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.

Automated summary

This paper discusses how to perform a post-processing task on a model parameter in an ill-posed inverse problem, where the parameter is only indirectly observed through noisy data. The authors propose a framework based on (deep) neural networks to formalize the reconstruction and post-processing steps as statistical estimation problems. By jointly training the networks against suitable supervised training data, an end-to-end task adapted reconstruction method is obtained. The suggested framework is generic and adaptable, allowing for customization of the inverse problem and post-processing task. The approach is demonstrated on joint tomographic image reconstruction, classification, and segmentation.