Theme 04 · Encoding and data
Turn gradient current, phase, and field maps into k-space data and reconstructed images.
Explore read, phase, and slice encoding; gradient moments; complex k-space; Fourier reconstruction; quantitative 3D susceptibility-to-field, GRE phase, QSM inversion; trajectories; and single-shot EPI artifacts.
03 / THREE ORTHOGONAL CONTROLS
Same physics.
Three jobs.
Read, phase, and slice are logical jobs attached to the prescribed image plane. The X/Y/Z labels in this first axial example name image coordinates—not a permanent promise that read = physical Gx, phase = physical Gy, and slice = physical Gz. For an oblique plane, the scanner combines all three fixed physical coil sets.
THE THREE TABS BELOW COMPARE ROLES; THEIR LEFT-TO-RIGHT ORDER IS NOT TIME. In the simplified 2D sequence taught here, RF plus the slice-select gradient first creates transverse signal in a slab. A later phase-encoding lobe creates a retained angle pattern. A read prephaser then prepares the start, and only afterward do the read gradient and ADC collect samples.
READOUT · ADC OPEN
Gread +20.0 mT/m
Different x positions have different receive-frequency offsets while whole-object I/Q samples are stored.
IMAGE COORDINATE FRAME
Axial · logical axes coincide with physical axes
Identity mapping: Gread = +1.00 Gx, Gphase = +1.00 Gy, and Gslice = +1.00 Gz.
Three-dimensional readout, phase-encoding, and slice-selection comparison
GX
GY
GZ
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
0 A · no added slope250 A teaching value400 A
With teaching efficiency η = 0.080 mT/m/A, 250 A produces 20.0 mT/m. During readout that increases frequency separation and moves farther through kx during a fixed time.
Follow what the current actually changes.
HELD FIXED: 220 mm FOV, 0.80 ms teaching interval, coil efficiency, RF bandwidth, and ideal sampling. Hardware current alone does not promise clinical resolution.
INTERACTIVE 3D · ONE 2D CARTESIAN REPETITION IN TRUE EVENT ORDER
Watch gradient-coil current become slice, phase, and readout encoding.
The translucent cylinder is the bore-side gradient former; its mint, coral, and violet paths are simplified physical Gx, Gy, and Gz winding cues. Moving bright beads show slowed conventional-current direction, not electrons. Inside, the oval and its true 3D voxel lattice are one digital object, the violet slab is the part given transverse signal, and the enlarged colored vectors show collective spin-packet phase—not individual protons. Scrub one event slowly, or use the B₀/Gx/Gy/Gz teaching buttons to isolate why field strength changes rotation rate and accumulated phase.
01 · ORIENT THE PRESCRIPTION
NO ENCODING GRADIENT YET
longitudinal magnetization · no transverse receive signal
Drag to orbit · pinch, wheel, or use + / − to zoom · tap an object to identify it
3D object voxel lattice
physical gradient winding cue
conventional-current marker
selected slab
linear ΔBz contour planes
RF transmit cue
phase-coded Mxy vectors
stored I + iQ sample
CONTROL IT WHILE THE 3D CAUSE STAYS VISIBLE
Event 01 · Orient · Axial plane
TEACHING FIELD ISOLATOR · PUSH ONE BUTTON
Not real scanner power controls
SEQUENCE MODE · Use the ordered events above. B₀ is still present continuously underneath every gradient and RF event.
Orientation defines the logical read, phase, and slice directions. No gradient current or RF excitation occurs yet.
01 · PRESCRIBED IMAGE PLANE
Axial example: logical read = physical X, logical phase = physical Y, and logical slice = physical Z. This simple one-to-one mapping changes when the plane rotates.
EVENT 01 / 07 · GEOMETRY PRESCRIBED
THE SCANNER DEFINES THREE LOGICAL DIRECTIONS
Read and phase lie inside the image plane. Slice points perpendicular to it. No RF pulse has tipped magnetization yet, so the transverse phase vectors are hidden and the receiver has nothing useful to read.
LOGICAL GRADIENT JOBOFForientation is geometry, not a gradient pulse
RF TRANSMITOFFno excitation yet
ADC / RECEIVER SAMPLINGCLOSEDno voltage sample is stored
CALCULATED K-SPACE ADDRESSkx 0 · ky 0an address from gradient area—not a body location
ACTIVE LOGICAL VECTOR → FIXED PHYSICAL COILS
No gradient command
Gx
Gy
Gz
The plane can be prescribed before any amplifier fires. When a logical gradient is requested, its direction cosines set the simultaneous Gx/Gy/Gz mixture.
LIVE PHYSICAL WINDING / CURRENT CUE
GX WINDING
GY WINDING
GZ WINDING
The colored paths are fixed physical conductor-pattern cues. No gradient amplifier is driving them in the orientation event. Exact amperes are not claimed because converting mT/m into A requires the scanner-specific coil efficiency and calibration.
WHY THE SLICE GRADIENT REVERSES AFTER RF
During a symmetric RF pulse, selected spins are tipped over a finite time while the slice gradient remains on. The gradient area after the pulse’s effective center leaves a position-dependent phase slope. A following opposite logical-slice area—about half the full selection plateau in this simplified case—cancels that slope. It does not undo excitation or select a second slice.
AUTOPLAY SPEED · DISPLAY ONLY
0.5× changes only this teaching animation’s wall-clock pace. It does not change gradient duration, slew rate, TE, TR, phase, k-space, or the clinical scan.
