## Working with mathematics and imaging

### Normen - an industrial and Applied Mathematics Bachelor alumnus and currently an Applied Mathematics Master's student

**After graduating cum laude from the Bachelor's programme in Industrial and Applied Mathematics at the University of Twente, Normen wanted to continue studying mathematics. So he opted for the Applied Mathematics Master's programme at the University of Twente. This programme offers two specializations. Normen is now following a path that is part of the Mathematical Systems Theory, Applied Analysis and Computational Science (SACS) specialization.**

An image: everyone thinks of an image as something you can use to visually store and show a given situation. But what is an image from a mathematical perspective? In mathematics we define an image by a function. This function has a certain domain as input, and this domain can be anything. For example, a square (2D), a human heart (3D), a clip of a beating heart (3D + time, also known as 4D).

In order to understand a function’s output, it is best if we go back to the image of a 2D rectangle. An HD television divides this domain (the rectangle) into 1920x1080 squares, also known as pixels. The function gives a trivalent output for each pixel: [*V*_{red}, *V*_{blue}, *V*_{green}] , in which the *V* stands for ratio. The television can use this output to ensure each pixel displays the desired colour, by "mixing" the three colours according to the ratio. You can imagine that in this way each image is entirely determined by the underlying function. This is why, in mathematics, we interpret the image with a function, to which we can then apply all sorts of tricks in order to improve the quality of the image. You can find a simple example of this below: during one of the Master's in Applied Mathematics classes, students transformed the image on the left into the image on the right by increasing the image’s *contrast*.

As you can imagine, this branch of mathematics has many applications in the medical world. At the moment a lot of research is being done in the area of ‘segmentation’, in which we use mathematics to automatically identify certain objects in an image. For example, we can automatically detect cancer cells in an image of blood taken through a microscope.

Every imaging technique currently in use in hospitals (MRI, PET, CT, etc.) is furnished with mathematical algorithms that adapt the signal that has been picked up to produce an image and make it clear. At the same time new imaging techniques are being discovered. Just Google MPI (Magnetic Particle Imaging) and you will see what I mean. For my thesis, I am going to try and apply the theory I learned in class to understand and improve the results these new imaging techniques produce. One of great benefits of this project is that I get to do a three-month internship with the Imaging Systems Laboratory at the University of California Berkeley!