1.    Quick review of math concepts. We make extensive use of complex numbers and the complex exponential function. We briefly review these topics.

2. Phase-contrast imaging in the electron microscope, including the contrast-transfer function (CTF).

3.    Fourier transforms(FTs) in one dimension, and in higher dimensions, are essential in dealing with the CTF and are key in 3D reconstruction algorithms. They are used in very many steps of computer image processing. We provide a quick but comprehensive introduction to FTs.

4.    Noise places important limits on the quality of cryo-EM results. We cover the basic principles of noise and random variables to help identify quantitatively what noise does and how its effects are best mitigated. 

5.    Estimation theory is the field of mathematics underlying important methods such as maximum-likelihood estimation.

6.  3D Reconstruction from 2D images is the central process in structure determination. Based on an understanding of CTFs, Fourier transforms and noise statistics, we are able to explain the details of 3D reconstruction programs. An understanding of the reconstruction process allows us to invent new imaging approaches and to understand problems such as preferred orientations and missing wedge effects in data acquisition.