Interactive MRI physics00 — 09
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, quantitative single-shot EPI artifacts, and an image reconstructed line by line.
↔ Drag to orbit · pinch or wheel to zoom · tap a label or object to explain it
The complete simplified patient—head through feet—rests on the table. Table motion changes which anatomy is near isocenter; it does not move through k-space and does not electronically select a slice.
SOLID physical hardware GLOW / ARROWS invisible field cue PLANE / CLOCKS / BOXES selected region or calculated data cue
At 0%, the coil layers sit where they are installed: concentrically around the bore.
S(k) = ∫ ρeff(r) e−i2π k·r dr
In plain words: treat the excited body as many tiny, equal-size regions. Each region contributes some receive-coil voltage. Gradients give those contributions a calculable phase—where they sit around one repeating cycle. At address k, the receiver adds every region and stores one pair of signed numbers, I and Q, called S(k). That pair describes the whole excited region; it is not one image pixel. Tap the equation for every symbol, unit, zero case, and a live numerical example.SEARCHABLE MRI DICTIONARY · 143 / 143 TERMS HAVE THE COMPLETE BEGINNER EXPLANATION
Every result starts without assumed MRI knowledge, then lets you go deeper into a labeled visual, scanner behavior, clinical consequence, numerical example, and connected terms.
BEGINNER-FIRST LEARNING · CHOOSE HOW FAR TO ZOOM
The same three-level control appears as a short entry card at every major topic. It changes the explanation layer—not the MRI physics or any simulated parameter.
START SIMPLEEach chapter now begins with one plain-language mental model. Use “Zoom in” on any card when you are ready.
THE WHOLE MRI EXAM · NOT ONLY RF, GRADIENTS, OR K-SPACE
An MRI scanner is a coordinated system: safety screening, a continuously energized main magnet, a moving patient table, shim and gradient coils, RF transmit/receive, monitoring, sequence control, digitization, corrections, reconstruction, storage, and display.
↔ Drag to orbit · pinch, wheel, or use +/− to zoom · tap an object or label for its full explanation
WHAT THIS VIEW CURRENTLY MEANSThe patient, table, local coil, and selected landmark are one moving setup. The magnet housing and yellow isocenter marker stay fixed.
SOLID SHAPES physical patient or hardwareGLOW / ARROWS / PLANE field, projected positioning-laser light, or selected-region cue—read its labelMOVING SQUARES stored-data flow—not particles traveling through the patient
Technologists screen the patient and every entering object, position anatomy and coils, choose a landmark, provide hearing protection and an alarm device, and use MR-conditional monitoring or gating when required.
The main magnet supplies B₀ continuously. The cryostat thermally supports the superconducting system; site infrastructure, shielding, and emergency procedures manage fields and rare abnormal events.
Fast survey images establish geometry. Field mapping and shim adjustments improve uniformity; reference scans can estimate coil sensitivity, center frequency, transmit behavior, and reconstruction corrections.
The controller schedules RF synthesizers/amplifiers, gradient amplifiers, transmit/receive protection, receiver bandwidth, and ADC sampling. Timing—not a single component—defines the acquisition.
Raw multi-channel I/Q data may be corrected for sampling and hardware behavior, calibrated, reconstructed, combined across coils, filtered, scaled, and packaged with geometry and protocol metadata.
The console and downstream systems show reconstructed images with orientation, scaling, annotations, and metadata. A displayed gray level is the end of a long weighted chain—not a direct photograph or universal tissue unit.
ρ EXPLAINED WITHOUT HIDDEN FACTORS · SOURCE → ECHO → COIL → IMAGE
Textbooks often reuse ρ for both proton density and the already-weighted signal distribution. This lab keeps those quantities separate, names every factor in the teaching product, and states the reference behind every “relative” number.
Most clinical proton MRI signal comes from hydrogen nuclei in mobile water and fat. Hydrogen locked in very rigid material can lose transverse coherence before the receiver can sample it; air has very few hydrogen nuclei. “Visible” never means visible light.
Here, an equal-size reference voxel with ρH = 1.00, complete recovery, no T₂* loss, a 90° excitation, and receive sensitivity 1.00 has ρeff = 1.00. Scanner gain and display windowing can rescale all image numbers, so there is no universal brightness or volt value.
One k-space measurement is not a pixel. After all complex samples are reconstructed, the ideal local complex value is proportional to ρeff. A magnitude image displays its size, usually after coil combination, scaling, filtering, and window/level.
Local mobile-¹H source, one T₁ recovery term, one T₂* survival term, ideal sin α excitation, and one relative receive-sensitivity number.
Flow/inflow, diffusion gradients, magnetization transfer, chemical exchange, contrast agents, fat/water phase, B₀ and B₁ nonuniformity, motion, multi-echo history, receive-channel combination, filters, noise, gain, and display window/level. These are named here rather than hidden inside “other factors,” but they are deliberately held out of this five-factor lab.
Brighter does not automatically mean “more protons.” First ask whether sequence timing, excitation, coil position, pathology, reconstruction, or display scaling also changed.
COORDINATE PRIMER · FIX THE NAMES BEFORE ENCODING
An axial example often pairs read with Gx, phase with Gy, and slice with Gz—but that pairing is not a law. Rotate the prescribed image plane and the scanner synthesizes each logical job by firing two or three physical gradient coils together.
WHY CARE · Confusing these naming systems can make an oblique image plane, artifact direction, or amplifier limit look wrong.[Gx Gy Gz]ᵀ = [+1.000 0.000 0.000]ᵀ × +40.0 mT/m
Each coefficient is a unitless direction cosine. Superscript ᵀ means “write this row as a column vector” (transpose)—it does not mean tesla here. Multiplying by +40.0 mT/m gives a real physical-coil command.00 / WHERE THE MAIN FIELD COMES FROM
A power supply first pushes electric current through many turns of superconducting wire. Those turns act together like one long solenoid: their magnetic fields add through the bore and return outside the magnet. After ramp-up, a closed superconducting path can keep that current circulating in persistent mode.
Drag to orbit · pinch, wheel, or use + / ↺ / − to zoom · tap any object · field lines are a map, not physical tubes
A controlled DC power supply applies voltage so current rises in the cold winding. “Current” means net electric charge crossing a wire section each second, measured in amperes. Yellow arrows follow the conductor in the conventional positive-charge direction; the wire does not move, and electrons do not jump between turns.
Each loop makes a magnetic field. Inside a long coil, neighbouring loop fields point mainly the same way and add; more amperes or more turns per metre gives a larger central field.
Once the target current is reached, an internal superconducting connection completes a very-low-resistance loop. The ramp supply can be disconnected while the established current continues.
The stable main field exists between scans. RF and gradient pulses switch during imaging; the superconducting main-magnet current normally does not pulse on and off for each patient.
