Theme 05 · Image consequences

Connect acquisition choices to voxel size, signal, folding, parallel imaging, blur, and ringing.

Manipulate MRI voxel dimensions, slice selection, k-space sampling, receive-coil sensitivity, SENSE acceleration, g-factor, SNR, and reconstruction artifacts.

View the complete lab

07 / RESOLUTION LAB

Shape the voxel
in all three axes.

Resolution is not a “megapixel” setting. It follows from field of view, sample count, and how far the acquisition reaches in k-space. Smaller voxels usually cost signal-to-noise, scan time, or both.

LIVE VOXEL PRESCRIPTION1.72 × 1.72 × 5.00 mm

03 · ONE RECONSTRUCTED VOXEL 1.72 × 1.72 × 5.00 mm The coral locator marks one true-scale bin; the isolated copy preserves its aspect ratio at a declared magnification.

Protocol builder

XReadoutfrequency

FOV220 mm

matrix128

1.72 mm

YPhaseencoded

FOV220 mm

matrix128

1.72 mm

ZSliceRF selected

RF BW2.0 kHz

Gz9.4 mT/m

5.00 mm

voxel volume14.77 mm³relative SNR proxy 1.00×

k-space extent±291 / ±291 m⁻¹Z uses slice profile

minimum phase encodes128≈ 25.6 s at TR 200 ms

REFERENCE PRESCRIPTIONWhat changed—and what it costs

This 220 mm, 128 × 128 reference produces 1.72 mm in-plane sampling. Move any control to see its direct consequence for detail, signal, and encoding time.

in-plane pixel area 1.00× signal proxy 1.00× phase-time proxy 1.00×

PIXEL / VOXEL SIZE

Δx = FOV_x / N_x ≈ 1 / (2k_x,max)

FIELD OF VIEW

FOV_x = 1 / Δk_x

2D SLICE THICKNESS

Δz = BW_RF / (γ̄ |G_z|)

Three-dimensional RF-selected slice

LOWER FREQUENCYHIGHER FREQUENCY

FIXED FREQUENCY COLOR KEY −160 kHz 0 +160 kHz z −120 → −51.1 kHz · z +120 → +51.1 kHz

RF-SELECTED SLAB 4.70 mm center z = 0.0 mm · axial plane

Drag to orbit · mint spins tip transverse inside the RF passband · ruler is 48 mm/world

FREQUENCY ADDRESS

Δf(z) = γ̄ G_z z

SELECTED THICKNESS

Δz = BW_RF / (γ̄ |G_z|)

Frequency-to-position map+GZ · CENTERED RF

Slice gradient |Gz|10.0 mT/m 41730 mT/m

RF transmit bandwidth2.0 kHz 0.53.256.0 kHz

RF center-frequency offset0.0 kHz −20on-resonance+20 kHz

CENTERED AXIAL SLICE

Gz maps position to frequency. The 2.0 kHz RF passband intersects that line around z = 0, exciting an ideal 4.70 mm slab.

frequency slope0.426 kHz/mm

RF passband−1.0 … +1.0 kHz

slice center0.0 mm

ideal thickness4.70 mm

BW_RF ↑

Thicker slab

A wider transmit band matches a larger interval of positions on the same frequency slope.

|G_z| ↑

Thinner slab

A steeper frequency slope maps the same RF bandwidth onto a narrower spatial interval.

f_RF ↕

Move the slice

Changing the RF center frequency moves the intersection without changing the ideal thickness.

NEXT · APPLY THIS PHYSICS Use three localizers to prescribe an anatomical plane, choose a receive coil, and run the virtual scan.

08 / SAMPLING & POINT RESPONSE

Change k-space.
The image answers.

Spatial resolution, field of view, ringing, and wraparound are different consequences of how k-space is sampled. Apply the operations directly to the complex data and inspect both the reconstructed image and the point-spread function that produced it.

Sampling prescription

CUTOFF / RESOLUTION

Δx ≈ 1 / (2k_x,max)

SPACING / FIELD OF VIEW

FOV_x = 1 / Δk_x

Retained k-space extent100% kmax 25% · blurred60%100% · full

Uniform sampling interval

k-space weighting

FULL REFERENCE

All encoded frequencies are retained at the native Δk. The PSF approaches one image pixel and no coherent wrap replicas are introduced.

nominal Δx × Δy3.44 × 3.44 mm

effective FOV220 × 220 mm

PSF FWHM1.0 × 1.0 px

retained samples4,096 / 4,096

Applied sampling mask100% RETAINED

k_yk_x

Mint points contribute to reconstruction. Coral lines are inside the selected extent but skipped by uniform undersampling.

