ECALL: Estimation-calibrated learning for joint deconvolution and kernel inference from unpaired data

Recovering an original image from a blurred observation when the blurring kernel is unknown is a classical inverse problem with applications in astronomy, microscopy, and medical imaging. This setting is often referred to as blind deconvolution, a highly ill-posed problem that suffers from significant non-uniqueness and typically requires strong prior assumptions or training data. While recent deep learning methods have achieved impressive performance, they generally rely on access to paired datasets of original and blurred images, which is often difficult to acquire. In this paper, we propose ECALL (Estimation-Calibrated Learning), a novel framework for joint kernel and signal recovery from unpaired data. ECALL operates without paired observations, instead using separate collections of original and blurred images. The method leverages moment-based constraints to estimate the kernel and simultaneously perform deconvolution. A key component of our approach is a loss function that incorporates cycle consistency with respect to the estimated kernel and a reconstruction operator, alongside terms that match statistical properties (e.g., moments) of the data distributions. We demonstrate the effectiveness of ECALL with numerical experiments, highlighting its potential in scenarios where paired data are unavailable.