Uncertainty Estimation for Learning-Based Classification of Corrupted Images

Image restoration tasks often admit a wide range of solutions that are equally consistent with the observed data. Quantifying this uncertainty is crucial for the robust interpretation of restored images and their reliable use in science and decision-making. We consider the classification of images reconstructed from noisy and corrupted measurements, with special attention to the quantification of uncertainty in the delivered classification results. We address this problem by constructing a Bayesian statistical approach that combines learning-based image priors and image classifiers with explicit image observation models specified during inference. Following the manifold hypothesis, we assume that the image prior is supported on a submanifold of the ambient space—which we learn from uncorrupted training data using a variational autoencoder—and use as a classifier a support vector machine operating in this low-dimensional representation. The observation model is incorporated during inference time through its likelihood function. Bayesian computation is then efficiently performed by leveraging variants of the unadjusted Langevin algorithm that operate directly on the submanifold and are robust to multimodality. This results in a robust image classification method that provides uncertainty estimates that are provably well-posed, derived from Bayesian decision theory rigorously and transparently, and which incorporate physical and instrumental aspects of the data acquisition process through Bayes’ theorem. We demonstrate the effectiveness of the proposed approach through experiments on the MNIST and CelebA datasets, where we achieve accurate uncertainty estimates, as measured by the expected calibration error, even in challenging image restoration problems with significant inherent uncertainty.