Image-space PSFℱ⁻¹{MASK}

positivenegative

The central lobe sets effective resolution; sidelobes produce ringing or displaced wrap replicas.

Resulting reconstructionREFERENCE

Full k-space extent and native spacing preserve the simulated image’s available detail and field of view.

Truncate extent

Removing outer k-space lowers the spatial-frequency cutoff. Image space is convolved with a broader sinc-like response: detail softens and sharp boundaries ring.

Increase spacing

Keeping every Rth point multiplies the effective Δk by R and reduces the encoded FOV by R. Repeated PSF peaks fold distant anatomy into the displayed FOV.

Apodize

A Hann or Hamming taper suppresses PSF sidelobes and Gibbs ringing, but broadens the central lobe. A calmer boundary is purchased with effective resolution.

QUANTITATIVE 3D MULTI-SLICE EXPERIMENT · EXCITATION-ONLY CROSS-TALK

When slices overlap, the neighbours change what this slice can signal.

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.

SLICE-NORMAL AXIS · NORMALIZED M / M₀ · DECLARED SLICE STACK, NOT ANATOMY Neighbour saturation changes the central slice by −15.3%

The upper plane is each slice's own excited |M_xy|. 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.

Three-dimensional multi-slice cross-talk model. Every number it shows is also printed in the readouts beside it.

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

Inter-slice gap0.0 mm 0 (contiguous)edge-to-edge distance10 mm

Slice thickness5.0 mm 2the gradient follows to hold it10 mm

Pulse time–bandwidth3.0 1 (blunt)profile sharpness12 (sharp)

Excitation flip angle80° 10°how hard each pulse saturates90°

TR500 ms 120shared by every slice in the loop4000 ms

Slices in the stack7 3odd counts keep a central slice15

Tissue T₁900 ms 200how fast a saturated edge recovers2500 ms

PER-SLOT SATURATION RECURRENCE

M ← E·r(z)·M + (1 − E), E = e^{−(TR/N)/T₁}

S = |∫ M_z⁻(z)·(m_x + i·m_y)(z) dz|

r(z) is the exact R_zz 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.

excited FWHM vs nominal5.61 mm / 5.00 mmexcited width is 12% wider than the slab it names

neighbour saturation at this slice5.8%excitation-weighted, one neighbour

slots since a neighbour fired3 of 7214 ms of T₁ recovery before this slice fires

signal vs the same slice alone84.7%15.3% lost to the neighbours

SAME GEOMETRY, BOTH ORDERS · CENTRAL SLICE ONLY

Sequential84.3%

Interleaved84.7%

Interleaving changes this slice by +0.4 percentage points here. That is this slice under these two cyclic orders, not a general rule.

01 · FINITE PROFILEThe excited width need not match the slab

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.

02 · NEIGHBOUR TIPS THESE SPINSSaturation arrives early

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.

03 · RECOVERY RACEOrder sets the recovery time

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.

04 · COHERENT SUMLess M_z usually means less echo

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.

MODEL BOUNDARY · WHAT THIS 3D EXPERIMENT DOES AND DOES NOT CLAIM

Quantitative: the slice profiles are an exact Bloch solution of one apodized sinc pulse, solved as successive exact rotations; the slice-select gradient is set from G = BW/(γ·Δz); the per-slot saturation recurrence is solved in closed form and independently verified against brute-force iteration; the signal is a coherent complex integral of m_x and m_y. Illustrative: the slab boxes, the separation of the three planes, and the acquisition-clock highlight are drawing choices, not measured geometry. Magnetization is normalized to M₀ and the axis is a declared slice stack, not anatomy. Omitted: magnetization transfer, diffusion and flow across slice boundaries, B₁ and B₀ inhomogeneity, gradient nonlinearity, refocusing pulses and their far wider saturation footprint, multiband and simultaneous-multi-slice excitation, in-plane encoding, noise, and every scanner timing overhead. Transverse magnetization is assumed perfectly spoiled between pulses and every slice is assumed to receive exactly TR/N. This is an excitation-only teaching model of one mechanism. It is not a scanner simulation, not a protocol, and not clinical guidance: its percentages describe this declared experiment and are not a prediction for any real sequence, and nothing here recommends a gap, an ordering, or a pulse for imaging a patient.

09 / PARALLEL IMAGING · RECEIVE SENSITIVITY AS ENCODING

Skip phase lines.
Let the coil array help unfold them.

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.

FROM THE RF LESSONLocal receive arraymultiple complex coil channels FROM THE SAMPLING LESSONKeep every Rth k_y linereduced FOV and coherent folding

HERESensitivity encodingcoil geometry separates folded positions

ADVANCED 3D + 2D EXPERIMENT · IMAGE-DOMAIN SENSE

A physical coil array supplies spatially distinct complex measurements.

