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So I'm trying to understand how MRI machines work. I understand all the concepts of it, the parts, what they do, how the machine works, etc. The part I'm having trouble with is the fourier transform business.

In the MRI machine, when the hydrogen atoms release the radio frequencies previously absorbed, the transceiver absorbs it and uses fourier transform to convert those signals from the time domain to the frequency domain.

The part I don't understand is, how does the fourier transform end up producing the image of the body. How do you go from math to a detailed picture of the brain??? Can anyone explain that?

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The MRI machine makes your head atoms vibrate at a different frequency from your toe atoms. Those atoms all vibrate at the same time, which is recorded, and then the FFT lets you extract the amplitude of each individual frequency, which is proportional to the amount of water in each slice. Then you do it again in the other two dimensions (left side vibrating at a different frequency from right side, chest vibrating at a different frequency from back) and you can put them all together to find the amount of water at each point. –  endolith Nov 28 '12 at 19:16
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The signals that we measure in MRI are a combination of signals from all over the object being imaged. It so happens that any signal (even if you simply make one up and draw a squiggle) is composed of a series of sine waves, each with an individual frequency and amplitude. The Fourier transform allows us to work out what those frequencies and amplitudes are. (That is to say, it converts the signal from the time domain into the frequency domain.)

Since we encode the signal with magnetic field gradients which make frequency and (rate of change of) phase relate to position, if we can separate out the frequencies we can say where we should plot the amplitudes on the image.

The amplitudes serves as the brightness levels in the image:

$Amplitude = I$

$(Phase, Frequency) = (x,y)$

$I(x,y) = I(x i + y j)$

Phase and Frequency help to find where in the image(which exact part of the MRI) has the particular amplitude measured by the technique of passing RF signals through the body and collecting the signal information.

The exact process is well explained with animations at: http://www.imaios.com/en/e-Courses/e-MRI/Signal-spatial-encoding

An RF pulse does not have one frequency only (for this, it would need to be of infinite duration). It covers a certain bandwith, which depends on the shape of the pulse and its duration.

The thickness of the slice can be varied by adjusting the bandwidth of the selective pulse and the amplitude of the slice selection gradient.

For a fixed amplitude gradient, the wider the bandwidth, the greater the number of protons excited and the thicker the slice For a fixed bandwidth, the stronger the gradient, the greater the variation of precession frequency in space and the thinner the slice

Moreover, the shape of the RF pulse in time will also determine the bandwidth profile in frequency, and thus the slice profile.

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ah this makes much more sense, thank you for the detailed explanation –  Richard Nov 23 '12 at 20:53
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