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 and an image reconstructed line by line.
The table carries the patient into the fixed bore. 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.
↔ Drag to orbit · pinch or wheel to zoom · tap a label or object to explain it
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 · 123 / 123 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.
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 coil length stays 4.0 m. The 3D winding shows 52 spaced turns to represent 16,000 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 loops 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
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(r,t) = B0 + G(t) · r
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.i The gradients are tiny beside B₀, but they are switched quickly and precisely. The audible knocking in MRI is gradient hardware moving under Lorentz force.
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
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.
Drag to orbit · mint cage is body RF · 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 sensitivity number is a normalized geometry teaching proxy, not a scanner specification. The shared B₀ control retunes the displayed ¹H carrier; it does not rescale this idealized coil geometry, B₁ field, SNR, relaxation, or SAR model. Actual values require coil- and patient-specific electromagnetic measurements.
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.
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.
Signals use simplified steady-state equations and illustrative relaxation values near 3 T. Real contrast also depends on sequence details, RF profiles, echo trains, magnetization transfer, flow, diffusion, coils, reconstruction, pathology, and field strength.
03 / THREE ORTHOGONAL CONTROLS
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.
Drag to orbit · pinch, wheel, or use + / ↺ / − · tap an object to identify it
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 oval is one digital volume. The violet slab is the part given transverse signal. Small white hands show the phase of spin packets inside that slab. They are not individual protons. Follow the numbered events; then rotate the prescribed plane and see the same logical jobs redistributed across physical Gx, Gy, and Gz.
Drag to orbit · pinch, wheel, or use + / − to zoom · tap an object to identify it
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 white transverse-phase hands 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.
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.
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 oval, points, phase hands, color bands, RF rings, and seven-address readout are an explanatory digital model—not anatomy, individual nuclei, a measured B field, or a clinical pulse-sequence prescription. 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 · plane normal is logical slice Z
Gphysical = R · Glogical
| physical | Read | Phase | Slice |
|---|---|---|---|
| Gx | +1.000 | 0.000 | 0.000 |
| Gy | 0.000 | +1.000 | 0.000 |
| Gz | 0.000 | 0.000 | +1.000 |
In an axial prescription, logical readout aligns with the physical X gradient. Only the Gx amplifier is needed for this +30.0 mT/m readout plateau.
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
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.
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 how far the sampled grid reaches into spatial frequency at a fixed 220 mm field of view. TE and TR change each tissue’s signal before encoding. They are separate causes, so the controls never pretend that contrast timing changes pixel size.
At fixed FOV and Δk, more samples extend to larger |k| and make smaller pixels. It also requires more phase-encoding repetitions in this one-line-per-TR model.
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.
One I + iQ sample is one whole-object spatial pattern. The inverse Fourier transform combines all acquired coefficients 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.
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.
Its signal-to-noise ratio is deliberately not estimated. In a scanner, smaller voxels usually contain less signal, and weak outer-k-space measurements may compete with random coil and receiver noise. More matrix is useful only when signal, scan time, coils, motion control, bandwidth, gradients, and reconstruction preserve the added information.
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
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)
i Delayed or filtered gradient response means the scanner may sample somewhere other than the commanded coordinate. Non-Cartesian reconstruction therefore often uses a measured trajectory.
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.
Drag to orbit · highlighted cell is one voxel
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 · highlighted rings fall inside the transmit passband
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.
09 / 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 OF 17
The scanner’s main field aligns proton magnetization. Gradient coils add controlled spatial slopes.