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Fourier transform is a mathematical operation which converts a time domain signal into a frequency domain signal 5.
Fourier transform is integral to all modern imaging, and is particularly important in MRI. The signal received at the detector (receiver coils in MRI, piezoelectric disc in ultrasound and detector array in CT) is a complex periodic signal made of a large number of constituent frequencies (i.e., bandwidth). This can be visualized as multiple sine and or cosine waves along a time-axis. Fourier transform represents the same data over a frequency-axis. A common example are the MR spectroscopy images in which different molecules are at different frequencies along the x-axis.
French mathematician Joseph Fourier developed a method of decomposing a wave function into a series of sine waves in the 19th century 5.