Bcenter ≈ μ0nI = μ0(N/L)I
4π × 10⁻⁷ T·m/A × 4,000 turns/m × 600 A ≈ 3.02 TAt 600 A and the selected winding density, the ideal long-solenoid estimate is 3.02 T. Raising current strengthens B₀ in direct proportion. Yellow arrowheads stay on the helical wire and mark conventional-current direction; they are not electrons and their display speed is not a measurement.
The winding length stays 1.50 m inside the 1.86 m package. The 3D winding shows 52 spaced turns to represent 6,000 effective turns. Move the slider: higher n visibly adds turns and closes their gaps; the drawing remains a compressed sample rather than literal manufacturer geometry.
This changes only how many direction-map curves the 3D lesson draws. It does not change current, B₀, tesla, gauss, fringe field, stored energy, or any clinical image.
At the center, increasing I or n increases the field almost linearly in this long-solenoid estimate. Clinically, a different B₀ changes the ¹H carrier frequency and many scanner/tissue behaviors; it is a magnet design and site platform, not a routine sequence knob.
Charge carriers have a tiny average drift inside the continuous wire. In a metal, negative electrons drift opposite the defined conventional-current arrow. They do not leap across the space between neighbouring windings; the wire, cryostat, and patient do not circulate.
Drawing lines connect the field direction at many points. They close into loops and crowd where the drawing represents stronger field. They are not strings, rays, current paths, or proton tracks.
Both measure magnetic flux density: 1 tesla = 10,000 gauss. Therefore 3 T = 30,000 G. Clinical MRI names B₀ in tesla; gauss is common in fringe-field discussions.
Ordinary resistance would turn sustained high current into continuous heat. Below its critical conditions, the magnet conductor can support persistent current with extremely small decay; cooling and protection remain essential.
3D FIELD BUILD-UP · MOVING CHARGE → CURRENT → LOOP → SOLENOID
Blue moving markers now represent groups of negative charge crossing a counting plane; increasing current shows more marker crossings and a stronger calculated field. They are not a literal census of electrons in the wire. Because electrons are negative, their average drift is opposite the yellow conventional-current arrow. Build from a straight segment to one loop, then watch the field shape lengthen as aligned turns are added.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object · animation pace and line count are teaching choices
The side view is kept two-dimensional because “crosses this plane during one second” is a measured rate. The rotatable model beside it shows the same conductor and circular field in 3D.
I = ΔQ / Δt = Ne|e| / Δt
Ampere measures charge crossing per second. Stationary charge gives no steady current-created B; current through the chosen geometry sets the calculated field.At 4.00 A, 4.00 coulombs of charge cross the chosen section each second—equivalent to about 2.50 × 10¹⁹ elementary electron charges per second. The conductor already contains vastly more carriers; this control changes net flow rate, not the mere existence of electrons.
More charge crossing per second strengthens every B contribution in direct proportion. The number of bright blue packets is a compressed rate display; their visible speed is deliberately not a microscopic drift-speed measurement.
Moving this control opens the many-loop stage. More equally oriented loops add more same-direction axial field at the center; it does not mean the 3D drawing represents a real MRI winding count.
QUANTITATIVE AXIAL CROSS-SECTION · SAME CURRENT AND WINDING CONTROLS
The 3D guides above are useful direction guides, but their spacing is not a measurement. This map calculates a field value at a grid of positions through the coil. Drag the red probe directly: moving it changes only the observation coordinate, never the magnet.
FINGER / POINTER: drag anywhere in the plot · KEYBOARD: focus the plot, then use arrow keys · the ordinary z and r sliders remain equivalent accessible controls
NEW QUANTITATIVE 3D LAB · MAPPED Δf → LOW-ORDER CORRECTION → RESIDUAL
A field map describes where proton resonance is above or below a chosen reference. Fit the standard first- and second-order spatial shapes over one spherical region, then compare the original map, the added correction, and what the fit cannot remove.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap a field volume · needles plot signed Δf along the B₀ axis; they are not tissue displacement
All three 3D volumes show the same coordinates and the same fixed hertz scale. Change only the fitted basis, region size, or local feature, then ask which spatial patterns the residual keeps.
The fit uses 257 equally weighted Cartesian teaching samples inside a 22 cm sphere. Changing the region changes which physical positions contribute; it does not resize a patient or claim an optimal clinical shim volume.
This synthetic local feature is not anatomy or a measured susceptibility field. Raising it adds spatial detail that eight broad harmonic shapes cannot reproduce exactly.
Δfres(r) = Δfmap(r) + Σ cj Φj(r)
Δf is proton off-resonance in hertz and Δf = γ̄ΔBz. Φj are explicit dimensionless X, Y, Z, Z², ZX, ZY, X²−Y², and 2XY shapes normalized at a 12 cm reference radius. cj are correction-field coefficients in hertz—not shim-coil current.Signs describe the field added by this fit in the stated coordinate convention. A real scanner maps calibrated coil currents and constraints to fields; this lesson intentionally stops at field coefficients.
The least-squares correction lowers RMS off-resonance from 19.31 Hz to 4.77 Hz over the same samples. The residual is not a failure of arithmetic: the local Gaussian shape is deliberately absent from the allowed basis.
Use the numbered journey in order. Every solid, sphere, arrow, field sheet, and ring can be tapped for its exact meaning. The jumps between body, tissue, atom, and nucleus are separate scale models—nothing literally swells inside the patient.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object to identify it · model sizes are not one continuous scale
The scanner surrounds the patient with B₀, gradient, and RF hardware. The yellow target marks one tiny tissue region inside the head. The next buttons magnify that region conceptually; the tissue does not move out of the body.
Patient, table, scanner bore, tissue water, fat, and many other materials.
A glowing target and connector make one future region easy to follow.
MRI never receives a ready-made picture from one location; it must excite and encode signals distributed through the body.
At 3.0 T, ¹H phase precesses at 127.73 million cycles per second. The 3D arrow is slowed enormously; its animation speed is only a trend cue.
The selected stage assigns this field slope to physical Z for slice selection, Y for phase encoding, or X for readout. Gradient current changes precession rate with position; it does not push atoms along that axis.
At 20.0 mT/m, a 2.0 kHz RF band selects an ideal 2.35 mm slab centered at z = 0 mm. Moving RF center frequency moves the slab; it does not move atoms.
f0 = γ̄B0
Δz = BWRF / (γ̄|Gslice|)
dk/dt = γ̄G(t)
vcoil(t) = −dΦ(t)/dt
01 / FROM FIELD TO POSITION
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.
Bz(x) = B0 + Gxx
Δf = γ̄Gxx → Δφ = 2πΔfΔt
kx = γ̄GxΔt → Δφ(x) = 2πkxx
PHASE ADDRESSAt x = +8.0 cm, Δf = +85.15 kHz accumulates +0.341 cycles (+122.6°) in 4.00 µs.
f0 = γ̄B0
42.58 MHz/T × 3.0 T = 127.73 MHzAt 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.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.