Current in shaped conductors creates a calibrated Bz slope
A gradient amplifier sends signed conventional current through one or more fixed winding sets. Reversing current reverses the slope; more current generally makes it steeper until hardware and safety limits intervene. mT/m measures the resulting field slope, not amperes. The moving beads show direction only, while exact current requires scanner-specific coil efficiency.
Limit which slab creates transverse signal
The logical slice gradient makes resonance frequency vary along the plane normal. RF excites its chosen frequency band, so only the matching slab is tipped. Without slice selection, signal from the larger excited volume would overlap in a 2D image.
Create an independent position-dependent angle pattern
A brief post-excitation gradient makes positions accumulate different transverse angles. The lobe then turns off, but the relative angles remain. Repeating a different signed area on later TRs supplies the independent ky measurements needed to separate positions along that direction.
Measure many kx addresses while ADC is open
The read gradient creates position-dependent frequency offsets and moves the calculated address through kx. The ADC stores a new whole-slice I + iQ coefficient at each dwell. Without it, one repetition would not efficiently sample a complete k-space row.
OPEN THE EXACT MEANING
MODEL BOUNDARY · The B₀/Gx/Gy/Gz buttons isolate concepts; they are not real scanner power buttons. An installed superconducting MRI main magnet is normally continuously energized, and changing its state is specialized engineering work—not a routine operator action. The isolated-gradient view assumes RF first prepared coherent transverse magnetization, removes the common B₀ carrier in a rotating-frame display, and exaggerates phase evolution. Gx, Gy, and Gz name the physical coordinate along which the small added longitudinal field changes; they do not make spins rotate about X, Y, or Z. The translucent former, loop/saddle paths, moving current beads, oval, voxel lattice, phase vectors, contour planes, RF rings, and seven-address readout are an explanatory digital model—not manufacturer winding CAD, electron drift speed, an electromagnetic field solution, anatomy, individual nuclei, a measured B field, or a clinical pulse-sequence prescription. Vector direction and cyclic mint→yellow→coral→violet color both encode wrapped phase from φ = 2π(kxx/FOV + kyy/FOV); color is not magnitude or tissue type. Real transverse-gradient windings are engineered distributed patterns, not these few saddle paths; exact current in amperes requires each scanner’s calibrated coil efficiency. The slow-motion percentage is an explanatory interpolation, not a quantitative RF envelope, ramp, slew-rate, TE, or TR clock. The “about half-area” slice-rephasing statement assumes a symmetric RF envelope on a flat selection gradient and refers to gradient-created phase from the effective RF center. Real pulse shapes, gradient ramps, , refocusing pulses, , , , and manufacturer implementation can change the exact waveform. In an axial plane the logical slice job can be physical Gz; in an oblique plane its reverse lobe reverses the required Gx/Gy/Gz mixture, not necessarily Gz alone.
READOUT VS PHASE · EVENT-BY-EVENT
After RF creates transverse signal, set one ky address. Then sample across kx.
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
Known phase-gradient area sets ky; read gradient plus ADC samples successive kx addresses.
The squares are calculated k-space addresses, not anatomy pixels. Here a “row” means seven data addresses with one shared ky label and changing kx labels. The sequence controller commands the gradient area; phase is the resulting angle pattern, not an agent that chooses.
current addresskx 0 · ky 0
stored this TR0 / 7 samples
current address
stored sample
same ky from one phase-gradient area
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
After RF has created transverse magnetization, the signed area under the phase gradient creates a known phase-versus-position ramp and therefore sets ky. Changing that commanded area on successive TRs sets different ky addresses; 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
physical components → yellow resultant
LOGICAL READOUT GRADIENT
+30.0 mT/m
Gx +20.6 · Gy +9.3 · Gz −19.7 mT/m
Drag to orbit · physical frame stays fixed · camera motion changes no gradient
COORDINATE TRANSFORM
Columns of R are the physical directions of logical readout, phase, and slice. The colored Gx, Gy, and Gz segments add head-to-tail inside the dashed projection box; the yellow vector is their sum. Those command vectors share one display scale, and R remains orthonormal.
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 | +0.687 | −0.508 | +0.520 |
| Gy | +0.310 | +0.852 | +0.423 |
| Gz | −0.657 | −0.129 | +0.742 |
PHYSICAL AMPLIFIER COMMANDS±40 mT/m per-axis model
GX
GY
GZ
DOUBLE OBLIQUE PLANE
The scanner synthesizes one +30.0 mT/m logical readout gradient by firing multiple fixed coils together: Gx +20.6, Gy +9.3, Gz −19.7 mT/m. Their vector sum points along the selected logical axis while readout, phase, and slice remain orthogonal.
logical vector magnitude30.0 mT/m
safe logical maximum58.2 mT/m
limiting physical channelGX · 52%
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
hue + direction = wrapped phase
equal-coordinate phase samples
white = coherent vector sum
GX MOMENT
kx = +170.3 m⁻¹
20 mm uniform column · 3.41 phase turns · coherent |ΣMxy| 8.9%
Drag to orbit · scroll to zoom · the sum bay faces the camera but is not a physical location
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
Build an image
from zero samples.
This is an interactive acquisition—not a prerecorded video and not an image being uncovered. Start with an intentionally empty reconstruction; each TR adds complex Fourier data from the whole slice, and the image is recalculated.
Δk · EXPLAIN EVERY WORD BEFORE USING THE FORMULA
Two numbered phase-pattern labels—and the exact numerical step between them.
A k-space “address” is not a place in the patient and has no width. It is a number attached to one measured complex sample. That number states how many gradient-created phase cycles occur per metre across the object for that sample.
