The plotted field goes from a positive peak, through zero and a negative peak, back to the next positive peak: one complete 360° oscillation. It is a cycle of the RF magnetic field—not a proton traveling in a circle.
Interactive MRI physics01 — 09
You don’t photograph a body.
You encode it.
Start with RF excitation and relaxation, then follow gradients as they turn position into frequency and phase—from body and local coils to an oblique trajectory through k-space and an image reconstructed line by line.
- B₀
- 3.0 T
- γ̄ / ¹H
- 42.58 MHz/T
- model
- Cartesian GRE
ƒEquation ?meaning · units · live what-if · why care
◎Tap object / explain viewidentify one 3D object in place or open the complete visual key
↔Change a controloutputs and cause/effect notes update together
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 B₀ = 3.0 T
LOCAL FIELD
Bz(r,t) = B0 + G(t) · r
0.5 Tclinical 1.5 / 3 T7.0 T
−400+40 mT/m
−12 cmisocenter+12 cm
ΔB+2.00 mT
Δf = γ̄ΔB+85.15 kHz
local f127.825 MHz
CARRIER LAW
f0 = γ̄B0
42.58 MHz/T × 3.0 T = 127.73 MHzB₀ CHANGESRF carrier127.73 MHz
B₀ DOES NOT SETGradient offset+85.15 kHz
B₀ DOES NOT SETOn-resonance flip92.0°
B₀ DOES NOT SETk-space speed1.064 × 10⁶ m⁻¹/s
At fixed G, position, B₁⁺, and pulse duration, changing B₀ retunes the MHz carrier. The gradient offset, ideal flip angle, and dk/dt stay unchanged.
Scope: this control retunes the field/RF frequency model. The TE/TR tissue constants remain the explicitly stated illustrative 3 T model; real coil fields, SAR, and relaxation are field-dependent.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 / RF TRANSMIT, RELAXATION & RECEIVE
Excite broadly.
Listen locally.
The transmit coil creates the rotating B₁⁺ field that tips magnetization. After excitation, the scanner stops transmitting and receive coils detect the tiny voltage induced by precessing transverse magnetization. A common 1.5 T / 3 T workflow is body-coil transmit with a close local array receiving.
RF WAVEFORM DECODER · THREE DIFFERENT SHAPES, THREE DIFFERENT JOBS
Frequency is horizontal spacing. B₁⁺ amplitude is height. Pulse duration is width.
The plots share the live controls below. They are separated because “a faster wave,” “a stronger pulse,” and “a larger received signal” do not mean the same thing.
Change B₀ → retune the carrier in this comparison
T means tesla, the unit of magnetic-field strength.
The slow outline connecting the peaks of thousands of fast carrier cycles. It is not a second radio wave and not a container; it summarizes how strongly the transmitter is driven over time.
The size of the complex voltage after the receiver mathematically removes the MHz carrier. Phase is stored too, even though this row shows only non-negative magnitude.
Three-dimensional RF transmit and receive model
TRANSMIT WINDOW · B₁⁺127.73 MHz
BODY COIL → B₁⁺ → SPINS
α = 92.0°
Local receive elements are detuned while the body coil transmits.
Drag to orbit · mint cage is body RF · coral loops are local elements
dead time / switch
TRANSMIT FLIP ANGLE
α = γ ∫ B₁⁺(t) dt
RECEIVED VOLTAGE
vRx(t) ∝ −dΦM/dt
2 µTRF field strength18 µT
0.20 mspulse area3.00 ms
close · 1 cmreceive geometryfar · 12 cm
RF waveformpower amplifier
→
body coilB₁⁺ transmit
→
magnetizationprecessing spins
→
local arrayB₁⁻ receive
→
preampsADC / k-space
BODY TRANSMIT · LOCAL RECEIVE
The large built-in body coil drives a broad, comparatively uniform B₁⁺ field. During transmit the nearby receive array is actively detuned; during reception the transmitter is isolated and the local elements feed low-noise preamplifiers.
predicted rectangular-pulse flip92.0°
receive sensitivity proxy1.00× local
excitation coveragebroad / uniform
receive coveragelocal · 8 channels
transmit safety focuswhole-body SAR
switch stateRx array detuned
The sensitivity number is a normalized geometry teaching proxy, not a scanner specification. The shared B₀ control retunes the displayed ¹H carrier; it does not rescale this idealized coil geometry, B₁ field, SNR, relaxation, or SAR model. Actual values require coil- and patient-specific electromagnetic measurements.
