编码为可训练神经网络的后处理。该方法在联合断层图像重建、分类以及联合断层图像重建分割上得到了验证。