It labels the phase pattern used when one whole-object I/Q sample was measured. It does not select a body point.
For example, 9.10 and 13.65 m⁻¹ are neighbors when no planned address lies between them.
13.65 − 9.10 = 4.55 m⁻¹. An address itself has no size; Δk is only the numerical separation.
One cycle is a full 360° change in the gradient-created phase pattern, not one RF carrier oscillation and not a proton orbit.
Number line + phase patterns + repeating image periodΔk 4.55 m⁻¹ · FOV 220 mm
Every spin’s complex contributionUNIFORM PHASE AT k = 0
Color is the phase of ρeff(x,y)e−i2πk·r. Here ρeff means the local echo signal after the sequence and receive-coil factors explained in the rho lab. 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) = ∫∫ ρeff(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 mT·ms/m
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 body transmit coil and slice-select gradient act together first: RF creates transverse magnetization only in the frequency-matched slab. No in-plane sample has been 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.
START · 0 OF 4096 COMPLEX SAMPLES
No acquired data means no MRI image.
The reconstruction canvas is intentionally empty. Press Run or drag the bottom progress slider; it controls the simulated acquisition and recalculates the image from only the samples acquired so far.
- 00No datano image yet
- 01Near k = 0broad shape + contrast
- 02Larger |k|edges + fine detail
- 03All rowscomplete ideal model
THE RULEA k-space sample does not paint one image pixel. Every acquired complex coefficient changes the calculation of every reconstructed pixel.
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 L = ln(1 + √(I² + Q²)); this compresses the preview only. Position in k-space is spatial frequency—not a location in the head.
Fourier reconstruction0% DATA
0 / 4096 COMPLEX SAMPLES
NO MRI IMAGE YET
Run the acquisition or drag the progress slider.
AR
Δx = 3.4 mm
0 samples: the canvas is deliberately empty because no image can be reconstructed yet.
QUANTITATIVE 3D SUSCEPTIBILITY · NONLOCAL FIELD · GRE PHASE · REGULARIZED INVERSION
A local material property
A local material property
creates a field beyond its boundary.
Magnetic susceptibility χ belongs to material, but the field perturbation measured by is a three-dimensional weighted sum of susceptibility everywhere. Follow a known synthetic source into its dipole field, GRE phase, and a deliberately imperfect reconstruction. The missing double cone in k-space is visible because it is the reason inversion needs a declared constraint.
01 · SUSCEPTIBILITY SOURCE
χ is assigned inside one finite 3D source.
The box is the entire periodic calculation array, not anatomy or a separate zero-padding buffer. B₀ direction sets the dipole-kernel orientation.
Interactive stage-specific 3D susceptibility field and QSM model
READ THE DEPTH · Each button replaces the central volume with one causal stage, so unrelated geometry cannot compete for space. Source geometry is finite; signed field glyphs extend outside its wire boundary. In the inversion stage, the double cone is a surface in 3D k-space where the forward kernel is zero—not an anatomical cone. Drag empty space to orbit, zoom, or tap an object for its exact role.
DRAWABLE SIGNAL PHANTOM · OBJECT → I + iQ K-SPACE → RECONSTRUCTION
Build the object.
Watch all of k-space respond.
Draw with a finger, add movable shapes, or start from a simple head or three “material buckets.” Every edit rebuilds the local complex MR-signal map, calculates its 2D Fourier transform, and reconstructs only the retained k-space support. Water, fat, muscle-like, brain-like, and custom choices are teaching signal models—not literal samples, diagnoses, or fixed tissue brightnesses.
Left–right and anterior–posterior are physical positions inside one slice. Slice thickness is held fixed. A true 3D acquisition would add many kz encodes; decorative depth would invent data this tool does not calculate.
For one isolated unchanged object, translation leaves ideal Fourier magnitude unchanged but adds a k-dependent phase ramp. That is why a magnitude-only k-space picture cannot tell the whole story.
The labels “water” and “fat” select declared example parameters. They do not paint an MRI brightness directly; the current sequence equation calculates the signal first.
Large smooth regions concentrate energy near k = 0. Small objects and sharp edges require faster spatial phase patterns farther from the center.
NEW · LINKED 3D VOLUME → SELECTED SLAB → EXACT 2D FOURIER INPUT
Move material through X, Y, and Z.
Move material through X, Y, and Z.
Watch the chosen slice change.
The translucent solids are synthetic material volumes. The violet slab is the part selected for this 2D acquisition. Its textured middle plane is copied from the exact complex source array used by the k-space calculation below—never a decorative preview. Move the selected volume along Z to make it enter, cross, or leave the slab.
Interactive three-dimensional material-volume and selected-slice model
HOW TO USE THE MODEL · Drag empty space to orbit the camera. Pinch, wheel, or use +/− to zoom. Use the six sliders to change physical geometry; camera movement never changes the slice or k-space. Tap a solid, plane, axis, tether, or intersection outline to identify it.
water / CSF-like
fat-like
muscle-like
brain-like
selected geometry / intersection
FAT/WATER PHASE → THREE-ECHO SEPARATION → KX PHASE RAMP → APPARENT X DISPLACEMENT
One frequency difference.
One frequency difference.
Three consequences you can separate.
Water-like and fat-like hydrogen nuclei can occupy one physical voxel while their molecular environments make them resonate at slightly different frequencies. That difference rotates their transverse complex vectors apart with echo time, lets multiple complex echoes encode water and fat separately, and can be mistaken for position during Cartesian readout. Every view below is driven by one quantitative state.
NEW QUANTITATIVE 3D MODEL · ONE SOLVED COMPLEX STATE
Follow echo time through phase evolution, one mixed voxel, and Dixon separation.