Transmit field
RF power at the Larmor frequency rotates magnetization. Amplitude and pulse duration set flip angle; spatial B₁⁺ variation makes flip angle nonuniform.
Receive sensitivity
Precessing transverse magnetization induces a tiny voltage. Close local elements couple strongly to nearby anatomy and admit less distant noise.
Isolation matters
Receive-only elements are detuned during the high-power transmit pulse. The transmit path is then isolated while low-noise preamplifiers listen for the echo.
Predicted relative signalSPIN ECHO · TR 500 / TE 15 ms
fatgraywhiteCSF
Recovery before RF / decay before echoT₁ RECOVERY + T₂ DECAY
IDEAL SPIN-ECHO SIGNAL
S = ρ(1 − e−TR/T₁)e−TE/T₂
The 180° pulse refocuses static dephasing, so ideal echo amplitude follows T₂ rather than T₂*.
200 msrecovery + scan time6000 ms
5 mstransverse decay200 ms
1°spin echo fixes 90°60°
WHITE MATTERρ 0.70 · T₁ 850 · T₂ 80 · T₂* 55 ms
CSF / FLUIDρ 1.00 · T₁ 4000 · T₂ 2000 · T₂* 400 ms
One repetition90° → 180° → ECHO → NEXT 90°
SHORT TR · SHORT TE
Short TR samples tissues before full longitudinal recovery, strengthening T₁ differences. Short TE limits T₂ decay, so the current spin echo is predominantly T₁ weighted.
white matter signal0.280
CSF / fluid signal0.114
pairwise contrast42.2%
dominant weightingT₁ weighted
2D scan-time proxy1:04 · 128 lines
echo fraction TE / TR3.0%
Signals use simplified steady-state equations and illustrative relaxation values near 3 T. Real contrast also depends on sequence details, RF profiles, echo trains, magnetization transfer, flow, diffusion, coils, reconstruction, pathology, and field strength.
03 / 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
READOUT VS PHASE · EVENT-BY-EVENT
Choose one row. Then read across it.
They are not two different kinds of magnetism. Both logical jobs use a gradient, and either gradient creates position-dependent frequency offsets while it is on. Their timing relative to the receiver is what makes their stored information different.
READOUT
Called “readout” because the receiver’s ADC reads many samples while the positive read gradient is on.
Gread commandOFF
ADC / receiverCLOSED
SPIN PHASE CLOCKS AT SEVEN LOGICAL ximage POSITIONShand angle = transverse phase
WHAT THE GRADIENT DOES RIGHT NOW
No x-dependent offset from Gread.
The MHz RF carrier sets the rotating reference. Readout uses much smaller position-dependent receive-frequency offsets around that carrier—not a change in RF amplitude.
PHASE ENCODE
Called “phase” because the lobe ends before ADC, yet its position-dependent phase pattern remains during readout.
Gphase commandOFF
phase memoryNONE YET
SPIN PHASE CLOCKS AT SEVEN LOGICAL yimage POSITIONSnot fixed physical Gy
WHAT THE GRADIENT DOES RIGHT NOW
No y-dependent offset from Gphase.
While the phase lobe is on it also creates frequency differences. After it turns off, those added frequency offsets vanish, but the accumulated angular differences remain.
THE TWO JOBS COMBINE
Phase chooses the row; readout fills points across it.
The squares are k-space addresses, not anatomy pixels. One TR normally acquires one horizontal row in this simplified 2D Cartesian example.
current addresskx 0 · ky 0
stored this TR0 / 7 samples
current address
stored sample
phase-selected row
WHY THE WORD “LOGICAL”?
The job stays named read or phase even when its physical coil recipe changes.
Logical axes belong to the prescribed image plane. Physical Gx, Gy, and Gz are fixed hardware channels. The scanner rotates the two logical commands into coil currents; the anatomy and image jobs do not have to line up with the bore.
