FOURIER TRANSFORMS &
K-SPACE
For my research to explore the whole MRI process I needed to investigate the point where the analogue signals are converted in a digital biomedical image that accurately represents internal anatomy. This is acheived through computational and mathematical processes: which seemed nothing more than magical to me. This is what I mean by the analogue-digital interface: where RF pulses become part of matter-energy and informational transformaitons. The mathematical tools used to do this include Fourier transforms (FTs) and their friends k-space and inverse Fourier transforms (IFT).
FTs are mathematical devices that decompose complex periodic functions into a sum of sine and cosine waves almost like a frequency un-mixing machine. FTs can take apart and identify the individual sine waves that make a complex signal. Metaphorically speaking, FTs unpick ‘a chord’ and identify the ‘individual notes’ that compose it. FTs operate from within a mess of analogue signals emerging from the body which they transform into a digital cross-section of the body. Highlights in the History of the Fourier Transform by Alejandro Domínguez (2016) explains that in mathematics an FT is “a function or signal as a superposition of sinusoids” (p. 53).
An FT will select multiple frequencies, adding them together until the total waveform resembles the signal. FTs are a widespread, specialist type of mathematics needed in signal analysis and whose step-by-step operations can only be depicted and revealed numerically. There will always be a limit to how closely we can represent their operations. FTs operate from within a mess of analogue signals emerging from the body which they transform into a digital cross-section of the body, taking apart and reconfiguring the different properties of a wave and arranging/rearranging these properties according to specific temporal and spatial configurations. An FT will analyse a signal by selecting multiple frequencies and re-combining them until the synthesised waveform resembles the signal. FTs are a widespread, specialist type of mathematics needed in signal analysis and whose step-by-step operations can only be depicted and revealed numerically. There will always be a limit to how closely we can represent their operations