The model uses a declared single fat peak at −3.5 ppm. Blue water and yellow fat arrows are transverse complex vectors—not molecules orbiting through the body. Opposed and in-phase echoes provide the two distinct water/fat basis states; a later opposed echo deliberately makes the solve overdetermined so model mismatch can leave a residual. The separation either uses the exact known local B₀ offset or omits it so you can expose leakage rather than hide the assumption.
Interactive quantitative three-dimensional fat–water and Dixon model
NET MAGNITUDE THROUGH ONE FAT–WATER CYCLETE 1.12 ms · |S| 0.200
READ THE GEOMETRY · In the first station, depth is echo time: blue water, yellow fat, and mint net endpoints trace one relative-phase cycle, while the bright cross-section is the selected TE. Arrow angle is complex phase and length is signal amplitude; these paths are signal evolution, not molecules moving through tissue. The opposed and in-phase voxel blocks use one shared linear 0-to-W+F display scale. The mixed-voxel and readout stations then separate TE cancellation from frequency-to-position displacement. The final station shows three measured complex echoes, an explicit shared-field correction, and the recovered water/fat amplitudes. Drag empty space to orbit; tap an object for its exact meaning.
WHY THIS LINKED VIEW REMAINS 2D · The 3D model above uses depth where the math genuinely has it: a complex plane, a finite voxel, a spatial boundary, and multiple echo/output stations. This ruler compares one signed frequency axis with one reconstructed readout axis. Decorative depth here would invent a third quantity. The source/k-space/image canvases below show the full live Fourier consequence.
Color identifies the assigned teaching material; brightness shows its current calculated signal magnitude. The yellow outline is the selected movable geometry and is not part of the Fourier input.
Brightness is the logarithm of the size √(I² + Q²), not raw voltage and not anatomy. The yellow square marks coefficients retained for the reconstruction; dim outer data still exists in the full calculation.
Color is the angle of each I + iQ coefficient. Very weak coefficients are dark because their phase is unstable and visually unhelpful. Moving geometry changes this view even when its magnitude pattern stays the same.
The inverse transform uses retained I and Q, not the displayed log-magnitude picture. Reducing k extent removes measured fine spatial patterns, so edges spread and ring even though the display still contains 128 bins per side.
MOVE THE WHOLE DRAWN SLICE OR ONE SELECTED MATERIAL OBJECT WHILE DIFFERENT KY LINES ARE MEASURED
Hold it still—or move it during the scan.
The editor above defines exact complex water/fat/muscle-like signal contributions—not a hidden photograph. Choose whether the entire slice moves together or only the yellow-selected geometry moves while the rest stays still. Then ask where that moving contribution was when each Cartesian k-space row was measured. A changing position can make the rows disagree and produce blur, ripples, or repeated-looking structure.
SUDDEN Y MOVE · COMPLETE ACQUISITION
The slice changes position after some rows already describe its old position.
Early and late I + iQ rows now carry different translation phase. One stationary image cannot satisfy both sets, so the reconstruction distributes the mismatch as structured artifact.
ONE SHARED CLOCK · LINE NUMBER j → POSITION Δr(j) → KY ADDRESS128 of 128 rows · j 128 → ky index +64 (+266.67 m⁻¹)
HOW TO READ / TOUCH THIS · Drag or tap the left position graph to scrub acquisition time directly. The line is the chosen moving source’s physical X or Y displacement at each line time: either the whole slice or one selected geometry. Right: each short horizontal mark is one ky row; its color records the position that existed when that row’s I + iQ samples were acquired. The moving yellow cursor is a time/row link—not a proton, RF wave, or sensor traveling through the patient.
ACQUIRED COEFFICIENT · TAP PANEL 02 TO CHOOSE ANOTHER
One k-space cell, opened into its actual I and Q arrows.
The left clock shows the complex coefficient if the source stayed at its starting position. The right clock shows the coefficient assigned to this row’s measurement time. These are calculated values from the current drawn source—not generic decorative arrows.
Stationary background · zero in whole-slice mode
Moving contribution · selected object or whole source
Total S = I + iQ · what one receiver sample stores
Held-total reference · dashed on the right clock
SELECTED K-SPACE ADDRESSindex (+12, +16)kx +50.00 · ky +66.67 cycles/m
WHEN THIS ROW IS SCHEDULEDj 80 of 128 · acquiredY +12.0 mm at that line time
STATIONARY BACKGROUND VECTORI 0.000 · Q 0.000|S| 0.000 · angle 0.0°
MOVING CONTRIBUTION · HELD → MOVEDcalculatingsame length · translation rotates its angle
TOTAL RECEIVER COEFFICIENT · HELD → MEASUREDcalculatingI and Q are relative Fourier-sum signal units
WHAT MOTION CHANGES AT THIS ONE ADDRESScalculatingtotal length and angle comparison
WHY THIS VIEW IS 2D · A demodulated complex sample has exactly two stored number coordinates: . Horizontal and vertical here mean those two numbers—not physical left/right or front/back in the patient. Arrow length is ; arrow direction is . Tap or drag the k-space change picture below, or focus it and use arrow keys, to inspect another coefficient. Selecting a coefficient changes only this microscope; it does not alter the scan or reconstruction.
WHAT THESE NUMBERS ARE MEASURED IN · This teaching source declares a local complex amplitude of 1.000 as a relative reference; it does not claim one proton or one volt. The forward discrete Fourier transform adds the modeled source-cell contributions, so these I, Q, and |S| values are reported in . Doubling every material's would double every displayed arrow and coefficient. A value such as 12.810 therefore means 12.810 times this model's declared local reference contribution after summation—not 12.810 volts, protons, tesla, or a directly calibrated clinical scanner reading.
