Many advanced technologies, from self-driving cars to processing medical images, rely on machine-learning to succeed. Technologies based on deep learning or deep neural networks have proven to be especially effective at learning from vast quantities of data, and yet our theoretical understanding of these tools has lagged behind. As we rely more on neural-network systems in our daily lives, it becomes even more important that we can guarantee their safety and reliability. One failure mode of current systems is that they can be confidently incorrect, and this can cause real-world harm. It would be better if our systems "know what they don't know" by maintaining an internal representation of their own uncertainty. In recent years, progress has been made in the mathematical study of "calibration" and "multicalibration," which establish a mathematical framework for uncertainty, probability, and fairness in problems having to do with categorical prediction such as classifiers and recommender systems. The goal of this project is to port these ideas into perceptual domains such as image or video processing. This project will result in the creation of new general-purpose neural networks for image-processing based on a new mathematical principle of learning "calibrated representations" of images, resulting in general-purpose systems that effectively "know what they don't know." This will enable more robust and reliable image-processing applications across wide sectors of research and