LOGICAL READ → PHYSICAL COILS
Gx
Gy
Gz
LOGICAL PHASE → PHYSICAL COILS
Gx
Gy
Gz
EVENT 01 / 08 · BEFORE IN-PLANE ENCODING
RF CREATES TRANSVERSE SIGNAL
The RF pulse tips magnetization so a receive signal can exist. RF transmit is not readout: the receiver is protected and ADC is closed during excitation. Neither in-plane logical gradient job has encoded a k-space row yet.
READOUT MOTION THROUGH K-SPACE
dkread/dt = γ̄ Gread(t)
A stronger read gradient crosses k-space faster. With dwell time fixed, that changes sample spacing and therefore FOV; with the acquisition prescription adjusted, it also affects bandwidth and distortion.
WHY CARE · This job sets read-direction sampling, bandwidth, chemical-shift displacement, and distortion behavior.
RETAINED PHASE PATTERN
Δφ(yimage) = 2π (−2 Δk) yimage
The signed area under the phase gradient sets ky. Changing that area on successive TRs chooses different rows; it does not move to a literal y location in the patient.
WHY CARE · Phase steps strongly affect scan time, phase FOV, wrap, motion ghosts, and the direction of many artifacts.Both cause Δf while on
A magnetic-field gradient changes local precession frequency. “Frequency encoding” and “phase encoding” describe how the sequence uses the accumulated effect, not two different gradient mechanisms.
ADC open versus ADC closed
Readout samples continuously while Gread is on. Phase encoding applies a lobe before sampling, closes it, and carries the retained phase ramp into the readout window.
One row per repetition
Readout collects many kx points in one ADC window. Conventional 2D Cartesian imaging repeats the TR with a new phase area to cover many ky rows, so the phase loop often dominates scan time.
MODEL BOUNDARY · Events are separated here so each cause is visible. Real pulse sequences often overlap the read prephaser and phase-encode lobe, use tens to thousands of samples, include finite ramps and spoilers, and may collect multiple ky lines per TR with echo trains or segmented readouts.
Three-dimensional oblique coordinate model
logical read / phase / slice frame
fixed physical Gx / Gy / Gz frame
LOGICAL READOUT GRADIENT
+30.0 mT/m
Gx +30.0 · Gy 0.0 · Gz 0.0 mT/m
Drag to orbit · plane normal is logical slice Z
COORDINATE TRANSFORM
Columns of R are the physical directions of logical readout, phase, and slice. Because R is orthonormal, the three logical axes remain mutually perpendicular.
Gphysical = R · Glogical
−90°physical X rotation+90°
−90°physical Y rotation+90°
−90°about slice normal+90°
−70reverse at 0+70 mT/m
| physical | Read | Phase | Slice |
|---|---|---|---|
| Gx | +1.000 | 0.000 | 0.000 |
| Gy | 0.000 | +1.000 | 0.000 |
| Gz | 0.000 | 0.000 | +1.000 |
PHYSICAL AMPLIFIER COMMANDS±40 mT/m per-axis model
GX
GY
GZ
AXES ALIGNED
In an axial prescription, logical readout aligns with the physical X gradient. Only the Gx amplifier is needed for this +30.0 mT/m readout plateau.
logical vector magnitude30.0 mT/m
safe logical maximum40.0 mT/m
limiting physical channelGX · 75%
Ideal per-axis amplitude example. Real systems also constrain slew rate, duty cycle, peripheral nerve stimulation, and vector-dependent safety limits.
04 / 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
Area, not height
A weak gradient held longer can create the same phase slope and k-space displacement as a short, strong gradient.
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.
05 / 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.
Every spin’s complex contributionUNIFORM PHASE AT k = 0
Color is the phase of ρ(x,y)e−i2πk·r. The receiver adds every colored contribution into one complex number.
The sample’s k-space addresskx 0 · ky 0
kx
ky
LOG |S(k)|LOWHIGH
The background is the phantom’s exact Fourier spectrum. The coral ring marks the single receiver sample inspected at left.