In Whole-slice mode every drawn contribution translates together. In Selected-geometry mode the yellow-selected material object moves over a stationary background; bright color mixing marks visual overlap, while the equation below—not screen-color blending—adds the actual I and Q arrays. Drag left/right or up/down to set the signed endpoint; arrow keys set ±1 mm and Shift + arrow sets ±5 mm. The model stays 2D because it does not invent rotation, deformation, or through-plane entry.
Whole-slice translation multiplies every coefficient by one unit-length phase factor, so its magnitude stays unchanged. If only one object moves, that object’s vector rotates while the stationary background vector does not; their total I + iQ length can then change. Color shows total angle change. Dark coral rows have not yet been acquired. Tap or drag anywhere in this square to open that exact cell in the I/Q microscope above.
This is calculated from the motion-modified complex rows—not a picture revealed underneath. A single constant displacement produces a clean shifted result; a line-dependent displacement can spread signal into repeats, blur, or ripples.
Mint is brighter and coral is darker than a held-still reconstruction made from the exact same partial row set. Matching row support isolates motion from the ordinary incompleteness of a half-finished scan.
BEFORE / AFTER CHANGE DETECTOR · COMPLEX SIGNAL, NOT A PHOTO OVERLAYMake one change.
Make one change.
See where its consequences go.
The three views subtract a pinned BEFORE state from the current state. This isolates what drawing, moving, changing material, changing TE/TR, or discarding outer k-space actually changes. It does not pretend that one k-space point belongs to one image pixel.
Press the button to copy the current complex source, full calculated k-space, retained support, and reconstructed magnitude into a temporary reference.
Drag or draw geometry, reassign water/fat-like material, change TE or TR, switch echo model, or reduce the retained k-space square.
Object-space change transforms into global ΔS across k-space. The final panel shows where retained-data reconstruction became brighter or darker.
REFERENCE READY · INITIAL THREE-BUCKET STATENo difference yet.
The current state and pinned BEFORE state are identical. Make one edit above; the difference views will update without hiding either full current k-space map.
BEFORE version 1 · spin echo · TE 20 ms · TR 2000 ms · extent 128²Brightness is |snow − sbefore|. Color is that complex change’s angle: a removed positive-real signal points 180° opposite an added one. Phase-only material changes therefore remain visible instead of being mistaken for “no change.”
Brightness is log(1 + |ΔI + iΔQ|) for the exact full-spectrum subtraction. Tap this square to inspect that address below. A local edit usually spreads across many coefficients; this is not a map of where the edit sits in the body.
Mint means the displayed reconstructed magnitude increased; coral means it decreased. This signed brightness difference is not the same operation as taking |ΔS|: inverse Fourier reconstruction is complex first, and magnitude is taken afterward.
CLICK TO EXPLAIN THE FOURIER RULE UNDER THIS SUBTRACTION
ΔS(k) = Snow(k) − Sbefore(k) = ℱ{snow(r) − sbefore(r)}
Fourier transformation is linear: subtracting the two source states first gives the same complex k-space change as transforming both and subtracting their I and Q values address by address.CHANGED SOURCE CELLS0 / 16,384cells whose local complex s changed
CHANGED FULL-K COEFFICIENTS0 / 16,384above numerical round-off threshold
CHANGED DISPLAY CELLS0 / 16,384signed reconstructed magnitude changed
SELECTED ADDRESSkx 0.00 · ky 0.00 m⁻¹same marker as both current k-space maps
ΔI · IN-PHASE CHANGE0.000current I minus BEFORE I · relative sum units
ΔQ · QUADRATURE CHANGE0.000current Q minus BEFORE Q · relative sum units
|ΔI + iΔQ|0.000length of the selected complex change vector
PARSEVAL ENERGY CHECKexact within round-offnormalized k-change energy equals source-change energy
No edit is present yet. After a local change, each k-space address compares the whole changed object with a different phase pattern, so many ΔI/ΔQ values can become nonzero.
A translated isolated object can keep the same current |S(k)| while its current phase changes. Complex subtraction still detects it because ΔS compares I and Q, not only two log-magnitude screenshots.
The current and BEFORE states retain the same k-space extent. If only extent changes, the source and full calculated ΔS stay zero while the reconstructed result changes because a different subset enters the inverse transform.
This is exact subtraction between two noise-free digital model states with perfect alignment. Clinical subtraction imaging also depends on motion registration, receiver noise, coil sensitivity, scaling, sequence timing, physiology, and validated reconstruction; color here is not a diagnosis.
ONE COEFFICIENT MICROSCOPE · TAP OR DRAG EITHER K-SPACE MAP
Where does one (I + iQ) number come from?
Tap or drag on either k-space square. The marker chooses one spatial-frequency address—not a place in the patient. Arrow keys move a focused marker by one address; Shift + arrow moves it by eight. The two pictures below then rebuild that coefficient from every nonzero source cell.
Brightness is that cell’s local signal magnitude. Color is its material phase plus the selected address’s position-dependent Fourier angle. Empty cells contribute exactly zero and remain dark.
Each short segment adds the total contribution from one source row; an empty row adds zero, so the path stays at the same point. The yellow arrow from the origin to the endpoint is the stored coefficient. A winding path means positive and negative I/Q parts partly cancel.
HIGHER-RESOLUTION FOURIER LAB · 64² → 128² → 256² COMPLEX SAMPLES
Keep the reconstruction.