BEGINNER GAME / THREE HIDDEN WATER BUCKETS
Three water buckets are hidden inside the bore. From the next room, choose a gradient vector, tune a narrow RF frequency offset, send a short probe, and listen for the echo. Find each bucket’s X, Y, and Z coordinate—then see why a frequency under one gradient selects a plane rather than a unique point.
Drag to orbit · solid ring = bore · blue plane = same-frequency locations · buckets reveal only after detection
With Gx on, frequency tells X but says nothing about Y or Z. Rotate the gradient job and repeat. Real MRI performs a much richer, systematic version with many phase and frequency encodings.
With G = 0, all three identical-water resonances sit near the same f₀. Frequency can say “water signal exists,” but not where it came from.
Δf = γ̄G·r. A measured frequency gives position along G, while an entire perpendicular plane has that same frequency.
Real imaging collects many known phase/frequency patterns. Reconstruction solves the mixed whole-object voltages for a spatial image; an operator does not hunt voxel by voxel.
02 / RF TRANSMIT, RELAXATION & RECEIVE
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
WHY THIS PLOT IS 2DLeft–right is elapsed time. Up–down is one field component or received-voltage magnitude. Neither axis is a patient direction, so a third spatial axis would be invented and misleading.
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.
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.
Before demodulation there is one rapidly alternating voltage. The receiver compares it with two synchronized reference waves; low-pass filtering leaves two slower signed voltages that preserve the signal’s magnitude and phase.
WHY THIS EXPLANATION USES 2DThe wave rows plot voltage versus time; the round I/Q view plots two mathematical receiver-reference components. I and Q are not scanner X and Y. Use “See this split in the 3D receiver” to locate the calculation in the hardware chain.
Drag to orbit · violet vectors are B₁⁺ · yellow arrow is net M · coral loops are local elements
α = γ ∫ B₁⁺(t) dt
vRx(t) ∝ −dΦM/dt
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.
The local sensitivity number uses the exact on-axis field falloff of an ideal 50 mm-radius circular loop, C(d)/C(3 cm) = [(1 + (d/5 cm)²)/(1 + (3/5)²)]−3/2. It is a normalized geometry lesson, not a scanner specification or a full spatial array map. The 3D ruler declares 1 world unit = 100 mm. The shared B₀ control retunes the displayed ¹H carrier; it does not rescale geometry, SNR, relaxation, or SAR. Actual values require coil- and patient-specific electromagnetic measurements.
LIVE 3D LOCAL-COIL RECEIVE JOURNEY · USES THE MODEL ABOVE
The body coil finishes the high-power B₁⁺ pulse. The transmitter is isolated, the receive-only local loops come out of their protected detuned state, and their low-noise paths are connected.
The programmed pulse ends and the transmit/receive switching network changes electrical connections after a short recovery interval.
Excited transverse magnetization exists in the patient. The local array is now electrically able to respond to it.
Switch timing is measured in seconds, commonly microseconds (µs). No image value has been measured yet.
The receive signal is tiny compared with transmit power. Isolation protects the preamplifier and prevents transmitter leakage from hiding early signal.
The RF pulse did not travel into the local coil as an image. It prepared magnetization; reception is a later electromagnetic induction event.
GdB = 20 log10(Vout/Vin) ↔ Vout = Vin·10GdB/20
Gain scales signal and existing input noise; it does not create localization or improve input SNR.CURRENT SETTINGSAt 40 dB, voltage is multiplied by 100 before digitization. A 128 kHz sample rate gives 7.81 µs between samples in this one-sample-per-dwell model.
Moving transverse magnetization distributed over position r.
One continuous signed RF voltage per element.
Two slower signed components per element.
Digital complex samples assigned to gradient-created addresses.
Aligned coil images become one relative magnitude image; tap for limits.
RF power at the Larmor frequency rotates magnetization. Amplitude and pulse duration set flip angle; spatial B₁⁺ variation makes flip angle nonuniform.
Precessing transverse magnetization induces a tiny voltage. Close local elements couple strongly to nearby anatomy and admit less distant noise.
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.
ADVANCED 3D MODEL · FINITE-DURATION RF · FULL BLOCH ROTATION PRODUCT
A shaped RF pulse and a slice-select gradient act at the same time. Position becomes resonance offset; the pulse spectrum tips a band of those positions; a signed gradient moment then removes most of the phase ramp left by a symmetric pulse. Every vector, curve, and number below comes from the same finite-step Bloch calculation.
The gradient does not cut tissue. It makes resonance frequency vary continuously along the selected axis.
The sinc spectrum predicts the low-flip response; the full Bloch product remains valid when the tip is large.
Magnitude can look acceptable while residual phase still matters for coherent downstream sequences.
The selected band is transverse while off-band magnetization remains near +Mz. Vector direction retains the complex phase that a magnitude-only slice drawing hides.
The quantitative RF waveform and slice-profile plots remain available if 3D rendering is unavailable.
Arrow base = physical position z. Arrow direction = magnetization M.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Mn+1(z) = R[−γ|Beff,n(z)|Δt] Mn(z)
Δf(z) = γ̄Gz + Δf₀ − ΔfRF · BW = TBW / T · Δznom = BW / (γ̄|G|) · Arephase = −GT/2Moving the RF carrier shifts the selected band; increasing |G| spreads the same RF bandwidth over a thinner physical slice.
Apodization reduces sidelobes but broadens the transition. The nominal thickness is not the exact FWHM of every pulse.
The dashed small-tip curve is center-scaled only to compare shape; the RMS metric uses its raw complex amplitude. Its error grows because Mz no longer remains approximately one.
A symmetric pulse accumulates approximately half the selection-gradient area after its effective excitation time; a negative half moment unwinds it.
This is a one-dimensional, single-band, nonadiabatic windowed-sinc pulse. It uses 256 midpoint RF samples and 181 spatial positions with exact norm-preserving rotations. Relaxation, flow, diffusion, chemical shift, multiband excitation, Shinnar–Le Roux pulse design, gradient ramps, concomitant fields, eddy currents, RF amplifier limits, and measured B₀/B₁ maps are not simulated. The post-RF rephaser is represented by its signed moment, so its duration and off-resonance evolution are omitted.
03 / SIGNAL, CONTRAST & MOTION
Start with the integrated virtual exam: choose receive hardware, acquire localizers, prescribe a slice, tune T₁/PD/T₂/FLAIR/DWI/GRE controls, and reconstruct modeled k-space. TE, TR, flip angle, and RF history shape the available signal. BOLD fMRI turns a delayed vascular oxygenation consequence into T₂*-weighted time-series change; ASL magnetically labels arterial blood to estimate perfusion; TOF turns replacement of saturated blood into bright-vessel magnitude; phase contrast turns coherent motion into signed velocity; diffusion sensitization probes microscopic displacement. These are distinct mechanisms—not interchangeable maps.
