MRI Physics: k-space and FOV

The spacing between k-space samples: FOV = 1/Δk (Δk in cycles/m). Smaller steps → larger FOV.
Each Δk step adds exactly one extra phase cycle across the FOV, so positions one FOV apart become indistinguishable.

How far out k-space is sampled: pixel size ≈ 1/W_k = FOV/N, where W_k = N·Δk is the total k-range.
Reaching further into outer k-space → finer pixels.

Pixel = 32/128 = 0.25 mm. Δk = 1/0.032 m = 31.25 cycles/m. W_k = 128 × 31.25 = 4000 cycles/m → 1/4000 m = 0.25 mm. Same grid seen from both sides.

Δkx = (γ/2π)·Gx·Δt, so FOV_x = 1/((γ/2π)·Gx·Δt), with Δt the dwell time between ADC samples.
Total readout T_acq = N·Δt sets the k-range W_kx = (γ/2π)·Gx·T_acq and thus the pixel size.

The step between phase-encode amplitudes sets the spacing: Δky = (γ/2π)·ΔGy·τ → FOV_y = 1/Δky. The largest |Gy| (number of steps) sets the resolution.

Extra outer samples at the same spacing → same FOV, finer resolution. Samples closer together over the same extent → larger FOV, same resolution.
Both cost more samples (and, in the phase direction, more time).

Δky doubles, so FOV_y halves: anything outside the new FOV wraps into the image (aliasing). Resolution is unchanged.
This is a spacing problem, not a loss of detail.

Signal outside the phase FOV gets the same phase labels as a position inside (phase repeats every 2π), so it is folded in.
Worked: FOV −10 to +10 mm (20 mm). Signal at +12 mm appears at +12 − 20 = −8 mm, on top of whatever is really there.
The readout direction rarely wraps (receiver filtering and oversampling), so wrap is mainly a phase-direction problem.

Same FOV, lower resolution: the image blurs. An abrupt cut-off also causes ringing (Gibbs ripples) next to sharp edges.

More displayed pixels and a smoother look (interpolation), but no new measured information and no real gain in resolution.

The image of a single point produced by your sampling: the inverse Fourier transform of the sampling mask (1 = acquired, 0 = missing, × any weighting).
It belongs to the acquisition, not the object, and every image is a sum of shifted PSFs (convolution). Broad main peak → blur. Side ripples → ringing. Extra peaks → aliased copies.

About π (half a cycle): adjacent pixels end up with opposite phase.
The largest step is N/2 × ΔGy·τ, so k_max ≈ 1/(2δy) and 2π·k_max·δy ≈ π. That is the finest pattern the grid can show, so the largest lobe sets the resolution.
Trap: 2π is the phase across the whole FOV from one ΔGy·τ step.

Only k = 0 → a perfectly uniform image (just the average brightness). The same value at every k → a single bright point.
A point in one domain is spread everywhere in the other. A fully, uniformly sampled grid has a PSF that is a single sharp point.

Less ringing (smaller side ripples) but a broader PSF, so more blur.
Smoother edges here mean less detail, not more.

It acquires a bit more than half of k-space and uses the mirror symmetry of opposite k-points (true for a real-valued image) to fill in the rest, using the central data to estimate the phase.
Simply zero-filling the missing half would not use this symmetry.

Samples recorded later have less signal, so k-space is weighted unevenly. If the outer (detail) part is recorded late, it is weakened → blur.
In multi-echo methods (e.g., turbo spin echo), the echo that fills the k-space centre sets the contrast.

A consistent phase mismatch between alternate (odd/even) k-space lines → a faint copy of the image shifted by half the FOV (N/2 ghost).
Structured duplication, not random noise.

Object shifted the whole time → a correct image, just shifted. Object moving partway through → k-space lines describe different positions → ghosts and blur, mostly in the phase direction.

Shifts the image (Fourier shift theorem); magnitudes are unchanged. Applied consistently to all data it gives a clean shift, not ghosts.

One corrupted k-space value is one stripe pattern added to the whole image → stripes everywhere.
Further from the centre → finer stripes; the direction of its offset sets the stripe orientation; its amplitude sets the strength.

Structured bands or a bright line across the image rather than random noise. A constant-frequency interferer appears as a line ("zipper") at one readout position.
The pattern of the artefact reveals which k-space data were disturbed.

Overflow: the signal exceeds the ADC range and is clipped (gain too high). Quantization: finite digital steps round off small differences (gain too low wastes range). Both corrupt k-space.

Fat resonates ~3.4 ppm below water. Because readout uses frequency as position, fat is drawn shifted along the readout direction relative to water from the same place.
The shift grows with B0 and with lower readout bandwidth.

SNR ∝ voxel volume, so halving the slice thickness halves the signal per voxel (≈ half the SNR) if everything else is unchanged.
Gain: better through-plane detail and less partial-volume mixing. More slices may be needed for the same coverage.

4× the averages (and ≈ 4× the time), because SNR ∝ √N_avg.
Averaging reduces random noise; it cannot fix wraparound or create contrast that is not there.

More coverage and larger (coarser) pixels; same number of phase encodes, so same scan time; more signal per voxel.
To keep the original resolution you would also need a larger matrix.

Faster scan (fewer TRs) but coarser pixels in the phase direction. The FOV is unchanged because the spacing is unchanged.

Each phase encode needs another TR; readout points are all collected within the same echo.
Fewer readout points mainly change readout durationandwidth, not the number of repetitions.

2D: T ≈ TR × N_phase × N_avg (× slice packages). 3D: T ≈ TR × N_y × N_z × N_avg.
Worked: 1.7 s × 128 = 218 s ≈ 3.6 min. Adding 128 partition encodes: 1.7 × 128 × 128 ≈ 27,850 s ≈ 7.7 h.

Interleaving: while slice 1 recovers during its TR, the scanner excites and reads other slices, then returns to slice 1.
Extra slices are "free" until N_slices × time per slice exceeds TR; then a longer TR or extra packages are needed. Longer TE → longer block per slice → fewer slices fit.

Real slice profiles have soft edges, so exciting one slice partly saturates its neighbour → signal loss.
Remedies: 1) gaps between slices (lose coverage); 2) interleaved order (odd then even, more recovery time between neighbours); 3) sharper RF pulses.

1) Enlarge the phase FOV or use phase oversampling (more phase encodes for the same resolution).
2) Put the phase direction along the object's shorter dimension (still covering it fully).
3) Saturation bands over the tissue outside the FOV.
Trap: a bigger matrix at the same too-small FOV does nothing for wrap.

1) A selective ~90° pulse + gradient tips Mz in the unwanted slab.
2) A spoiler gradient scrambles that Mxy.
3) The imaging excitation follows quickly and finds Mz ≈ 0 there → no signal.
It is temporary (T1 recovery), so it is repeated every TR.

Cover the tissue that causes the problem (e.g., wrap) while keeping the band's soft edge out of the target; otherwise wanted tissue next to it goes dark.
Each band adds RF pulses: more SAR, hardware load and preparation time.

No. Isotropic means equal size in all three directions; 3D makes thin partitions possible, but FOV and matrix must be chosen to make them equal.

Gradient heating: R·I^2 in the gradient coils (cooling, duty cycle). RF hardware limits: peak/average power of amplifier, cables, coil. SAR: RF energy absorbed by the sample (W/kg).
Three different checks; more saturation or refocusing pulses raise RF load and SAR.