Fast & Parallel Imaging

Why Conventional Imaging Is Slow

In a conventional spin-warp acquisition, every repetition collects exactly one line of k-space. The phase-encode gradient is stepped to a new value on each TR, so a matrix with $N_y$ phase-encode lines requires $N_y$ excitations. The total scan time is the brutal product

With $N_y = 256$ and $\\mathrm{TR} = 2500$ ms, a single average takes over 10 minutes. Physiologic motion, patient comfort, and functional or dynamic studies all demand far shorter times. The strategies below attack the problem from three angles: fill k-space faster per excitation (EPI), acquire fewer lines and reconstruct the rest (parallel imaging, partial Fourier, compressed sensing), and image multiple slices at once (simultaneous multi-slice).

Echo-Planar Imaging: All of K-Space in One Shot

Echo-planar imaging (EPI), introduced by Peter Mansfield in 1977, traverses all of k-space after a single RF excitation. Following one excitation pulse, a rapidly oscillating readout gradient sweeps back and forth across $k_x$, drawing a zig-zag raster. Between each readout lobe, a small phase-encode blip advances the trajectory by one line in $k_y$. The entire image, perhaps 64 to 128 lines, is collected within a single $T_2^{*}$ decay envelope, typically 30 to 100 ms.

Because every line is acquired under the same exponential $T_2^{*}$ decay, lines collected late carry less signal and are blurred relative to early lines. The center of k-space, which dominates contrast, is acquired at the effective echo time $\\mathrm{TE}_{\\mathrm{eff}}$, the moment the trajectory crosses $k_y = 0$.

Geometric Distortion and Bandwidth Along the Phase-Encode Axis

The defining weakness of single-shot EPI is its tiny bandwidth per pixel in the phase-encode direction. The time between adjacent phase-encode samples is the full echo spacing $\\Delta t_{\\mathrm{esp}}$ (often 0.5 to 1 ms), whereas in the readout direction adjacent samples are only microseconds apart. The phase-encode pixel bandwidth is therefore

For $N_y = 128$ and $\\Delta t_{\\mathrm{esp}} = 0.8$ ms, $\\mathrm{BW}_{\\mathrm{PE}} \\approx 10$ Hz per pixel, roughly two orders of magnitude narrower than the per-pixel readout bandwidth (commonly 1 to 2 kHz). The ratio is set entirely by the timing: the readout dwell time between adjacent $k_x$ samples is only a few microseconds, whereas a full echo spacing separates adjacent $k_y$ samples. Off-resonance spins, especially at air-tissue interfaces and from fat, accumulate phase between successive phase-encode samples that maps to large geometric shifts. A B0 offset of $\\Delta f$ displaces a pixel by $\\Delta f / \\mathrm{BW}_{\\mathrm{PE}}$ pixels, so even a few hundred hertz of fat-water shift moves fat by tens of pixels, which is why EPI of the orbitofrontal cortex and brainstem warps so badly and why fat must be aggressively suppressed.

The Nyquist N/2 Ghost

Because alternate readout lines are acquired with the gradient running in opposite directions, even and odd lines suffer slightly different timing, eddy-current, and group-delay errors. After Fourier transform these alternating-line inconsistencies produce a replica of the object shifted by half the field of view in the phase-encode direction, the classic Nyquist or N/2 ghost. It is corrected with a reference scan (three navigator lines acquired without phase encoding) that measures and removes the even/odd phase difference.

Parallel Imaging: Coils as Spatial Encoders

Parallel imaging exploits a second source of spatial information that gradients never touch: the spatially varying sensitivity of each receive coil in a phased array. A coil near the right side of the head sees right-sided signal more strongly than left-sided signal. This sensitivity map C_\\ell(\\mathbf{r}) acts as an extra encoding function, letting us skip phase-encode lines (undersample k-space) and recover the missing information from the redundancy across $L$ coils.

Skipping lines by an integer acceleration factor $R$ widens the k-space sample spacing, which shrinks the effective field of view and causes the image to alias: pixels separated by $\\mathrm{FOV}/R$ fold on top of one another. The reconstruction task is to unfold these superimposed pixels.

SENSE: Unfolding in the Image Domain

Sensitivity Encoding (SENSE, Pruessmann 1999) works after the Fourier transform. For $R = 2$, each pixel in the aliased single-coil image is the sum of two true pixels weighted by that coil's sensitivities. Stacking the measured aliased values across all coils gives a small linear system per pixel:

SENSE requires accurate, artifact-free sensitivity maps (usually from a quick low-resolution prescan or built-in body coil reference). If the maps are wrong or the prescribed FOV is smaller than the anatomy, residual central aliasing artifacts appear.

