# K-space exploration

**Denis Hoa**

The readout MR signal is stored in K-space which is equivalent to a Fourier plane. To go from a k-space data to an image requires using a 2D inverse Fourier Transform.

Frequency-encoding and phase-encoding are done so that data is spatially encoded by differences in frequency and phase, amenable to analysis by Fourier transform. In k-space, fx-coordinates (horizontal spatial frequencies) and fy-coordinates (vertical spatial frequencies) of the Fourier plane are replaced by kx and ky-coordinates.

A classic spin echo sequence fills the k-space line by line. Here is the explanation of the k-space trajectory :

- 90° RF pulse + Slice-selection gradient : location at origin (center) of k-space
**Negative and strong phase-encoding gradient**: moves to the lower bound of k-space**Positive frequency-encoding gradient (dephasing lobe)**: moves to the right bound, location at the lower right corner**180° RF pulse + Slice-selection gradient : moves to the opposite location, location at the upper left corner****Positive frequency-encoding gradient + Data acquisition**: moves to the right + acquire MR signal**Repetition for each line with increasing phase-encoding gradient strength (negative to positive intensity).**The amount of gradient phase change between adjacent line is constant. This results in a sequential (line by line) filling of k-space from top to bottom.

The k-space location (kx and ky coordinates) of data is governed by the accumulated effect of gradient events and excitation pulses. Here are the rules for moving in k-space:

- The initial RF excitation pulse (with the slice-selection gradient) is the beginning of the sequence: location is at the center of k-space.
- A 180° RF pulse causes a jump to the opposite location.
- The greater the net strength of the phase-encoding gradient (or the longer the gradient is on), the farther from the k-space origin the data belong, in the upper direction if the gradient is positive or in the lower direction if the gradient is negative. As the duration of phase-encoding gradient is most often constant, the strength of the phase-encoding gradient governs the location on the vertical axis (ky-coordinate).
- The longer the frequency-encoding gradient is on (or the greater the net strength of the gradient is), the farther from the k-space origin the data belong, in the right direction if the gradient is positive or in the left direction if the gradient is negative. As the strength of the frequency-encoding gradient is most often constant, the duration of the frequency-encoding gradient governs the location on the horizontal axis (kx-coordinate)

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