Localize and prescribe first; then use the same quantitative sequence controls to acquire T₁, proton-density, T₂, FLAIR, DWI, or GRE contrast.
BOLD fMRI is delayed and indirect: a vascular state changes susceptibility-related T₂* weighting, then EPI samples volumes at TR.
ASL couples label duration, ATT, PLD, T₁ survival, control−label difference, and model-based CBF.
TOF couples slab-normal transport to spoiled-GRE history, source magnitude, directional preparation, and MIP.
Phase contrast subtracts two complex measurements, decodes one component, and exposes VENC aliasing and area integration.
Microscopic displacement between lobes leaves a phase distribution; a wider distribution reduces the coherent echo.
Repeat directions to estimate anisotropy; one direction and one b-value are not a tensor fit or tract map.
3D ECHO FORMATION · ONE VOXEL IN THE ROTATING FRAME
Follow the same transverse ensemble through excitation, dephasing, a refocusing action, rephasing, the echo, and renewed dephasing. Switch mechanisms without changing the tissue or field spread.
A 90° excitation is represented as complete. Every visible spin packet begins along +X′ in a frame rotating near the carrier; the fast MHz carrier itself is removed.
The rotating-frame echo model needs WebGL. The synchronized timeline, equations, metrics, and controls remain available.
The pulse or gradient command is a cause. The phase fan converges later. Vertical screen position is a teaching state—not another scanner coordinate.
This teaching scrubber changes the instant being inspected. It does not change the prescribed TE, tissue, or field spread.
Longer TE gives both irreversible T₂ relaxation and unrefocused static offsets more time to reduce the gradient echo. The ideal spin echo removes the static-offset term only at its echo center.
A wider static frequency distribution fans phase faster. The 180° pulse reverses its ordering; a gradient-polarity reversal does not.
SSE(TE) / S₀ = e−TE/T₂
The 180° pulse removes the static-offset phase factor at TE; it does not restore the T₂ envelope.1/T₂* = 1/T₂ + 1/T₂′ · T₂′ = 1/(πΔfFWHM)
1/T₂* = 1/80 + 1/79.6 ms → T₂* = 39.9 msWHY THE TWO ECHOES DIFFER ·At TE 80 ms, intrinsic T₂ leaves 36.8%. A 4.0 Hz static spread reduces the gradient echo to 13.5%, while an ideal 180° pulse removes that static factor at the spin-echo center.
Fast packets move ahead and slow packets fall behind. An ideal 180° pulse swaps that phase ordering, so continued evolution brings them together at TE.
The visible packet arrows shorten as the modeled coherent contribution survives. The refocusing pulse changes future phase evolution; it does not add lost coherence or reset time.
A gradient echo cancels the deliberate gradient moment at TE but leaves static B₀ offsets accumulating, so its ideal envelope follows T₂*.
LIVE SEQUENCE · T₁ SPIN ECHOTR 500 · TE 15 ms
HEAD TEACHING STARTBrain, ventricles, and posterior fossa. Acquire the synthetic localizers, prescribe, then run the virtual scan.
Receive proxy = mean analytic sensitivity over one central axial elliptical mask. Nominal channel count is a hardware label, not a multiplier. A posterior partner changes the modeled sensitivity term. This does not predict clinical SNR or usable acceleration.
Acquire the survey first. Localizers reveal internal synthetic anatomy relative to scanner coordinates; the external landmark alone does not prescribe a slice.
ONE 3D MODEL, THREE REPARAMETERISATIONSAll three surveys and every prescribed slice sample the same code-native 3D tissue field in patient millimetres: R/L positive toward L, A/P positive toward P, H/F positive toward F, measured from each region's modelled isocentre. Axial reads (R/L, A/P) at a fixed H/F, sagittal reads (A/P, H/F) at a fixed R/L, coronal reads (R/L, H/F) at a fixed A/P. R→L runs left to right; axial A→P and sagittal/coronal H→F run top to bottom. Positive slice position points toward L, P, or F for sagittal, coronal, or axial prescriptions, and its limit is that region's own half-extent along the selected normal. Each survey is a soft-maximum (power-mean, p = 3) projection of tissue proton density ρ through a 15 mm slab. Every displayed scout voxel uses 2 × 2 in-plane midpoint samples and seven through-slab samples—28 samples total. It is a declared projection, not a simulated pulse sequence, so it carries no TR, TE, TI, flip angle, relaxation, diffusion, coil weighting, or noise. Survey brightness uses a declared display window (ρ 0.56–1.02, γ 0.90); the prescription overlay is drawn in true millimetres at each survey's own scale.
AT THE TEACHING STARTChange slice thickness, FOV, matrix, or NEX. This readout separates resolution, SNR, coverage, and time so “brighter” is not mistaken for “better.”
NO PERIODIC MOTIONThe k-space rows remain mutually consistent. Choose Cardiac or Respiratory, then compare gating on and off in the reconstructed image.
CONTROLS CHANGEDThis result preserves the acquired state. Run the scan again to apply the new sequence or prescription.
WHAT IS QUANTITATIVETR, TE, TI, flip angle, b/ADC signal equations; FOV/matrix pixel size; voxel volume; slice-stack coverage; one-line-per-TR base time; SNR scaling with voxel volume and √NEX while other factors are held fixed; the exact Fourier translation phase applied to each modeled kᵧ line; the Fourier transform of the displayed center-slice synthetic source; and the plane geometry itself — every survey, overlay, embedded 3D slice, and acquired image is the same 3D field sampled through one millimetre coordinate map.
WHAT IS ILLUSTRATIVEThe ten study buttons are starting states, not protocols. Analytic coil sensitivity, deterministic complex k-space noise, the cardiac/respiratory waveform, trigger acceptance and wall-clock proxy, whole-slice rigid displacement, procedural anatomy, derived display shells, localizer projection/window, partial-volume quadrature, preview resolution, per-region scene scale, nominal channel labels, and compressed playback are teaching models. Slice thickness can improve this model's SNR and increase through-plane averaging; fixed display windowing means it does not guarantee a brighter image.
WHAT IS OMITTEDThe other prescribed slices and repeated NEX lines are not separately reconstructed. The anatomy is hand-declared—not segmented, measured, atlas-derived, registered, diagnostic, or population-normal. Motion is one rigid phase-axis translation, not organ deformation, flow, temporal cine reconstruction, navigator design, arrhythmia classification, prospective controller behavior, or a patient-specific quiet window. Also omitted: measured B₀/B₁/coil maps, channel noise covariance, bandwidth changes, parallel reconstruction, safety screening/limits, calibration, and vendor protocol logic. Nothing here recommends a clinical protocol.
