Episode 18: Medical imaging and Fourier analysis

This week the topic was Fourier analysis. We interviewed Heather Williams, who’s a medical physicist and works with Positron Emission Tomography (PET) scanners, as well as other medical scanning devices. We talked about her work and how maths is important in converting data from the scanner into images that can be used to diagnose patients.

Interesting links:

• PET scanners on the NHS website
• Being a Medical Physicist on the NHS careers website
• Central Manchester University Hospitals, Heather's employer
• Shape of the sine and cosine graphs at BBC Bitesize
• An interactive guide to the Fourier transform at BetterExplained.com
• XKCD comic 'Fourier'

Puzzle: If a function is made by adding sin(x) + cos(x), what’s the maximum value attained by this function?

Solution:

This is a periodic function, which repeats every 180 degrees (or π radians). Its maximum value is the square root of two, or √2 = 1.414213..., which it first reaches at a value of 45 degrees, or π/4. The function varies between √2 and -√2, and it looks like a sin curve. The function can also be written as √ 2 + sin(θ + π/4). For a graph of the function, and more detail, input it into Wolfram Alpha.