Area, not height
A weak gradient held longer can create the same phase slope and k-space displacement as a short, strong gradient.
Interactive MRI physics01 — 08
You don’t photograph a body.
You encode it.
MRI gradients turn position into frequency and phase. Follow that encoding from the scanner bore to a trajectory through k-space—and watch an image emerge line by line.
- B₀
- 3.0 T
- γ̄ / ¹H
- 42.58 MHz/T
- model
- Cartesian GRE
Interactive 3D scanner
Scanner coordinate frame
↔ Drag to rotate
THE CENTRAL RELATION
s(k) = ∫ ρ(r) e−i2π k·r dr
The received signal at one k-space coordinate is a weighted sum from the entire excited object.01 / FROM FIELD TO POSITION
A controlled tilt in the magnetic field.
A gradient coil adds a small, nearly linear variation to the much larger main field. Proton frequency now carries an address: where a spin sits determines how fast its phase turns.
x = +8.0 cm
Live field model
¹H at 3 T
LOCAL FIELD
Bz(r,t) = B0 + G(t) · r
−400+40 mT/m
−12 cmisocenter+12 cm
ΔB+2.00 mT
Δf = γ̄ΔB+85.15 kHz
local f127.825 MHz
i The gradients are tiny beside B₀, but they are switched quickly and precisely. The audible knocking in MRI is gradient hardware moving under Lorentz force.
02 / THREE ORTHOGONAL CONTROLS
Same physics.
Three jobs.
“X, Y, and Z” are logical axes chosen by the pulse sequence. The scanner combines all three physical coil sets to rotate an oblique slice, so readout does not have to mean anatomical left–right.
GX · ON DURING ADC
Position becomes frequency.
During signal readout, Gx makes spins at different x positions precess at different frequencies. Sampling through time walks continuously across one row of k-space.
FREQUENCY OFFSETΔf(x) = γ̄ Gx x
gradient actionContinuous plateau
moves throughkx within a line
resolution set by±kx,max
03 / GRADIENT MOMENT & SPIN PHASE
The area under G
becomes phase.
Gradient amplitude alone does not set a k-space coordinate. Its signed time integral does. Build one lobe, add an opposite rewinder, and watch a three-dimensional spin ensemble wind into a phase pattern—or return coherently to k = 0.
Three-dimensional spin phase ensemble
transverse magnetization vectors
equal-phase planes
GX MOMENT
kx = +170.3 m⁻¹
Phase is drawn in the transverse x–y plane; B₀ points along physical z.
Drag to orbit · scroll to zoom
ZEROTH GRADIENT MOMENT
M0(t) = ∫0t G(t′) dt′
POSITION IN K-SPACE
k(t) = γ̄ M0(t)
SPIN PHASE
φ(r,t) = 2π k(t) · r
Signed gradient area and accumulated kUNBALANCED
−300+30 mT/m
00.601.20 ms
none100% · k = 0120%
COHERENT SIGNAL FROM THE COLUMN8.9%
The positive Gx area moves the sample to +kx. Spins separated along x retain different phases after the lobe turns off, so their vector sum is small.
net M₀+4.00 mT·ms/m
k coordinate+170.3 m⁻¹
phase across 20 mm+3.41 turns
Polarity sets direction
Changing the sign of G reverses the phase ramp and moves to the opposite side of k-space along the selected logical axis.
Balanced area refocuses
For stationary spins in this ideal model, an equal opposite lobe cancels M₀. The phase ramp unwinds and the coherent signal returns at k = 0.
04 / LIVE CARTESIAN ACQUISITION
Watch k-space
become an image.
Each TR selects a slice, applies one phase-encoding moment, then samples an echo under the readout gradient. Repeat with a new Gy area and the raw data matrix fills one row at a time.
Pulse sequence / one TRTR 01 / 64
ExciteSelect zSet kyTraverse kx
Raw signal / k-spacekx −32 · ky +31
kx
ky
Brightness shows log signal magnitude. Position in k-space is spatial frequency—not a location in the head.
Fourier reconstruction0% DATA
AR
Δx = 3.4 mm
Press run to acquire the first phase-encoded line.
Near k = 0
Contrast & broad shape
Slow spatial variation. High signal energy. Acquired at the echo center.Large |k|
Edges & fine detail
Rapid spatial variation. Extending farther raises the ideal resolution limit.Sample spacing Δk
Field of view
Closer k-space samples encode a wider unaliased FOV: FOV = 1 / Δk.05 / TRAJECTORY STUDIO
Gradients draw
the path.
A gradient is velocity in k-space. Hold it constant and k moves in a straight line; reverse it and the path turns back; rotate two gradients together and the path spirals. Compare five encoding strategies generated from that one rule.
