  Image-to-image regression is an important learning task, used frequently in
biological imaging. Current algorithms, however, do not generally offer
statistical guarantees that protect against a model's mistakes and
hallucinations. To address this, we develop uncertainty quantification
techniques with rigorous statistical guarantees for image-to-image regression
problems. In particular, we show how to derive uncertainty intervals around
each pixel that are guaranteed to contain the true value with a user-specified
confidence probability. Our methods work in conjunction with any base machine
learning model, such as a neural network, and endow it with formal mathematical
guarantees -- regardless of the true unknown data distribution or choice of
model. Furthermore, they are simple to implement and computationally
inexpensive. We evaluate our procedure on three image-to-image regression
tasks: quantitative phase microscopy, accelerated magnetic resonance imaging,
and super-resolution transmission electron microscopy of a Drosophila
melanogaster brain.