GRAPPA: Filling K-Space from Autocalibration

GeneRalized Autocalibrating Partial Parallel Acquisition (GRAPPA, Griswold 2002) operates before the Fourier transform and needs no explicit sensitivity maps. A small block of fully sampled autocalibration signal (ACS) lines is acquired near the center of k-space. From these the algorithm learns linear weights that predict a missing k-space point in one coil from a neighborhood of acquired points across all coils. Those weights are then applied throughout k-space to synthesize the skipped lines, after which each coil image is reconstructed and combined.

| Property | SENSE | GRAPPA | | --- | --- | --- | | Domain | Image domain (post-FT) | K-space domain (pre-FT) | | Calibration | Explicit coil sensitivity maps | Autocalibration (ACS) lines | | Output | Single unfolded image | Full k-space per coil, then combine | | Failure mode | Central residual aliasing if FOV or maps wrong | Diffuse noise / coherent residual ghosting | | Robustness to map error | Lower | Higher |

Acceleration Factor R and the g-Factor SNR Penalty

Two effects degrade SNR when you accelerate. First, acquiring $R$ times fewer lines collects less signal, costing a factor of $\\sqrt{R}$ exactly as fewer averages would. Second, the unfolding inversion amplifies noise locally whenever coil sensitivities are not well separated at the folded pixels. This spatially varying penalty is the geometry factor, or g-factor:

Because $g \\geq 1$, the loss is always worse than the ideal $\\sqrt{R}$. The g-factor is highest in the center of the body where coil sensitivities overlap most, which is why accelerated images often show a noisy central region. The g-factor grows rapidly as $R$ approaches the number of independent coils, so practical in-plane factors are $R = 2$ to $3$ for an 8-channel array and up to $R = 4$ for 32-channel arrays.

Partial Fourier: Exploiting Hermitian Symmetry

For a perfectly real-valued object, k-space is Hermitian symmetric: $S(-\\mathbf{k}) = S^{*}(\\mathbf{k})$. In principle one need only sample half of k-space and fill the rest by conjugation. Real images are never perfectly real because B0 inhomogeneity, flow, and chemical shift introduce slowly varying phase, so we acquire slightly more than half (typically 60 to 75 percent, often called 5/8 or 6/8) of the lines, reserving a central band to estimate and correct that low-spatial-frequency phase.

The standard reconstruction is homodyne detection: the central symmetric region estimates a phase map, the asymmetric region is high-pass weighted, and the conjugate-symmetry assumption fills the unmeasured half. Because only a fraction $f$ of k-space is actually measured, the SNR penalty is

A 6/8 partial Fourier scan ($f = 0.75$) retains about $\\sqrt{0.75} \\approx 0.87$ of the SNR while cutting roughly a quarter of the acquisition time. Partial Fourier in the phase-encode direction shortens scan time; partial Fourier in the readout (partial echo) shortens TE, useful for the short echoes needed in MR angiography and single-shot DWI.

Simultaneous Multi-Slice and CAIPIRINHA

Parallel imaging accelerates within a slice; simultaneous multi-slice (SMS, or multiband) accelerates across slices. A composite multiband RF pulse excites two to eight slices at once. Their overlapping signals are then separated using the same coil-sensitivity mathematics as SENSE/GRAPPA, but applied in the slice direction. Crucially, because no k-space lines are skipped, SMS incurs no $\\sqrt{R}$ signal loss from undersampling, only a g-factor-like penalty from the slice unaliasing.

To make adjacent slices easier to separate, CAIPIRINHA (Controlled Aliasing In Parallel Imaging Results IN Higher Acceleration) imposes slice-dependent phase shifts during excitation, shifting each simultaneously excited slice by a fraction of the FOV relative to its neighbors. This increases the distance between aliased copies and dramatically lowers the g-factor. Multiband factors of 2 to 4 are now routine in diffusion and resting-state fMRI, cutting whole-brain coverage times by the same factor.

Compressed Sensing: Sparsity Plus Incoherence

Compressed sensing (CS) breaks from the Nyquist criterion entirely. It rests on three pillars. First, sparsity: MR images are compressible in some transform domain (wavelets, finite differences / total variation, or temporal Fourier for dynamic data), meaning most transform coefficients are near zero. Second, incoherent sampling: k-space is undersampled with a pseudo-random pattern (denser at the center) so that the aliasing appears as incoherent, noise-like artifact rather than coherent fold-over. Third, nonlinear iterative reconstruction that enforces both data consistency and sparsity.

The $\\ell_1$ penalty promotes sparse solutions while the data-consistency term keeps the answer faithful to the measured samples. Iterative solvers (for example iterative soft thresholding or ADMM) alternate between enforcing measured data and thresholding small transform coefficients. CS combines naturally with parallel imaging (CS-SENSE, $\\ell_1$-SPIRiT) and is the engine behind clinical accelerations of 4 to 8 fold for free-breathing cardiac and dynamic contrast-enhanced studies. The interactive tool below lets you undersample k-space and watch coherent versus incoherent aliasing emerge.

Putting It Together

Modern protocols stack these techniques. A clinical brain DWI might use single-shot EPI for speed, GRAPPA $R = 2$ to shorten the echo train and reduce distortion, 6/8 partial Fourier to shorten TE, and multiband 2 to cover the whole brain quickly. Each layer multiplies the time savings but also compounds the SNR and artifact penalties, so the radiographer must balance net acceleration against image quality for the clinical question at hand.

| Technique | What it skips | Speed gain | Dominant cost | | --- | --- | --- | --- | | EPI | Nothing; one-shot readout | Whole image per excitation | Distortion, T2* blur, N/2 ghost | | SENSE / GRAPPA | Phase-encode lines (in-plane) | Factor R | g times square root of R SNR loss | | Partial Fourier | Half of k-space via symmetry | Up to about 2x | Square root of f SNR loss, edge blur | | SMS / CAIPIRINHA | Nothing; multiple slices at once | Factor MB | Slice g-factor, higher SAR | | Compressed sensing | Random k-space samples | 4 to 8x | Iterative recon, possible detail loss |