Tomography is a technique that produces cross-sectional images of an object. It derives from the Greek words tómos (section) and gráphó (to write). The idea behind the approach is to reveal internal structures of a certain object without physically opening or cutting it. Therefore, as a non-invasive technique, it is very convenient for a wide range of applications, such as medical and industrial tomography. For example, it can be used to check for physical defects or the contents of drainage pipes. Archeologists also use tomography to study mummies in their sarcophagi. Biologists study the complex social interactions of insects in their colonies without modifying their behaviour and researchers apply the principles of tomography to reveal charged particle structures in the upper atmosphere. These are just some examples of tomography and, although they may differ in methods, instruments and goals they share the same mathematical principles as the bases for the reconstruction of the final tomographic images. The following brief tutorial aims to provide a better understanding of the concepts behind tomography. It is a basic tutorial to demonstrate how tomography works using the “Sum along Ray” principle. The tutorial has four stages that will illustrate very simple concepts and give an introduction to some difficulties and challenges when deploying tomography as a tool.

Level 1

We will start using tomography by scanning a simple object. The object (sample) to be scanned is represented by a 2x2 matrix, and a unique density value (subsample) is associated with each pixel. The discretisation of the object through a grid is necessary as computers work with finite arithmetic.

example — Figure 1: The objected to be inverted (sample) as a matrix 2x2 pixels. Each pixel is denoted as subsample and contains an arbitrary density value. Rays, which give the observation values, are indicated as green arrows. Measurements are indicated in boxes at the end of the arrow.

The image obtained through tomography is the result of a mathematical interpretation of the data retrieved from measurements (or observations) received by detectors. Green lines represent rays that cross the object and provide the measure of the sum (integrated) of the density values along the ray path. Sum values are reported on the side of the grid in correspondence with the ray’s arrow.

INSTRUCTIONS:

The small squares represent different subsamples within a larger sample (the large square) being studied.
The numbers within the squares represent the values within the squares (e.g. tissue densities in biological sample or electron densities in the ionosphere).
You can left click on your mouse to step through the tutorial.
The green lines represent the direction of the ray.
The sum along ray value is given at the end of the arrow.
Left click on the Reset button to start again.

Level 2

Rays cross the object and provide a measurement that is the sum of density values along the ray path. At this point, we assume that density values inside the object are discretized according to the grid. In reality, measurements come from an object whose densities are defined continuously in space and in time.

Level 2 gives examples of how tomographic measurements relate to subsample values. When the subsample values change the final sum (tomographic measurements) also change.

INSTRUCTIONS:

Left click on your mouse on any of the squares to step through the tutorial.
A minimum of one square must be selected (e.g. Bottom left square).
Left click on the Reset button to start again.

Level 3

It should now be clear that observations are a collection of rays that measure integrated densities along the ray path. The job of tomography is to elaborate sets of observations in order to reconstruct a picture of the object that accurately represents the physical characteristics of that object. The mathematical operation used is called inversion and it is at the heart of all tomographic techniques. An appropriate analogy can be made with respect to the well-known game Sudoku.

Level 3 gives you the opportunity to perform manual inversions when given the sum along ray values.

INSTRUCTIONS:

Left click on any of the blank squares to enter and increase the subsample values.
Right click on any of the squares to reduce the subsample values.
Progress through your manual inversions until you calculate all the subsample values.
The traffic light guide at the top right of the tutorial will indicate RED-wrong value, YELLOW close but not quite correct and GREEN will indicate that you have calculated the correct subsamples values based on the tomographic measurements you were given.
The New Example button will provide other sets of tomographic values for you to invert.
Left click on the Reset button to start again.

Level 4

Ideally observations need to be numerous enough in order to cross all the cells in the grid. But this is usually not sufficient to provide a good reconstruction. In this level you will get an understanding of the importance of the ray’s arrangement. It is not always possible to select the rays we need for tomography however on this level, you will be free to select the rays you prefer to use.

This will be the most complex level for you to enjoy. Here you can build your own tomography system which will automatically invert the subsamples for you. Your aim will be to use the minimum number of rays to calculate the subsample values. You will be able to select more than the minimum number of rays but the result will be less efficient and a message will ask you to try again.

INSTRUCTIONS:

Place the cursor outside the sample then left click and drag the mouse over the path you want the ray to follow.
You can choose any path.
When you think you have enough rays left click the INVERT button to calculate the subsample values.
The Undo button removes the last ray.
The Reset button will remove all the rays in one go.
Left click on the Play Again button to create a new game.
Left click on the Reset button to start again.

This tutorial is a very simplified version of how real applications work. However, it should now be clear how tomography works and how important ray arrangements are. Some real life applications may not have enough observations to cross all the cells in the grid. Furthermore, to obtain reconstructions with good resolution, the number of cells in the grid should be as many as possible. In the real world thousands of pixels form the grid and this complicates the inversion operation. It should be clear however as the number of rays has to increase considerably to cover all the cells in the grid. In addition, with real objects, these principles are then applied to a three-dimensional grid which increases the number of voxels (3D pixels) and associated complexities.