Chapter 16: Spatial Encoding & k-Space

The signal from the previous chapter carries no information about where in the body it came from: in a uniform static field, every proton precesses at the same Larmor frequency. Encoding position is the job of gradient fields — and gradient-based MRI, though different from CT, returns to the same Fourier transform.

Frequency encoding with gradients

Superimpose on the main field a weak field whose strength varies linearly with position. Applying an $x$ gradient $G_x$ makes the instantaneous Larmor frequency a linear function of position:

$$\omega(x) = \gamma (B_0 + G_x\, x)$$

Now simply resolving the received signal into frequencies maps each frequency component to an $x$ coordinate. The $G_x$ applied during readout is the frequency-encode gradient. For the other axis, a gradient $G_y$ is applied briefly before readout to give each row a position-dependent phase shift (phase encoding). Repeating with a different $G_y$ amplitude each time collects the two-dimensional information.

k-space

The phase a gradient imparts to the signal over time is the time integral of the gradient. Define that integral as a new coordinate $\mathbf{k}$:

$$k_x(t) = \frac{\gamma}{2\pi}\int_0^t G_x(\tau)\, d\tau, \qquad k_y(t) = \frac{\gamma}{2\pi}\int_0^t G_y(\tau)\, d\tau$$

Then the signal received at time $t$ is exactly the 2D Fourier transform of the transverse magnetization $m(x,y)$, sampled at the point $(k_x, k_y)$:

$$s(k_x, k_y) = \iint m(x, y)\, e^{-i 2\pi (k_x x + k_y y)}\, dx\, dy$$

So MRI measures not the image itself but the values of its Fourier transform (k-space). Manipulating the gradients is nothing other than tracing a pen across k-space: the readout gradient draws a line in the $k_x$ direction, and the phase encode selects the starting $k_y$. Standard 2DFT scans k-space one horizontal line at a time to fill the grid.

Reconstruction is just a 2D inverse Fourier transform of the collected k-space — no ramp filter, no backprojection as in CT. The sample spacing $\Delta k$ sets the field of view (FOV = $1/\Delta k$); the maximum $k_{\max}$ sets the spatial resolution ($\approx 1/2k_{\max}$).

Simulation: filling k-space

k-space (center), the 2D Fourier transform of the true image (left), is filled one phase-encode line at a time, with the inverse transform (right) shown at each step. Recall the scan-and-sinogram growth in Chapter 2 — this is its k-space counterpart. The central lines bring in the overall grayscale; the outer lines sharpen the edges. Toggle "center-out order" to scan from the center outward and see the contrast settle early.

Simulation: what lives where in k-space

Reconstruct from only part of k-space. "Center only" keeps the low frequencies: blurred, but the contrast survives. "Periphery only" keeps the high frequencies: edges appear and flat regions vanish. Move the radius slider to see how contrast and resolution are split. This "center = contrast, periphery = resolution" structure is the key to the scan-time reductions in the coming chapters.

Key points

Gradient fields make the Larmor frequency a function of position, turning the received signal into samples of the image's Fourier transform. That measurement space is k-space, and manipulating gradients traverses it. Reconstruction is a single 2D inverse Fourier transform. The center of k-space carries low frequencies and contrast; the periphery carries high frequencies, edges, and resolution. Sample spacing sets the FOV; the maximum frequency sets the resolution. The next chapter shows how to design the contrast of that signal through pulse sequences and the parameters TR and TE.

References

• Ljunggren S. A simple graphical representation of Fourier-based imaging methods. Journal of Magnetic Resonance 54, 338–343 (1983).
• Twieg DB. The k-trajectory formulation of the NMR imaging process. Medical Physics 10, 610–621 (1983).
• Nishimura DG. Principles of Magnetic Resonance Imaging, Chapters 5–6. Stanford University (2010).