Add detail, contrast, voxel SNR, and time.
This is a second, independent reconstruction built from a 256 × 256 digital teaching phantom. Matrix changes sampled detail and in-plane voxel size at a fixed 220 mm field of view. TE and TR change tissue signal. Slice thickness changes voxel volume. Receiver bandwidth changes admitted noise and readout time. NEX repeats and averages the complex measurements. Every control enters the calculated I/Q noise or Fourier reconstruction, so smaller pixels can gain detail while losing confidence.
At fixed FOV, more samples reach larger |k| and make smaller in-plane voxels. Smaller voxels contain less contributing material, so their signal competes less strongly with noise.
Longer echo time waits longer before k-space center is sampled. Short-T₂ tissues lose more coherent spin-echo signal than long-T₂ fluid.
Longer repetition time lets more longitudinal magnetization recover before the next RF pulse and lengthens this simplified scan-time estimate.
A thicker 2D slice combines more material per voxel and usually raises SNR, but can mix structures together through the slice.
Wider listening bandwidth accepts more frequency noise and shortens the ADC window. It also reduces read-direction off-resonance shift in the separate artifact lab.
Repeating the same encoding and averaging complex I/Q reduces random noise by √NEX, while scan time grows directly with NEX.
Every acquired I and Q number contains wanted voltage plus random electrical variation. The noise control declares the baseline amount before voxel, bandwidth, and averaging factors.
One I + iQ sample is one whole-object spatial pattern. The inverse Fourier transform combines all acquired coefficients—including their noise—into every reconstructed pixel.
ADVANCED COMPLETE · 65,536 OF 65,536 COMPLEX SAMPLES
All sampled spatial-frequency patterns now contribute at 256 × 256.
The full matrix contains four times as many samples along each axis—and sixteen times as many complex coefficients—as the 64 × 64 beginner reconstruction. Every stored coefficient contributes to every output pixel; no photograph is uncovered from underneath.
This first panel is a known digital input used to test the math—not an image secretly revealed during acquisition. Its gray levels use the spin-echo signal equation for five illustrative tissue classes.
Mint pixels are acquired I + iQ coefficients. Coral rows are planned but not yet acquired; the dark outer area lies beyond the selected matrix. Log magnitude makes weak coefficients visible but does not discard their stored phase.
nominal acquired interval 0.86 mm
At full 256 × 256 support the finest digital phantom targets are better separated. This is ideal sampling detail, not guaranteed clinical diagnostic resolution or a patient image.
SNR IMAGE MICROSCOPE · ONE FAIR A/B COMPARISON
How far does the wanted image stand above random receiver variation?
Signal-to-noise ratio, or SNR, is a comparison—not a substance inside the patient. The numerator is a declared repeatable wanted signal. The denominator is the standard deviation σ (“sigma”): the typical spread of random measurements around their mean. A ratio of 20 : 1 means the declared signal is twenty times that noise spread. It does not mean that 1 of every 20 pixels is noise.
In this lab, the numerator is the brightest current ideal tissue signal after TE/TR weighting. Real scanners must state how and where signal is measured.
σ is one standard deviation of reconstructed I or Q before magnitude. It has the same relative signal unit as the numerator, so their units cancel in the ratio.
Higher SNR usually makes weak boundaries and small intensity differences more believable. It does not by itself create contrast, prevent blur, or prove diagnostic quality.
Each noisy k-space I/Q coefficient enters every output pixel through the inverse Fourier transform. The grain is reconstructed from raw-data variation; it is not sprinkled over a finished image.
CLEAN MODEL
This is not the hidden source or a promise of a perfect scan. It uses the same selected matrix, TE, TR, and acquired ky rows as panel B, but mathematically removes only the added receiver-noise values.
ACTUAL LAB RESULT
This is an exact duplicate of the main reconstruction above. The stable random pattern prevents flicker while a slider changes its scale; a new real acquisition would contain a different noise realization.
DIFFERENCE × AUTO
The map subtracts the noise-free magnitude value from the noisy magnitude value at every display location. It stretches the largest absolute change to full color, so its brightness is not on the same scale as A or B. Read the numbers below for actual size.
DECLARED SIGNAL NUMERATORcalculatingbrightest ideal TE/TR-weighted tissue value
EXPECTED NOISE DENOMINATORcalculatingone σ in I or Q before magnitude
SNR TEACHING RATIOcalculatingsignal ÷ noise; complete support only
VISIBLE MAGNITUDE DIFFERENCE RMScalculatingroot-mean-square of B − A in this one fixed realization
LARGEST MAGNITUDE CHANGEcalculatingsets the difference map’s full-color display scale
MATCHED FOURIER SUPPORTcalculatingsame acquired addresses in clean and noisy panels
CLICKABLE SNR DEFINITION · CHANGE THE SIGNAL NUMERATOR IN A WORKED EXAMPLE
SNR = μsignal ÷ σnoise
μ (“mu”) means the declared average or repeatable signal measurement. σ (“sigma”) means the standard deviation—the typical random spread measured with the same signal scale. The ratio has no unit because relative-signal units divide by the same relative-signal units.WHY AVERAGING HELPS · REPEAT THE SAME MEASUREMENT
The target stays put. Random error lands above or below it.
The dots are a standardized statistical illustration, not readings secretly taken from one image pixel. Their vertical scatter uses the current calculated noise-to-signal ratio. Purple dots are the NEX repeats currently averaged; outlined dots show repeats that are available but not selected.
Which knob changed signal, noise, detail, or time?