Interactive k-space trajectory
commanded k(t)
delayed response
CURRENT SAMPLE
k = (−1.00, −1.00, 0.00)
Drag to orbit · scroll to zoom
MULTI-SHOT · RECTILINEAR
One echo, one row.
A prephaser moves to −kx. The readout gradient crosses one horizontal line while ADC samples. A new Gy area selects the next ky on the following TR.
THE TRAJECTORY LAWDirection of G sets direction of travel. Its magnitude sets speed in m⁻¹/s.
dkdt= γ̄ G(t)
Representative gradient commandSHOT 01 / 15
ideal10 µs20 µs
i Delayed or filtered gradient response means the scanner may sample somewhere other than the commanded coordinate. Non-Cartesian reconstruction therefore often uses a measured trajectory.
coverage / excitation1 k-space row
center crossingsonce per echo
reconstructiondirect FFT
Gradient amplitude
Controls instantaneous speed through k-space. Stronger readout gradient covers more spatial frequency per unit time.
Slew rate
Limits how sharply a path can turn. Fast switching also drives acoustic noise and peripheral nerve-stimulation constraints.
Sampling window
The path may move while the receiver is off. Only coordinates visited during ADC become acquired data samples.
06 / RESOLUTION LAB
Shape the voxel
in all three axes.
Resolution is not a “megapixel” setting. It follows from field of view, sample count, and how far the acquisition reaches in k-space. Smaller voxels usually cost signal-to-noise, scan time, or both.
3D ENCODED VOLUME1.72 × 1.72 × 5.00 mm
Drag to orbit · highlighted cell is one voxel
Protocol builder
XReadoutfrequency
1.72 mm
YPhaseencoded
1.72 mm
ZSliceRF selected
5.00 mm
voxel volume14.77 mm³relative SNR proxy 1.00×
k-space extent±291 / ±291 m⁻¹Z uses slice profile
minimum phase encodes128≈ 25.6 s at TR 200 ms
PIXEL / VOXEL SIZE
Δx = FOVx / Nx ≈ 1 / (2kx,max)
FIELD OF VIEW
FOVx = 1 / Δkx
2D SLICE THICKNESS
Δz = BWRF / (γ̄ |Gz|)
07 / SAMPLING & POINT RESPONSE
Change k-space.
The image answers.
Spatial resolution, field of view, ringing, and wraparound are different consequences of how k-space is sampled. Apply the operations directly to the complex data and inspect both the reconstructed image and the point-spread function that produced it.
Sampling prescription
CUTOFF / RESOLUTION
Δx ≈ 1 / (2kx,max)
SPACING / FIELD OF VIEW
FOVx = 1 / Δkx
25% · blurred60%100% · full
FULL REFERENCE
All encoded frequencies are retained at the native Δk. The PSF approaches one image pixel and no coherent wrap replicas are introduced.
nominal Δx × Δy3.44 × 3.44 mm
effective FOV220 × 220 mm
PSF FWHM1.0 × 1.0 px
retained samples4,096 / 4,096
Applied sampling mask100% RETAINED
Mint points contribute to reconstruction. Coral lines are inside the selected extent but skipped by uniform undersampling.
Image-space PSFℱ⁻¹{MASK}
positivenegative
The central lobe sets effective resolution; sidelobes produce ringing or displaced wrap replicas.
Resulting reconstructionREFERENCE
AR
Full k-space extent and native spacing preserve the simulated image’s available detail and field of view.
Truncate extent
Removing outer k-space lowers the spatial-frequency cutoff. Image space is convolved with a broader sinc-like response: detail softens and sharp boundaries ring.
Increase spacing
Keeping every Rth point multiplies the effective Δk by R and reduces the encoded FOV by R. Repeated PSF peaks fold distant anatomy into the displayed FOV.
Apodize
A Hann or Hamming taper suppresses PSF sidelobes and Gibbs ringing, but broadens the central lobe. A calmer boundary is purchased with effective resolution.
08 / THE COMPLETE CHAIN
From current waveform to one image voxel.
- 01
Gradient current
Amplifiers drive X, Y, Z coil windings. Current and geometry create a controlled field slope in T/m.
I(t) → G(t) - 02
Spin phase
The local field changes Larmor frequency. Accumulated phase records the gradient’s area through time.
φ(r,t) = γ r·∫G dt - 03
k-space sample
The receiver sums every transverse spin at the current spatial-frequency coordinate.
k(t) = γ̄∫G dt - 04
Image estimate
An inverse Fourier transform separates the superposed spatial frequencies into locations.
ρ̂(r) = ℱ⁻¹{s(k)}
REFERENCE DESK