ONE RECEIVER SAMPLE
S(kx,ky) = ∫∫ ρ(x,y)e−i2π(kxx+kyy) dxdy
one k-space coordinate is a weighted sum from the entire excited slice
−32 Δkcenter+31 Δk
−32 Δkcenter+31 Δk
COHERENT DC SAMPLE
At k = 0 the encoding phase is identical everywhere. All positive spin density adds coherently, producing the large center coefficient that represents the object’s average signal—not a center pixel.
|S(k)| / |S(0)|100.00%
sample phase0.0°
normalized complex sample+1.000 + i0.000
required gradient momentMx 0.000 · My 0.000
phase cycles / FOVX 0 · Y 0
encoded wavelengthuniform phase
GY PREPHASEky = 0 set · Gx / ADC idle
READY · SELECT k OR SWEEP
RF, gradients, receiver, and signalTE 6 ms · TR 30 ms
The event block is expanded so short RF and gradient operations remain visible; the broken segment compresses idle recovery before the next RF pulse.
Integrated gradient pathkx 0.0 · ky 0.0 m⁻¹
Violet is unsampled prephasing. Mint is the portion stored by the ADC. The yellow ring is kx = 0—the echo center on this ky row.
RF + GZ · SLICE EXCITATION
BODY TX ON · RX ARRAY DETUNED
+GZ · RF BAND SELECTS ZThe transmit coil tips spins in the RF-selected slice. No in-plane k-space sample is recorded yet.
TIME IN TR0.00 / 30 ms
K-SPACE COORDINATEkx 0.0 · ky 0.0 m⁻¹
ADC STATEOFF · 0 / 64 SAMPLES
RECEIVE MAGNITUDE0.0% · BEFORE ECHO
THE ACQUISITION CHAIN
k(t) = γ̄∫G(τ)dτ → S(k) → ℱ−1
gradients choose the Fourier address; the ADC stores the complex receiver voltage only while its gate is open
160sets Δk + pixel320 mm
64receiver bandwidth256 kHz
−32Gy moment+31
5RF center → echo25 ms
15next RF + scan time100 ms
20illustrative tissue160 ms
REFERENCE PRESCRIPTION
A 220 mm FOV sampled at 64 readout points produces 3.44 mm pixels and Δk = 4.55 m⁻¹. Scrub the timeline or change a parameter to see the dependent quantities move together.
Δk = 1 / FOV4.55 m⁻¹
nominal kx,max145.5 m⁻¹
ideal Δx = FOV / 643.44 mm
ADC dwell / window7.81 µs / 0.50 ms
readout Gx13.7 mT/m
Gy moment−1.708 mT·ms/m
signal magnitude left at echo by T₂*89.7%
64-line scan proxy1.92 s
Ideal one-line-per-TR Cartesian GRE model with 64 complex samples and no ramp sampling. “Total sample rate” is used here because vendor bandwidth displays may instead report Hz/pixel. Here T₂* controls only the slow decay of transverse signal magnitude with time; use the TE/TR lab above for the full spoiled-GRE steady state.
BEFORE THE FIRST TR
Frequency is not amplitude.
Open the definitions below, then press Run. This explanation will follow RF transmit, gradients, receive sampling, and reconstruction through each repetition.
RF carrier frequency≈ 127.73 MHz at 3.0 T+
How fast the transmit field oscillates and the received voltage alternates. It is tuned near the ¹H Larmor resonance. It may be offset or shaped to select a slice, but it is not the waveform height.
RF amplitude · B₁⁺µT · slow strength outline+
How strong the transmit field is. Together with pulse duration it sets flip angle. The drawn outline is simply B₁⁺ strength versus time; it cannot display the tens to hundreds of millions of carrier cycles per second at this scale.
Gradient · Gx, Gy, GzmT/m · field slope+
Not a radio wave. A gradient slightly changes local Larmor frequency with position. Its signed area sets retained phase and the k address; while it remains on, its amplitude sets how many inverse metres that numerical address changes per second.
Received signalcomplex voltage · I + iQ+
The coil detects a tiny RF voltage near the carrier. The receiver removes that fast carrier and stores a complex sample: magnitude says how much coherent signal arrived and phase preserves spatial encoding.
Image brightness|inverse Fourier transform|+
Not raw RF amplitude and not a k-space location. Reconstruction combines every acquired complex sample, and magnitude display maps the resulting voxel signal to brightness.
Pulse sequence / one TRTR 01 / 64
ExciteSelect zSet kyTraverse kx
Raw signal / k-spacekx −32 · ky +31
kx
ky
Each acquired cell stores S = I + iQ. Its displayed brightness is = log(1 + √(I² + Q²)); this compresses the preview only. 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.06 / TRAJECTORY STUDIO
Gradients draw
the path.
