Three Dimensional Electron Microscopy/Fourier transforms

What You Need To Know About A Fourier Transform
'''By David J DeRosier

Professor Emeritus, Department of Biology & Rosenstiel Basic Medical Sciences Research Center, Brandeis University'''

Introduction
Most students of molecular electron microscopy keep well away from learning about Fourier transforms. At schools whose aim is to train students in molecular electron microscopy, however, the gathered class must sit through a lecture or two on the Fourier transform. A mathematical lecture on the topic is usually more satisfying to the faculty than it is to students, who use the occasion to day dream or simply sleep having been up half the previous night at some bar. I guess it is not obvious why such a mathematical operation would be of interest to those who simply wants to know the molecular architecture of some cellular machine.

My aim is to tell you why you want a Fourier transform of your electron micrographs, what you can learn from a Fourier transform, how to think about a Fourier transform without having to waddle through the mathematics, and how to generate a Fourier transform when you want one. Since microscopy and image analysis are visual, I am presenting many of the lessons as pictures. I do not intend to prove the properties of various transforms but rather to show the results. There are lots of books on the theory.

Background on Fourier Transforms
Fourier Transforms are a tool used to analyze complicated data. They are, in essence, a mathematical function that transforms spatial or time based data into amplitudes and frequencies. This allows for the data to be analyzed at a glance, and more importantly makes it very easy to adjust features of the signal. Uses of the Fourier transforms include processing and analysis of sound, video, images, and other large, complex sources of data. The technology transition from analog technologies to digital has ushered in an increased use of this technique. Since the original algorithm for the transform is very long and computer resource intensive, a computer friendly version known as the Fast Fourier Transform (FFT) was independently invented by both J.W. Cooley and J.W. Tukey in 1965.

In transmission electron microscopy image analysis, fourier transforms are heavily utilized to remove the low resolution data from a collected image. Removing the low resolution data, which is the center of the image produced by a fast fourier transforms, is a way of increasing the contrast on images without losing much of the identifying information. It is often necessary to remove some of the high resolution data, which is found at the outside of the fast fourier transform, since the method of collection in a tunneling electron microscope often distorts the data. The need to do this step varies with the quality of the image recorded. For example, having a direct detector instead of film will greatly increase image quality and will allow more of the data transformed to be kept, which leads to a higher resolution 3D reconstruction.

Taking data from an electron microscopy image and transforming also allows for greater ease in 3D reconstruction. Fourier transforms are completely and easily reversible even after images have been processed, so frequently 3D reconstructions are generated with the fourier transforms of 2D images.

Fourier Transforms involve many of different types of math, but the most common element that is seen in fourier transform math is complex numbers. Complex numbers are usually seen in some form or another but a common form of this is. Now i stands for imaginary number. You might be asking yourself why involve such numbers or complex math altogether. Well its simple really, complex numbers make things go together really well. Rather than laying hundreds of equations and piling on thousands upon thousands of different mathematical proofs for you to see and understand, complex numbers shorten it and make it look a bit more elegant. Complex numbers is just one element that shows up in fourier equations and transforms.

Figure 1: Fourier Series formula, Purdue University.

Figure 1 listed above gives an apt description of what the formula for Fourier series looks like. Its expressed in sines and cosines, which are commonly found in frequencies. Fourier series and transforms allow any type of data to be converted into frequencies and amplitudes and it also allows us to delete certain frequencies away from the data. This allows us to clean up most of the data or image in our case. This is extremely helpful in electron microscopy as, many images taken from a EM are littered with aberrations or under focusing on the image. The fourier transform allows us to convert the images we see in to frequencies or amplitudes and allows us to modify or “delete” parts of the frequency away in order to clear up the image and correct any aberrations seen or found in the data. Then running an inverse of the fourier transform, turns the frequency back into an image with the corrections made.