S = ρH(1 − e−TR/T₁)e−TE/T₂
The 180° pulse refocuses static dephasing, so ideal echo amplitude follows T₂ rather than T₂*.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.
MODEL BOUNDARY · SYNTHETIC ANATOMY, NOT A PATIENT OR PROTOCOL RECOMMENDATION. Every pixel belongs to one ideal nonexchanging tissue compartment with illustrative values near 3 T. Spin echo uses ρ(1−e−TR/T₁)e−TE/T₂; FLAIR adds one ideal inversion-prepared repeated cycle; DWI multiplies the same spin-echo baseline by e−b·ADC; GRE uses the spoiled steady state and T₂*. The grayscale uses one fixed relative-signal display mapping, so numeric comparisons remain available without per-preset auto-windowing. Real anatomy and contrast also depend on field strength, sequence family, echo trains and k-space ordering, inversion and fat-suppression profiles, diffusion direction, T₂ shine-through, perfusion, flow, exchange, coils, noise, artifacts, reconstruction, pathology, safety limits, and vendor timing constraints.
ADVANCED 3D MODEL · T₁ SATURATION ACROSS REPETITIONS
Follow one ideally spoiled gradient-echo experiment pulse by pulse. The vector shortens inside a reference sphere, the history plot records the exact recurrence, and the Ernst protractor shows which flip angle maximizes ideal steady-state signal for the selected tissue and TR.
The experiment begins at equilibrium: Mz⁻ = 1 before the first RF pulse. The translucent sphere is only the M₀ reference boundary.
The quantitative history plot remains available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Important: these curves show magnetization state, not TE-weighted signal. TE scales the echo readout after each pulse; it does not change convergence toward the spoiled-GRE steady state.
Mn+1− = 1 − (1 − Mn− cosα)E₁
E₁ = e−TR/T₁ · fixed point Mss− = (1 − E₁)/(1 − E₁ cosα) · αE = cos−1(E₁)The first repetition starts at equilibrium. Later pulses begin from the recovery left by every earlier pulse.
Instantaneous ideal RF creates Mxy = Mz⁻ sinα and leaves Mz⁺ = Mz⁻ cosα.
T₂* reduces the measured echo. The recurrence then applies coherent Mxy → 0 before the next pulse without claiming a gradient- or RF-spoiling mechanism.
During TR, Mz returns toward M₀. Repeating the same map approaches one fixed point at rate E₁ cosα.
One normalized isochromat per tissue is shown in the rotating frame, with instantaneous nonselective RF, perfect loss of coherent transverse magnetization between repetitions, monoexponential T₁ recovery, and magnitude-only T₂* echo weighting. T₁ recovery during the short TE is folded into the repetition map. The model excludes coherent/balanced SSFP, RF phase cycling details, B₁ error, slice profile, inflow and flow, magnetization transfer, diffusion, exchange, multiple compartments, off-resonance banding, imaging gradients, noise, and reconstruction. Z/B₀ is the spin-frame reference here, not a patient-orientation claim.
ADVANCED 3D MODEL · INVERSION RECOVERY FAMILY
An inversion preparation sends longitudinal magnetization below zero. Tissues recover at different T₁ rates, so a readout placed at the right inversion time can null fat, null fluid, or preserve the sign that ordinary magnitude display folds away.
Efficiency controls the effective longitudinal inversion; it is not drawn as a fictitious hard-pulse angle.
The finite-TR fixed cycle shifts that crossing from the familiar long-TR T₁ ln 2 result.
A phase-sensitive display preserves negative-versus-positive recovery; magnitude folds both sides above zero.
The five declared materials rise through the shared zero plane at different times; fat is at its finite-TR crossing near TI 243 ms.
The signed TI-response plot remains available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Upper plot: signed Mz immediately before the readout. Lower plot: the chosen signed or magnitude echo after flip-angle, proton-density, and TE weighting.
Mz(TI) = 1 − cinvETR − (1 − cinv)ETI1 − cinvcos α ETR
cinv = 1 − 2η · ETI = e−TI/T₁ · S = ρ Mz(TI) sin α e−TE/T₂(or T₂*)η = 1 maps +Mz to −Mz. Lower η reduces that effective longitudinal displacement and changes the null.
Fat crosses zero before white matter, gray matter, and CSF in this declared 3 T teaching table.
The readout flip converts Mz into Mxy. Spin echo uses T₂; gradient echo uses T₂* in this model.
Signed mode retains polarity; magnitude mode applies an absolute value after the same physical signal calculation.
Every material is one normalized, nonexchanging compartment with monoexponential T₁, T₂, and T₂* constants. Inversion and readout are instantaneous and spatially uniform. η is an effective longitudinal action, not a literal pulse angle or a model of adiabatic-pulse dynamics. The repeated mode contains one readout per TR with exact fixed-cycle history; fully recovered mode forces Mpre = M₀. The colored lanes and tiles are declared material samples, not anatomy or a diagnostic image. “STIR” and “FLAIR” presets isolate their nulling mechanism; they are not scanner protocols. “Signed / PSIR-like” preserves ideal signal polarity but does not model a reference acquisition, phase errors, or PSIR reconstruction. Excluded effects include slice profile, B₁ variation, finite pulse duration, magnetization transfer, exchange, multi-component fat, CSF flow, FSE echo trains, stimulated echoes, k-space ordering, coils, noise, motion, SAR, parallel imaging, and vendor timing constraints.
ADVANCED 3D MODEL · FSE / TSE / RARE · EXTENDED PHASE GRAPH
A fast spin-echo train is not one spin echo copied many times. Every refocusing pulse mixes transverse and longitudinal configuration states. Those coherent and stimulated-echo pathways determine each echo; assigning different echoes to phase-encode lines then changes contrast, point spread, and scan time.
EPG order k records accumulated gradient phase area. It is dimensionless pathway bookkeeping—not an imaging ky coordinate.
Sub-180° pulses store part of the history longitudinally and recall it later as stimulated echoes.
The resulting complex echo weighting is an MTF; its inverse transform is the phase-direction PSF.
EPG CONFIGURATION ORDER k tracks dephasing pathways inside one modeled compartment. IMAGING ky is a commanded spatial-frequency address acquired on one echo. The dashed bridge is a data-flow link, not an equality.
The receiver samples only F₀ at 80 ms. Other transverse and longitudinal orders preserve hidden history for later echoes.
The exact echo-train, MTF, and PSF plots remain available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Ωkafter RF = T(α,φ)Ωkbefore RF · Fk+ → Fk+1+
Sn = ρF0(n·ESP) · W(ky) = Se(ky)/maxe|Se| · PSF = DFT−1{W}Relaxation scales F states by E₂ and Z states by E₁; recovery adds only to equilibrium Z₀.