Here is a spatial-phase address, not a physical position or a moving particle. A gradient changes that address over time: hold one component constant and the address follows a straight line; reverse it and the address turns back; vary two components together and the address can spiral. Compare five encoding strategies built from that rule.
WHAT DOES k MEAN?
Count full turns of relative spin phase across distance.
k = 100 m⁻¹ means the gradient-created phase pattern completes 100 turns per metre. Therefore two fixed positions 10 mm apart differ by one full 360° turn at that instant.
The marker is the current data address where ADC stores a whole-object complex sample. It is not a proton, voxel, anatomical location, RF carrier cycle, or signal amplitude.
FIXED POSITION rk = +100 m⁻¹
0 mm
0° 2.5 mm
90° 5.0 mm
180° 7.5 mm
270° 10 mm
360° = 0°
0° 2.5 mm
90° 5.0 mm
180° 7.5 mm
270° 10 mm
360° = 0°
10 mm of position → 1 complete phase cycle
WHY A GRADIENT “MOVES” k
dk/dt = γ̄G(t)
dk/dt is the rate at which the spatial-phase address changes, in m⁻¹/s. G(t) is the field slope in T/m. Positive G moves toward +k, negative G toward −k, and G = 0 holds the address still.
WHY CARE · The visited extent sets potential detail, spacing sets FOV, and a wrong address from delay or miscalibration creates blur, ghosts, or geometric distortion.
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 LAWG is a field slope. Its direction chooses which k coordinate changes; its magnitude sets the address-change rate 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 how rapidly the numerical k address changes. A stronger readout gradient covers more spatial-frequency address per unit time; nothing physically flies through the patient.
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.
07 / 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
REFERENCE PRESCRIPTIONWhat changed—and what it costs
This 220 mm, 128 × 128 reference produces 1.72 mm in-plane sampling. Move any control to see its direct consequence for detail, signal, and encoding time.
in-plane pixel area 1.00×
signal proxy 1.00×
phase-time proxy 1.00×
PIXEL / VOXEL SIZE
Δx = FOVx / Nx ≈ 1 / (2kx,max)
FIELD OF VIEW
FOVx = 1 / Δkx
2D SLICE THICKNESS
Δz = BWRF / (γ̄ |Gz|)
Three-dimensional RF-selected slice
LOWER FREQUENCYHIGHER FREQUENCY
RF-SELECTED SLAB
4.70 mm
center z = 0.0 mm · axial plane
Drag to orbit · highlighted rings fall inside the transmit passband
FREQUENCY ADDRESS
Δf(z) = γ̄ Gz z
SELECTED THICKNESS
Δz = BWRF / (γ̄ |Gz|)
Frequency-to-position map+GZ · CENTERED RF
41730 mT/m
0.53.256.0 kHz
−20on-resonance+20 kHz
CENTERED AXIAL SLICE
Gz maps position to frequency. The 2.0 kHz RF passband intersects that line around z = 0, exciting an ideal 4.70 mm slab.
frequency slope0.426 kHz/mm
RF passband−1.0 … +1.0 kHz
slice center0.0 mm
ideal thickness4.70 mm
Thicker slab
A wider transmit band matches a larger interval of positions on the same frequency slope.
Thinner slab
A steeper frequency slope maps the same RF bandwidth onto a narrower spatial interval.
Move the slice
Changing the RF center frequency moves the intersection without changing the ideal thickness.
08 / 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.
09 / 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
Continue into the physics.
Clinical overviewK-space in the clinic↗
Interactive-teaching reviewK-space tutorial and image consequences↗
Open reviewCMR physics for clinicians↗
ISMRM courseSpatial encoding in MRI↗
Trajectory reviewEPI and spiral imaging↗
Signal derivationGradient phase and k-space↗
Cartesian acquisitionPrephasing, readout, and ADC trajectories↗
Oblique coordinatesLogical-to-physical gradient rotation↗
RF coil reviewBody transmit and local receive↗
Relaxation modelSpoiled GRE steady-state signal↗
Slice-selection modelRF bandwidth, Gz, and slice position↗
Artifact reviewTruncation, aliasing, and filtering↗