Phase Encoding Intro II
I understand the 2-pixel example, but I still can't put it all together with the whole image and frequency encoding. Can you help?
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Note to reader: Please make sure you understand the 2-pixel phase-encoding example in the last Q&A before proceeding on to this more complicated example.
We expand our previous 2-pixel Q&A example to a 6-pixel one, showing how Fourier transformation is used to sort out overlapping signal contributions in two dimensions. Below is a 2x3 array of pixels (A-F) at the center of a brain image with frequency- and phase-encoding directions as designated.
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In this simple experiment we again play out only two phase-encoding steps. In Step 0 the PE gradient is off and the signals from all pixels remain in phase. In Step 2 we assume the pixels in the second row (B, D, and F) all acquire a phase shift of 180° relative to row 1. After each phase-encoding step, a single aggregate MR signal is recorded reflecting contributions from each of the six pixels. Hence in this simple example with only two phase-encoding steps, exactly 2 MR signals are recorded from the slice.
For a real-life MR image, the process described in the 6-pixel example is repeated using 128-256+ phase-encoding steps. The simultaneous solution of multiple equations is not quite so straightforward but can still be accomplished by advanced mathematical techniques. (Note that here the phase-encoding direction is left-to-right).
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The first of these signals is acquired when there is no phase shift between the two rows. Fourier transformation of this MR signal generates a spectrum revealing three frequencies (ω1, ω2, and ω3), whose amplitudes (A+B), (C+D), and (E+F) are the sum of the pixel values in each frequency column. At this stage we only know the sums of the pixel values for each frequency column; we do not know the individual values for any cell.
When we record the MR signal from the whole sample after Step 1 (where the second row has been phase-shifted by 180°, the Fourier spectrum has changed. Again three frequencies can be separated, but this time their magnitudes are (A−B), (C−D), and (E−F). We still are not able to assign unique pixel values with this single measurement, because now only the differences in pixel values for each frequency column are known.
Because the Fourier Transform has sorted out frequencies into three columns, the corresponding phase change information can now be decoded. As in the 2-pixel example, and algebraic manipulation of the data for each frequency column using data from each phase-shift allows us to determine A, B, C, D, E, and F uniquely. A real MR image may have hundreds of pixels in each dimension, and hence requires hundreds of phase-encoding steps to decode, but the basic principles are the same.
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References
Felmlee JP, Morin RL, Salutz JR, Lund GB. Magnetic resonance imaging phase encoding: a pictorial essay. Radiographics 1989; 9:717-722.
Wald L. MR image encoding. (From MIT OpenCourseWare http://ocw.mit.edu)
Felmlee JP, Morin RL, Salutz JR, Lund GB. Magnetic resonance imaging phase encoding: a pictorial essay. Radiographics 1989; 9:717-722.
Wald L. MR image encoding. (From MIT OpenCourseWare http://ocw.mit.edu)
Related Questions
I understand frequency-encoding, but I just don't get phase-encoding. Can you explain?
What is a Fourier transform?
I think I see how to calculate pixel values in your simple examples, but how do you do this in a real image with thousands of pixels?
I understand frequency-encoding, but I just don't get phase-encoding. Can you explain?
What is a Fourier transform?
I think I see how to calculate pixel values in your simple examples, but how do you do this in a real image with thousands of pixels?