A 180° CPMG pulse swaps transverse pathways. Lower angles also create Z storage and later stimulated echoes.
The receiver sees the signed complex sum at order zero, not the hidden higher-order states individually.
Center ky largely sets effective contrast; variation over ky determines the PSF, blur, ringing, and possible phase modulation.
The EPG engine is on-resonance, nonselective, single-compartment, and monoexponential. RF pulses are instantaneous and spatially uniform; every half-echo interval receives one identical positive unit gradient-order shift. CPMG and CP specify ideal RF-axis relationships. The “declared ramp” is a transparent exponential 180°-to-floor teaching schedule—not a vendor optimizer, prescribed-signal algorithm, or claim of an executable protocol. The imaging panel separately maps the solved echo train into declared grouped one-dimensional phase-encode bands across 128 lines. It is not the EPG configuration axis, a universal view order, or a two-dimensional anatomy reconstruction. TR is used only for scan-time accounting because every displayed train begins at equilibrium; inter-train saturation is excluded. Also excluded are slice profile and selective-pulse crushers, B₁/B₀ distributions, finite RF duration, chemical shift, multiple compartments, exchange, magnetization transfer, diffusion, flow, motion, noise, coils, parallel imaging, partial Fourier, echo sharing, ramp sampling, gradient limits, SAR calculation, and vendor-specific constraints. Signal, MTF, PSF, and timing values are educational simulations, not clinical recommendations.
ADVANCED 3D MODEL · COHERENT BALANCED STEADY STATE
The spoiled-GRE lesson deliberately destroys transverse coherence. Here the net gradient area returns to zero every TR, so coherence survives, off-resonance phase accumulates, and the repeating vector cycle creates bright passbands separated by dark signal nulls.
The next pulse remembers mainly longitudinal recovery.
The next RF pulse receives a coherent 3D vector with longitudinal and transverse history.
Signal nulls repeat every 1/TR and move when RF phase cycling changes.
The analytic fixed point repeats after free precession, T₁/T₂ relaxation, RF phase advance, and the next ideal RF rotation.
The quantitative frequency profile remains available if 3D rendering is unavailable.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
Read magnitude and phase together: a stopband is a signal minimum with an abrupt phase transition, not a dark stripe painted at a fixed location.
M+ss = Rx(α)[D(β)M+ss + c]
β = 2πΔf·TR − Δφ · E₁ = e−TR/T₁ · E₂ = e−TR/T₂ · solve (I − RxD)M = RxcPosition-dependent gradient phase is rewound within each TR. Residual off-resonance phase is not.
Both transverse coherence and longitudinal recovery enter the following RF rotation, so the state is three-dimensional.
The steady state is solved exactly; the optional pulse history shows why it is not established after the first excitation.
RF phase cycling translates the periodic response. Separate phase-cycled acquisitions can reduce, but do not prevent, the underlying off-resonance sensitivity.
Each displayed frequency uses one normalized isochromat with monoexponential T₁/T₂ relaxation, instantaneous nonselective RF, a constant phase increment, and perfectly zero net gradient area each TR. TE is fixed at TR/2. Vectors use an RF-phase-aligned receiver frame that distributes the programmed phase increment uniformly through each TR; the physical laboratory-frame azimuth is not plotted. The 3D frequency–I/Q manifold maps a declared linear off-resonance ramp to independent steady states: frequency is the long axis, I/Q occupy each perpendicular plane, and radius is relative echo magnitude. These are data coordinates, not anatomy, a measured B₀ map, or a reconstructed patient image. Four-cycle RMS combines four separate magnitude acquisitions and has no single coherent magnetization vector. The model excludes intravoxel frequency distributions, slice profile, B₁ error, finite RF duration, eddy-current imbalance, gradient first-moment flow effects, motion, exchange, magnetization transfer, diffusion, chemical multi-peaks, noise, coil sensitivity, k-space ordering, transient catalyzation, SAR calculation, and reconstruction. Real bSSFP names and implementations vary by vendor.
ADVANCED 3D BOLD fMRI · NEURAL INPUT → HEMODYNAMICS → T₂* SIGNAL
Drive one declared cortical region, follow the extended Balloon-model states, and watch changing deoxyhemoglobin alter local susceptibility, spin coherence, gradient-echo magnitude, and the sampled EPI time series. Neural input, physiology, MR contrast, acquisition, and statistical interpretation remain separate objects.
The grayscale background is fixed synthetic anatomy. The colored patch displays the latest TR-sampled volume, while the chart cursor retains the continuous state; neither is a statistical activation map or patient image.
READ THE 3D CHAIN · The translucent brain and voxel are spatial context. Yellow pulses declare modeled neural input. Mint vessel flow and coral deoxyhemoglobin beads are hidden physiological states. Distorted field rings and spin arrows expose an MR consequence that is normally invisible. The final volume cards are discrete samples taken at TR; they are not neurons, blood cells, or proof of activation. Drag to orbit, use the visible camera controls, or tap an object for its exact meaning.
ADVANCED 3D ARTERIAL SPIN LABELING · RF LABEL → TRANSIT → DELIVERY → CBF
Run one quantitative pseudo-continuous ASL experiment from the neck to a cortical voxel. A control/label RF preparation creates the difference, arterial transit delays delivery, blood T₁ erases part of the tag, and a paired subtraction exposes a signal only a few tenths of one percent of M₀ before the consensus single-PLD equation estimates cerebral blood flow.
The control and label anatomy share one magnitude scale. Their difference uses a separately declared amplified display so the tiny modeled perfusion signal is visible; the CBF panel applies the displayed equation to that synthetic difference.
READ THE 3D CHAIN · The lower plane represents a pCASL RF-plus-gradient preparation. Moving beads are synchronized spin-packet cues: coral means label difference, fading toward gray as T₁ erases it. The vessel tree is schematic physical transport; the highlighted cortical voxel is the modeled delivery region. Control and label cards are separate acquisitions, while the difference card is a calculation. Drag to orbit, use the camera controls, or tap any object for its exact role and boundary.
ADVANCED 3D TIME-OF-FLIGHT MRA · TRANSPORT × REPEATED SPOILED GRE
Follow one constant-velocity blood cohort across a hard 3D excitation slab. Physical transport decides where each RF event meets the cohort; the exact spoiled-GRE recurrence decides its remaining Mz; and a calculated maximum-intensity projection shows how that depth-dependent signal becomes a bright-blood angiographic display.
Why the steps are real in this model: the plot freezes a continuous plug-flow column at one RF instant. Spins that entered during the same TR share an RF count. A real voxel, slice profile, pulsatile velocity distribution, and asynchronous entry can smooth this ideal stroboscopic pattern.
