In Chapter 2 we introduced the Kullback–Leibler (KL) divergence that measures the dissimilarity between two distributions. This turns out to be a special case of the family of f-divergences between probability distributions, introduced by Csiszár. Like KL divergence, f-divergences satisfy a number of useful properties: operational significance, invariance to bijective transformations, data-processing inequality, variational representations (à la Donsker–Varadhan), and local behavior.

The purpose of Chapter 7 is to establish these properties and prepare the ground for applications in subsequent chapters. The important highlight is a joint-range theorem of Harremoës and Vajda, which gives the sharpest possible comparison inequality between arbitrary f-divergences (and puts an end to a long sequence of results starting from Pinsker’s inequality – Theorem 7.10).

From unconditional synthesis to conditional synthesis, GANs have shown that they are a perfect fit for such problems thanks to their ability to learn probability distributions. For unconditional synthesis, the objective is to stochastically generate MR images of target contrast. Conditional synthesis refers to the case where the model learns nonlinear mapping to the different MR tissue contrasts without altering the physiological information. Furthermore, by merging collaborative information of multiple contrast images, missing data imputation among many different domains is also effectively solved with GANs. Although promising results are seen, development in the area is still at its early stage. Interesting research directions are proposed from prior work, including the application of more advanced methods and rigorous validation in clinical settings. In effect, MRI image synthesis techniques should be able to reduce the burden of costly MR scans, benefiting both patients and hospitals.