MATRIXcalculatingdetail support + voxel volume + phase-line time
SLICE Δzcalculatingthrough-plane volume + partial-volume tradeoff
RECEIVER BWcalculatingnoise spectrum + ADC readout duration
NEXcalculatingrandom-noise confidence + repeated time
BASELINE NOISEcalculatingraw I/Q variation before Fourier reconstruction
TEcalculatingT₂ survival changes numerator, not pixel size
TRcalculatingT₁ recovery changes signal and planned time
PROGRESScalculatingmissing spatial patterns are not the same as noise
A clean image can still fail to separate two similar tissues.
SNR compares one declared signal with random spread. Contrast-to-noise asks whether the difference between two signals is large compared with that spread. If two tissues both measure near 0.70, excellent SNR can make both stable while their boundary remains faint.
“The SNR is 20” is incomplete without saying how it was measured.
Magnitude processing, multi-coil combination, parallel imaging, filtering, spatially varying coil sensitivity, and the chosen signal/noise regions change the statistic. This site therefore labels its value a bright-reference complex-channel teaching proxy, not a scanner-certified clinical SNR.
Low SNR can make available detail untrustworthy.
A matrix may support a small nominal pixel while grain hides a weak structure or makes an apparent edge unstable. More NEX can improve random-noise confidence but costs time; thicker voxels collect more signal but mix anatomy; narrower bandwidth reduces admitted noise but changes other readout tradeoffs.
CLICK A TERM
DETAIL MICROSCOPE · THE SAME PHYSICAL SQUARE ON BOTH IMAGES
Do not hunt for the extra detail.
Put it side by side.
Drag either mint square with a mouse or finger. Both squares stay locked to the same place. The left magnifier shows the declared digital source; the right magnifier shows what the currently acquired I + iQ data can reconstruct there. Arrow keys move a focused square; Shift + arrow moves it farther.
This is not scanner output. It is the answer key supplied to the Fourier calculation, magnified with hard square display bins so its tiny targets are visible.
SAME WINDOW · LIVE MEASUREMENT LEDGER
34.38 mm square near the lower-left teaching targets
PHYSICAL WINDOW WIDTH34.38 × 34.38 mmsame on source and result
ACQUIRED INTERVAL0.86 mm220 mm FOV ÷ matrix
INTERVALS ACROSS WINDOW40.0 cellsyellow grid on the result
SCREEN DISPLAY BIN0.86 mmfixed 256-bin output grid
CLICK FORMULA · CHANGE N IN THE LIVE EXAMPLE
nominal acquired interval = FOV ÷ N
220 mm ÷ 256 acquired samples = 0.86 mm. This number says how the sampled width is divided; it does not prove that a 0.86 mm object is visibly resolved.
The yellow lines mark the 256 × 256 acquisition intervals. At this setting one acquired interval and one displayed bin have the same width.
All 65,536 complex samples are present. Compare the circle gaps and line edges with the known source; similarity here follows the ideal sampled Fourier data, not a hidden photograph.
Display grid ≠ acquired detail ≠ measured sharpness
This teaching result always uses 256 display bins across 220 mm. With a 64 matrix, each acquired interval spans four display bins in each direction; the extra in-between gray values are calculated from the same 64 × 64 Fourier data. They make a smoother-looking screen, but cannot invent missing outer-k-space patterns.
A point response must stay narrow enough
An ideal mathematical point becomes the system’s point-spread function, or PSF. Finite k-space, filters, gradient errors, relaxation during readout, motion, off-resonance, and reconstruction can broaden that response. Two tiny objects blur together when their broadened responses overlap too much—even if the screen pixels are smaller.
Receiver noise is now visible and adjustable
The lab adds a declared teaching noise level to acquired I and Q before reconstruction, then scales its effect from voxel volume, receiver bandwidth, and NEX. Compare 64² with 256²: the larger matrix can carry finer patterns, but its smaller voxels have less signal capacity. Add averages to recover confidence and watch the planned time grow.
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
full commanded paths
traced to cursor
full delayed response
every path drawn
CURRENT ADC SAMPLE
k = (−145.5, −145.5, 0.0) m⁻¹
= (−32.00, −32.00, 0.00) Δk · t = 0.000 ms
Drag to orbit · scroll to zoom · tap an object to identify it
MULTI-SHOT · RECTILINEAR
One echo, one row.
A prephaser sets the negative-kx starting address. The readout gradient traverses one constant-ky line while ADC samples. On the following repetition, a different signed phase-gradient area creates a different retained phase ramp and sets the next ky value.
THE TRAJECTORY LAWG is a field slope in tesla per metre. Its direction chooses which k component changes; its magnitude sets the address-change rate in m⁻¹/s. Everything below is that one integral evaluated at ADC instants.
dkdt= γ̄ G(t)
Δk = 1/FOV4.545 m⁻¹
kmax = N/2FOV145.5 m⁻¹
Δx = FOV/N3.44 mm
dwell Δt = 1/(N·BW)15.63 µs
Gread = Δk/γ̄Δt6.83 mT/m
readout N·Δt1.000 ms
One dwell of 6.83 mT/m advances k by exactly one Δk = 4.545 m⁻¹, so 64 dwells span 2kmax = 290.9 m⁻¹ and resolve 3.44 mm.
