Basics of Image acquisition

Basics of Image acquisition#

Topics overview#

  • k‑space trajectories: Cartesian, radial, spiral, PROPELLER, 3‑D shells

  • Encoding: frequency‑encode, phase‑encode, slice‑select; partial Fourier

  • Sampling parameters: FOV, matrix size, receiver bandwidth, dwell time, echo train length

  • Readout schemes: single‑shot, segmented, turbo factor / ETL

  • Averaging & noise: NEX/NSA, parallel‑receive SNR behavior

  • Reconstruction: Fourier transform, gridding, NUFFT, density compensation

  • Coil combination: sum‑of‑squares, adaptive combine, ESPIRiT maps

  • Parallel imaging: SENSE, GRAPPA, CAIPIRINHA

  • Compressed sensing / low‑rank: k‑t, XD‑GRASP

  • Corrections: motion/phase, eddy‑currents, \(B_0\) drift

  • Artifact mitigation & QC: ghosts, Gibbs ringing, geometric distortion; basic checks

Historical notes (FID and early NMR)#

  • 1946 (CW NMR) — Bloch, Purcell et al. used fixed‑frequency \(B_1\) and swept \(B_0\) through resonance, observing “resonance absorption” / “nuclear induction signal.”

  • 1948 — Bloembergen, Purcell, Pound reported small oscillations (“wiggles”) flanking the main absorption after passing resonance.

  • Hahn (pulsed NMR) — Held \(B_0\) constant, applied pulsed \(B_1\) at the Larmor frequency; observed the free induction decay (FID).

MRI differs from other modalities because the signal is generated by the patient; localization identifies where signals originate.

Spatial localization#

Three components:

  • Slice selection (SS)

  • Frequency encoding (FE)

  • Phase encoding (PE)

Important

Having two gradients on at the same time means the vector sum of gradients; it is not equivalent to turning them on independently.

Slice selection#

  • Apply \(G_{\text{slice}}\) during the RF pulse. RF bandwidth and gradient strength set the slice thickness: - Wider RF bandwidth or weaker gradient → thicker slice. - Narrower RF bandwidth or stronger gradient → thinner slice.

  • After the RF, apply an opposite‑polarity rephasing lobe on \(G_{\text{slice}}\) to undo slice‑selection dephasing within the slice.

Frequency encoding (readout)#

  • During signal acquisition at \(\mathrm{TE}\), apply \(G_{\text{read}}\) so position maps to frequency.

  • The ADC samples with dwell time \(\Delta t\); receiver bandwidth \(\mathrm{RBW} \approx 1/\Delta t\) (per complex sample).

  • 1‑D Fourier transform along readout reconstructs the spatial profile in the read direction.

Phase encoding#

  • A brief gradient \(G_{\text{phase}}\) before readout imparts a position‑dependent phase that persists (spins see the same \(B_0\) before/after the lobe).

  • Repeating the sequence with different PE amplitudes fills different rows of k‑space.

k‑space formalism#

Define k‑space coordinates from gradients:

\[\mathbf{k}(t) = \frac{\gamma}{2\pi} \int_0^{t} \mathbf{G}(\tau)\, d\tau \quad\Rightarrow\quad s(t) = \int \rho(\mathbf{r})\, e^{-i 2\pi \mathbf{k}(t)\cdot \mathbf{r}}\, d\mathbf{r}\]

Key properties:

  • Central k‑space (low spatial frequencies) encodes contrast; acquired with smaller PE amplitudes, higher signal coherence.

  • Outer k‑space (high spatial frequencies) encodes edges / resolution; larger PE amplitudes.

  • Symmetries permit partial Fourier (half‑scan) if conjugate symmetry holds after phase correction (e.g., homodyne, POCS).

Sampling & imaging relations#

Let readout matrix be \(N_x\), FOV be \(\mathrm{FOV}_x\), and read gradient be constant during acquisition.

  • k‑space sample spacing: \(\Delta k_x = 1/\mathrm{FOV}_x\)

  • Resolution: \(\Delta x \approx \mathrm{FOV}_x / N_x\)

  • Maximum spatial frequency: \(k_{x,\max} \approx N_x/(2\,\mathrm{FOV}_x)\)

  • Dwell time vs bandwidth: \(\Delta t \approx 1/\mathrm{RBW}\) (per complex sample)

Readout schemes#

  • Single‑shot: entire k‑space (or a plane/segment) in one shot (e.g., EPI); fast, sensitive to distortions and \(T_2^*\).

  • Segmented: k‑space split across shots; reduced distortion, longer scan.

  • Turbo/fast spin echo (FSE/TSE): multiple echoes per TR; echo train length (ETL) / turbo factor is number of k‑space lines per TR.

Averaging & noise#

  • NEX/NSA (number of excitations / signal averages): SNR scales as \(\sqrt{\text{NEX}}\).

  • Parallel receive arrays: SNR depends on coil geometry and g‑factor; combination method matters.

Coil combination#

  • Sum‑of‑squares (SoS): simple, robust magnitude combination.

  • Adaptive combine: SNR‑optimal with estimated noise covariance / sensitivity.

  • ESPIRiT / sensitivity maps: explicit coil sensitivity modeling for parallel imaging.

Parallel imaging#

  • SENSE: image‑domain unfolding using sensitivity maps.

  • GRAPPA: k‑space interpolation using autocalibration lines (ACS).

  • CAIPIRINHA: controlled aliasing pattern to improve conditioning / g‑factor in 2‑D/3‑D.

Compressed sensing & low‑rank#

  • CS (k‑t): incoherent undersampling + sparsity (e.g., temporal TV, wavelets).

  • Low‑rank / subspace: e.g., XD‑GRASP for motion‑resolved reconstruction.

Non‑Cartesian recon#

  • Radial / spiral / PROPELLER / 3‑D shells require gridding or NUFFT with density compensation before FFT.

  • Trajectory calibration and system delays affect image quality; correct with field probes, trajectory measurement, or self‑navigation.

Corrections & QC#

  • Motion/phase correction: navigator echoes, PROPELLER, retrospective registration, phase stabilization.

  • Eddy‑current correction: pre‑emphasis, post‑hoc modeling, bipolar gradients.

  • :math:`B_0` drift correction: reference navigators, field monitoring.

  • Artifacts: - Ghosting (EPI Nyquist): odd/even echo phase mismatch; calibrate/phase‑correct. - Gibbs ringing: insufficient high‑frequency sampling; mitigate with apodization or higher matrix. - Geometric distortion: \(B_0\) inhomogeneity in EPI; reduce bandwidth per pixel in phase‑encode, use parallel imaging, field maps.

  • Basic QC: check noise level, ghost‑to‑signal ratio, NPS/PSF behavior, gradient timing, and k‑space center stability.

Workflow summary#

  1. Design trajectory and encoding (SS, PE table, readout).

  2. Acquire data with appropriate sampling (dwell time, RBW, FOV, matrix).

  3. Preprocess (DC/phase corrections, eddy/\(B_0\) drift fixes).

  4. Reconstruct: - Cartesian: FFT (with coil combine / parallel imaging). - Non‑Cartesian: gridding/NUFFT + FFT (then coil combine, PI, CS/low‑rank as needed).

  5. Validate/QC and mitigate artifacts where necessary.