Uncertainty quantification in inverse medical imaging tasks with deep learning has received little attention. However, deep models trained on large data sets tend to hallucinate and create artifacts in the reconstructed output that are not anatomically present. We use a randomly initialized convolutional network as parameterization of the reconstructed image and perform gradient descent to match the observation, which is known as deep image prior. In this case, the reconstruction does not suffer from hallucinations as no prior training is performed. We extend this to a Bayesian approach with Monte Carlo dropout to quantify both aleatoric and epistemic uncertainty. The presented method is evaluated on the task of denoising different medical imaging modalities. The experimental results show that our approach yields well-calibrated uncertainty. That is, the predictive uncertainty correlates with the predictive error. This allows for reliable uncertainty estimates and can tackle the problem of hallucinations and artifacts in inverse medical imaging tasks.
Cold posteriors have been reported to perform better in practice in the context of Bayesian deep learning (Wenzel et al., 2020). In variational inference, it is common to employ only a partially tempered posterior by scaling the complexity term in the log-evidence lower bound (ELBO). In this work, we first derive the ELBO for a fully tempered posterior in meanfield variational inference and subsequently use Bayesian optimization to automatically find the optimal posterior temperature. Choosing an appropriate posterior temperature leads to better predictive performance and improved uncertainty calibration, which we demonstrate for the task of denoising medical X-ray images.
We present Posterior Temperature Optimized Bayesian Inverse Models (POTOBIM), an unsupervised Bayesian approach to inverse problems in medical imaging using mean-field variational inference with a fully tempered posterior. Bayesian methods exhibit useful properties for approaching inverse tasks, such as tomographic reconstruction or image denoising. A suitable prior distribution introduces regularization, which is needed to solve the ill-posed problem and reduces overfitting the data. In practice, however, this often results in a suboptimal posterior temperature, and the full potential of the Bayesian approach is not being exploited. In POTOBIM, we optimize both the parameters of the prior distribution and the posterior temperature with respect to reconstruction accuracy using Bayesian optimization with Gaussian process regression. Our method is extensively evaluated on four different inverse tasks on a variety of modalities with images from public data sets and we demonstrate that an optimized posterior temperature outperforms both non-Bayesian and Bayesian approaches without temperature optimization. The use of an optimized prior distribution and posterior temperature leads to improved accuracy and uncertainty estimation and we show that it is sufficient to find these hyperparameters per task domain. Well-tempered posteriors yield calibrated uncertainty, which increases the reliability in the predictions. Our source code is publicly available at github.com/Cardio-AI/mfvi-dip-mia.
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