160 mmΔk 4.545 m⁻¹420 mm
200 Hz/pxdwell 15.63 µs2400 Hz/px
MATRIX N
64 read samples · 64 phase steps
8Nyquist needs πN/2 = 101160
14.0 turns per interleaf24
4partition 7.50 mm32
40 mmΔkz 8.333 m⁻¹240 mm
Commanded gradient + delayed physical responseSHOT 01 / 64
solid · commanded
dashed · delayed physical
−2.00 dwell= 0.00 µs · pure time shift+2.00 dwell
−2.00 dwellno effect while Gy is off during ADC+2.00 dwell
iThe delay is a pure time shift of the physical gradient: Gactual(t) = Gcommand(t − τ), so kactual(t) = kcommand(t − τ). ADC timestamps never move. Unchecked, the reconstruction labels each sample with the coordinate the sequence commanded — which is what a scanner does when it has not measured its own trajectory.
Peak per-axis |G| 25.6 mT/m of 40 · peak per-axis slew 179 T/m/s of 180 · within declared hardware.
THE FULL PRESCRIBED DATASET · INDEPENDENT OF THE SAMPLE CURSOR
Where data landed · what one point becomes · what the object becomes.
Signal is generated where the gradients actually reached, then placed where reconstruction assumes. The mask and centred-point response describe the coordinates used for reconstruction; with a trajectory mismatch, that point response is not a universal convolution kernel.
support covered100.0%
largest gap0.00 Δk
effective R1.00×
FWHM read1.00 px
FWHM phase1.00 px
peak sidelobe−91 dB
RMS vs object0.0%
viewmagnitude
coordinates usedcommanded
Fully sampled Cartesian data reconstructs the numerical object exactly: the point spread function is one pixel wide and its sidelobes sit below −60 dB.
ADVANCED LAB / QUANTITATIVE SINGLE-SHOT EPI
One fast zigzag.
Three different image consequences.
The trajectory above shows where EPI samples. This lab calculates what those samples contain. Every ky line is acquired at a different time, so a declared Δf(x,y) map writes line-dependent complex phase; finite T₂* weights the echo train; and an odd/even phase mismatch creates an exact FOV/2 ghost. The 3D object, time-colored path, k-space plane, reconstruction, plots, and metrics all share one 64 × 64 complex calculation.
01object + Δf(x,y)physical slice and static off-resonance
→
02echo trainone ky line every ESP
→
03complex k-spacephase and envelope enter the samples
→
04inverse Fourier imagedistortion · blur · N/2 ghost
CAUSAL 3D VIEW · X = READ · Z = PHYSICAL +PHASE
Compare · separate displacement, blur, and ghost
The mint height field is the exact solved |IFFT{S}|. The violet contour comes from the undistorted input, field-scaled arrows show the local B₀ displacement predictor on the declared physical scale, and the coral surface isolates the analytic FOV/2 ghost component.
Quantitative causal single-shot EPI 3D model
horizontalread x / kx
upwardphase y / ky
depthsignal magnitude
coral layerisolated ghost
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
SAME ARRAYS · HONEST 2D READOUTS
not extra anatomy and not decorative screenshots
LINE TIME + DECLARED T₂* ENVELOPE
ky = 0 at t = 0 · alternating read direction already reordered
exp(−|t|/T₂*) line weight
odd echo
even echo
displayed acquisition state
−ky,maxline 64 / 64+ky,max
0.30 msphase BW 22.32 Hz/pixel1.50 ms
0 Hzmax shift +3.58 px160 Hz
20 msedge weight 0.666140 ms
−30°ghost/main 6.99%+30°
PHASE-ENCODE POLARITYchanges line-time sign, not the Δf map
THE TWO EXACT CHECKSfor this declared single-shot model
BWPE,pix=1 / (Ny · ESP)
Δypix=p · Δf / BWPE,pix
|ghost| / |main|=|tan(φ/2)| ?
PHASE BANDWIDTH / PIXEL22.32 Hz/pix1 / (64 × 0.70 ms)
max signed B₀ displacement+3.58 px+12.32 mm in a 220 mm FOV
echo-train readout44.10 ms63 line intervals around ky = 0
analytic FOV/2 ghost / main6.99%|tan(8.0° / 2)|
minimum edge-echo weight0.666declared symmetric T₂* envelope
magnitude RMS change0.1890relative to synthetic reference energy
largest |Δf| in solved map80 Hzanalytic teaching map · not measured B₀
complex samples4,09664 echoes × 64 ADC points
Geometric distortion
Each object point accumulates a different phase before its ky line is sampled. A uniform Δf becomes a pure cyclic phase-direction shift; a spatially varying field produces local compression, stretching, and pile-up.
Phase-direction blur
Outer ky lines are measured farther from the echo center and receive less weight in this declared symmetric envelope. That apodizes ky and broadens the phase-direction point response.
Nyquist N/2 ghost
Alternating-line phase modulation splits the object into a main coefficient and a copy shifted by exactly half the phase FOV. The printed amplitude ratio is independent of the phantom.
MODEL BOUNDARYQuantitative complex teaching signal—not a clinical EPI simulator.
This is one noiseless, fully sampled, single-shot Cartesian gradient-echo EPI readout. It assumes an ideal instantaneous excitation; linear ky order; ideal ramps and phase blips; perfect correction of alternating read direction into a Cartesian grid; one synthetic real-valued object; one static analytic Δf map; a symmetric exp(−|t|/T₂*) line envelope; and one spatially constant odd/even phase mismatch. It omits chemical species, through-voxel dephasing, nonlinear gradients, eddy-current spatial terms, readout delay, concomitant fields, motion, flow, diffusion weighting, spin-echo refocusing, partial Fourier, parallel imaging, SMS, multishot segmentation, coil sensitivities, noise, field-map correction, and patient prediction. Cyclic wrap is intrinsic to the finite discrete Fourier model.
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.