Photonic Imaging with Statistical Guarantees: From Multiscale Testing to Multiscale Estimation

In this chapter we discuss how to obtain statistical guarantees in photonic imaging. We start with an introduction to hypothesis testing in the context of imaging, more precisely we describe how to test if there is signal in a specific region of interest (RoI) or just noise. Afterwards we extend this approach to a family of RoIs and examine the occurring problems such as inflation of type I error and dependency issues. We discuss how to control the family-wise error rate by different modifications, and provide a connection to extreme value theory. Afterwards we present possible extension to inverse problems. Moving from testing to estimation, we finally introduce a method which constructs an estimator of the desired quantity of interest with automatic smoothness guarantees.