READ THE 3D CAUSAL CHAIN · Each stage button replaces the central geometry so unrelated spaces cannot compete for room. The transparent box is a selected slab, not scanner hardware. A physical vessel crosses it at the chosen angle; vector length and color encode pre-RF Mz; rings locate one cohort at successive RF instants; the coral plane is an ideal 90° preparation outside the positive slab face; and Volume → MIP keeps the calculated source stack visibly separate from its max-over-Y projection. Drag to orbit, zoom, or tap any object for its exact meaning.
ADVANCED 3D PHASE-CONTRAST FLOW · BALANCED M₀, OPPOSITE M₁
Follow one steady through-plane velocity component from a transparent vessel, through two opposite bipolar encodings, into two measured I + iQ vectors and a wrapped velocity map. The physical flow, gradient moments, phase subtraction, VENC, aliasing, and integrated volume flow all share one exact state.
READ THE 3D CAUSAL CHAIN · Each stage button replaces the central geometry. The tube contains the physical velocity profile. The two depth-separated moment volumes are acquisition-time diagrams—not gradients located in the vessel. The volumetric I/Q clocks show SA, SB, conjugation, and their wrapped product. The final relief surfaces use height and hue from the same signed 61 × 61 velocity arrays integrated for mL/s; violet marks wrapped samples. Drag to orbit, use visible zoom controls, or tap an object for its exact meaning.
ADVANCED 3D DIFFUSION EXPERIMENT · PULSED-GRADIENT SPIN ECHO
Orbit a magnified displacement-space model. The first gradient labels position with phase; the 180° pulse reverses that phase ordering; the matched second lobe cancels it only if a spin has not changed position along the selected direction.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object to identify it · molecular displacement is magnified, not voxel scale
GRADIENT WEIGHTING32.6 mT/m, δ 20 ms, and Δ 40 ms produce b = 1014 s/mm² in this ideal rectangular-lobe model.
DIRECTIONAL QUESTIONg = [+1.00, +0.00, +0.00], parallel to the tensor’s fixed X principal axis.
AXIALLY SYMMETRIC TEACHING TENSORD∥ is 3.75× D⊥. The ellipsoid is longer along physical X; it is a voxel-scale statistical model, not a drawn axon.
b = (γGδ)²(Δ − δ/3)
S(b,g) / S₀ = e−bDapp(g)
γ uses radians per second per tesla inside b. Dapp = gᵀDg probes one direction through the tensor. The effective finite-lobe diffusion-time term is Δ − δ/3 = 33.3 ms.G along g creates a position-dependent frequency offset. Integrating for δ writes phase proportional to g·r.
Between the lobe centers, molecules change position statistically. The 3D paths magnify micrometre-scale displacement so direction can be seen.
After the 180° reversal, the matched lobe cancels stationary phase. Motion along g leaves phase proportional to the projected displacement.
The coil receives one vector sum. Broad residual phases cancel more, so larger bDapp lowers the ideal coherent echo.
QUANTITATIVE 3D DTI · MULTI-DIRECTION ACQUISITION → SIX-PARAMETER FIT
Continue directly from the pulsed-gradient experiment. Every diffusion-weighted direction supplies one value of gᵀDg. Combine non-collinear measurements, fit all six independent tensor elements, diagonalize D, and inspect where a single tensor fails.
READ THE 3D SPACES · The source and fitted ellipsoids live in one voxel’s physical coordinate frame. Gradient arrows are encoding directions, not water motion. The direction shell is a measurement design, not k-space. Drag to orbit, zoom, or tap an object for its exact role.
03 / THREE ORTHOGONAL CONTROLS
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.
GX · ON DURING ADC
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.
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.
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.
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.
Drag to orbit · pinch, wheel, or use + / − to zoom · tap an object to identify it
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Called “readout” because the receiver’s ADC reads many samples while the positive read gradient is on.
Called “phase” because the lobe ends before ADC, yet its position-dependent phase pattern remains during readout.
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.
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.
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.
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.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.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.
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.
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.
Drag to orbit · physical frame stays fixed · camera motion changes no gradient
Gphysical = R · Glogical
| 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 |
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.
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
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.
Drag to orbit · scroll to zoom · the sum bay faces the camera but is not a physical location
M0(t) = ∫0t G(t′) dt′
k(t) = γ̄ M0(t)
φ(r,t) = 2π k(t) · r
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.
A weak gradient held longer can create the same phase slope and k-space displacement as a short, strong gradient.
Changing the sign of G reverses the phase ramp and moves to the opposite side of k-space along the selected logical axis.
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
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.
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.
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 background is the phantom’s exact Fourier spectrum. The coral ring marks the single receiver sample inspected at left.
S(kx,ky) = ∫∫ ρeff(x,y)e−i2π(kxx+kyy) dxdy
one k-space coordinate is a weighted sum from the entire excited sliceAt 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.
The event block is expanded so short RF and gradient operations remain visible; the broken segment compresses idle recovery before the next RF pulse.
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.
The 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.
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 openA 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.
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.
Open the definitions below, then press Run. This explanation will follow RF transmit, gradients, receive sampling, and reconstruction through each repetition.
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.
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.
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.
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.
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.
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.
THE RULEA k-space sample does not paint one image pixel. Every acquired complex coefficient changes the calculation of every reconstructed pixel.
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.
0 samples: the canvas is deliberately empty because no image can be reconstructed yet.
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.
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
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.
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.
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-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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.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.
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
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.
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.
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
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.
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.
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.
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.
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.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.
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.
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.
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.
DETAIL MICROSCOPE · THE SAME PHYSICAL SQUARE ON BOTH IMAGES
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.
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.
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.
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.
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
Large |k|
Sample spacing Δk
06 / TRAJECTORY STUDIO
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.
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.
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.
Drag to orbit · scroll to zoom · tap an object to identify it
MULTI-SHOT · RECTILINEAR
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.
dkdt= γ̄ G(t)
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.
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.
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.
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
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.
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.
DRAG TO ORBIT · + / − TO ZOOM · TAP AN OBJECT TO IDENTIFY IT
BWPE,pix=1 / (Ny · ESP)
Δypix=p · Δf / BWPE,pix
|ghost| / |main|=|tan(φ/2)| ?
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.
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.
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.
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.
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.
Limits how sharply a path can turn. Fast switching also drives acoustic noise and peripheral nerve-stimulation constraints.
The path may move while the receiver is off. Only coordinates visited during ADC become acquired data samples.
07 / RESOLUTION LAB
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.
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.
Δx = FOVx / Nx ≈ 1 / (2kx,max)
FOVx = 1 / Δkx
Δz = BWRF / (γ̄ |Gz|)
Drag to orbit · mint spins tip transverse inside the RF passband · ruler is 48 mm/world
Δf(z) = γ̄ Gz z
Δz = BWRF / (γ̄ |Gz|)
Gz maps position to frequency. The 2.0 kHz RF passband intersects that line around z = 0, exciting an ideal 4.70 mm slab.
