Abstract
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.
Highlights
The overall goal in inverse problems is to determine model parameters such that model predictions match measured data to sufficient accuracy
There is an extensive theory that guarantees that these techniques are statistically consistent, but it comes with two critical drawbacks that has prevented widespread usage of Markov chain Monte Carlo (MCMC) techniques in imaging
A key aspect for the implementation of the joint task adapted reconstruction method in (21) is that both decision rules are given by trainable neural networks, which after joint training forms a single intertwined neural network
Summary
The overall goal in inverse problems is to determine model parameters such that model predictions match measured data to sufficient accuracy. Such problems arises in various scientific disciplines. One example is biomedical imaging where the image is the “model parameter” that needs to be determined from data acquired using an imaging device like a tomographic scanner or a microscope. A key element in realizing this role is the ability to solve the inverse problem of calibrating parameters in a mathematical model of the system so that simulations match benchmark data [8]
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