A wider transmit band matches a larger interval of positions on the same frequency slope.
A steeper frequency slope maps the same RF bandwidth onto a narrower spatial interval.
Changing the RF center frequency moves the intersection without changing the ideal thickness.
08 / SAMPLING & POINT RESPONSE
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.
Δx ≈ 1 / (2kx,max)
FOVx = 1 / Δkx
All encoded frequencies are retained at the native Δk. The PSF approaches one image pixel and no coherent wrap replicas are introduced.
Mint points contribute to reconstruction. Coral lines are inside the selected extent but skipped by uniform undersampling.
The central lobe sets effective resolution; sidelobes produce ringing or displaced wrap replicas.
Full k-space extent and native spacing preserve the simulated image’s available detail and field of view.
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.
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.
A Hann or Hamming taper suppresses PSF sidelobes and Gibbs ringing, but broadens the central lobe. A calmer boundary is purchased with effective resolution.
QUANTITATIVE 3D MULTI-SLICE EXPERIMENT · EXCITATION-ONLY CROSS-TALK
Slice selection has a second half that a single-slice picture cannot show. A real pulse has a finite profile, so every slice also tips spins that belong to the slices beside it. Those spins are already part-saturated when their own turn comes—and how far they have recovered depends on how long ago the neighbour fired, which is the acquisition order.
The upper plane is each slice's own excited |Mxy|. The lower plane compares the longitudinal magnetization the central slice actually starts from against the same slice acquired alone; the coral band between them is what the neighbours took.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object to identify it · the number over each slab is its acquisition slot within one TR
M ← E·r(z)·M + (1 − E), E = e−(TR/N)/T₁
S = |∫ Mz−(z)·(mx + i·my)(z) dz|
r(z) is the exact Rzz of the pulse's rotation, so the longitudinal update is exact once transverse magnetization is spoiled. The signal is a coherent integral: the two transverse components are summed separately along z and the magnitude taken once. One slot lasts 71.4 ms.A time-limited RF pulse cannot have a rectangular frequency response. Its excited width and its shoulders are set by the time–bandwidth product: a blunt pulse spills well past the thickness it names, a sharp one barely does. Move the time–bandwidth slider and watch the printed FWHM cross the nominal value.
When a neighbour fires, spins in the overlap lose longitudinal magnetization. They are not the neighbour's spins or this slice's spins—they belong to both profiles, which is why no single-slice picture can show this.
Those spins recover with T₁ until this slice's own turn. Sequential ordering leaves the worst-served slice one slot; interleaving leaves it at least two, once there are five or more slices. Here that timing shift is worth far less than the gap itself—compare the two numbers rather than assuming.
The coil integrates one complex quantity across the slab. Whatever the neighbours removed mostly never becomes transverse magnetization, so the slice normally reads darker than the same slice acquired alone. Not always: the profile shoulders carry a signed phase, and suppressing an opposing shoulder can leave the coherent sum a fraction of a percent higher.
09 / PARALLEL IMAGING · RECEIVE SENSITIVITY AS ENCODING
Uniformly skipping phase-encode lines reduces the encoded field of view and folds distant positions together. A receive array supplies additional spatial information because every element has a different complex sensitivity map. SENSE can separate the folded positions—but geometry and noise determine the cost.
ADVANCED 3D + 2D EXPERIMENT · IMAGE-DOMAIN SENSE
Solid loops and the source slice are physical geometry. Calculated quantities remain explicitly labeled mathematical geometry: |Cc| and |ρCc| become displaced 3D surfaces with cyclic phase hue; folded source positions converge on one reduced-FOV pixel; the local complex system opens into N phasor rows; solved ρ values become paired bars; and g−1 becomes a capped, calibrated terrain beside a separate √R bar. The 2D panels preserve exact image-map views of the same arrays.
Drag to orbit · zoom · tap an object · camera motion changes no data · calculated height scales are declared in-scene
DISPLAY SELECTION ONLYEvery channel remains in the SENSE calculation. This control chooses which physical loop, sensitivity map, and aliased coil image are highlighted.
SPATIAL DISTINCTNESSA broader proxy makes channels look more alike across the slice. This is a teaching sensitivity width, not loop diameter or a measured B₁⁻ map.
NOISE COVARIANCE ΨThe model uses equal variance and one shared correlation coefficient. Real arrays require measured channel covariance and coupling-aware calibration.
The noiseless SENSE result matches the synthetic source to numerical precision. The g-map reports statistical noise amplification; it is not hidden by the clean display.
Sc(k) = ∫ Cc(r)ρ(r)e−i2πk·rdr
Cc is complex receive sensitivity; c names the channel. The signal remains I + iQ.a = Cρ + n
a is 8 × 1 · C is 8 × 2 · ρ is 2 × 1ρ̂ = (CᴴΨ−1C)−1CᴴΨ−1a
H means conjugate transpose. Magnitude-only coil maps are not enough for this equation.SNRR = SNRfull / (g√R)
√R comes from fewer samples; g ≥ 1 comes from coil geometry and noise covariance.At x = +6.9 mm, two source positions 110 mm apart fold together. Each row below is one coil’s complex sensitivity to those positions and its resulting aliased I + iQ measurement.
| coil | ρ₁ | ρ₂ | ac |
|---|
A coil has smooth sensitivity across the slice; it does not illuminate one isolated region like a flashlight.
The displayed phase FOV is half-sized and stretched here for comparison. This is coherent folding from increased Δky, not truncation blur.
The clean result proves algebraic separation in the ideal model. It does not erase the statistical noise cost shown beside it.
Dark means little extra geometry penalty; yellow marks less distinguishable sensitivity rows and larger noise amplification. Printed values keep the map readable without color.
Nearby transverse magnetization induces a different complex voltage in each receive element.
Δky grows by R, nominal phase encoding time falls to 1/R, and R positions fold together.
SENSE solves one local N × R system at every reduced-FOV image location.
√R is global; g varies spatially. More coils do not guarantee low g unless their maps add distinct information.
10 / THE COMPLETE CHAIN
Amplifiers drive X, Y, Z coil windings. Current and geometry create a controlled field slope in T/m.
I(t) → G(t)The local field changes Larmor frequency. Accumulated phase records the gradient’s area through time.
φ(r,t) = γ r·∫G dtThe receiver sums every transverse spin at the current spatial-frequency coordinate.
k(t) = γ̄∫G dtAn inverse Fourier transform separates the superposed spatial frequencies into locations.
ρ̂(r) = ℱ⁻¹{S(k)}REFERENCE DESK
STEP 1
The scanner’s main field aligns proton magnetization. Gradient coils add controlled spatial slopes.