Solid loops and the source slice are physical geometry. Calculated quantities remain explicitly labeled mathematical geometry: |C_c| and |ρC_c| 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.

OPEN A CONCEPT ?

Interactive 3D receive-array and sensitivity model

07 / 07 · GEOMETRY COST MEASURE WHAT UNFOLDING COSTS The g-factor map shows extra spatially varying noise amplification. The separate √R term is the unavoidable loss from collecting fewer phase-encode lines.

solid loop · receive hardware height + lightness · magnitude; cyclic hue · phase planes · source and reduced phase FOV colored marks · folded sources / recovered ρ

Drag to orbit · zoom · tap an object · camera motion changes no data · calculated height scales are declared in-scene

Phase-encode reduction R

R = 2 retains every second k_y line. The encoded phase FOV becomes 110 mm, so two positions 110 mm apart fold into each displayed location.

Independent receive channels N

Eight channels provide eight complex equations for two folded source positions. Channel count helps only when the sensitivity rows are spatially distinct.

Array geometry

The ring samples the object from distinct directions, improving separation along the undersampled phase axis.

Inspect receive channelCoil 1 / 8 channel 1changes the highlighted mapchannel 8

DISPLAY SELECTION ONLYEvery channel remains in the SENSE calculation. This control chooses which physical loop, sensitivity map, and aliased coil image are highlighted.

Analytic sensitivity width85 mm 45 mm · localanalytic |C_c| falloff145 mm · broad

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.

Off-diagonal noise correlation0.05 0 · Ψ = Ishared channel noise0.45

NOISE COVARIANCE ΨThe model uses equal variance and one shared correlation coefficient. Real arrays require measured channel covariance and coupling-aware calibration.

phase lines / time proxy50%½ the phase encodes

reduced phase FOV110.0 mmfull FOV 220 mm

mean / peak g1.24 / 1.35unitless geometry penalty

summary using mean g57.2%1 / (1.24 × √2) · not mean local SNR

FULL-RANK TEACHING SYSTEM Eight complex measurements can separate two unknown positions.

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.

01 · EACH COIL MEASURES ITS OWN COMPLEX WEIGHTING

S_c(k) = ∫ C_c(r)ρ(r)e^−i2πk·rdr

C_c is complex receive sensitivity; c names the channel. The signal remains I + iQ.

02 · ONE ALIASED LOCATION MIXES R TRUE POSITIONS

a = Cρ + n

a is 8 × 1 · C is 8 × 2 · ρ is 2 × 1

03 · SENSE USES NOISE-WEIGHTED COMPLEX INVERSION

ρ̂ = (CᴴΨ⁻¹C)⁻¹CᴴΨ⁻¹a

H means conjugate transpose. Magnitude-only coil maps are not enough for this equation.

04 · SPEED HAS TWO DISTINCT SNR COSTS

SNR_R = SNR_full / (g√R)

√R comes from fewer samples; g ≥ 1 comes from coil geometry and noise covariance.

ONE FOLDED PIXEL · OPEN THE LINEAR SYSTEM

The selected image location is a vector of coil measurements—not one brightness.

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.

Complex sensitivity matrix C and aliased signal a at the probe location
coil	ρ₁	ρ₂	a_c

A · SELECTED COMPLEX COIL MAPCOIL 1 · |C₁(r)| + PHASE CONTOURS

larger |C_c|curves · phase

A coil has smooth sensitivity across the slice; it does not illuminate one isolated region like a flashlight.

B · SELECTED COIL AFTER k_y SKIPPINGR2 · TWO POSITIONS SUPERIMPOSED

The displayed phase FOV is half-sized and stretched here for comparison. This is coherent folding from increased Δk_y, not truncation blur.

C · SENSE UNFOLDED MAGNITUDENOISELESS SOLVE · ERROR < 10⁻⁹

The clean result proves algebraic separation in the ideal model. It does not erase the statistical noise cost shown beside it.

D · GEOMETRY-FACTOR MAPg MEAN 1.24 · PEAK 1.35

g = 1g ≥ 1.35

Dark means little extra geometry penalty; yellow marks less distinguishable sensitivity rows and larger noise amplification. Printed values keep the map readable without color.

01 · PHYSICALN loops surround the slice.

Nearby transverse magnetization induces a different complex voltage in each receive element.

02 · ENCODINGEvery Rth phase line is retained.

Δk_y grows by R, nominal phase encoding time falls to 1/R, and R positions fold together.

03 · INVERSIONComplex sensitivity rows separate the fold.

SENSE solves one local N × R system at every reduced-FOV image location.

04 · CONSEQUENCEFewer samples and conditioning reduce SNR.

√R is global; g varies spatially. More coils do not guarantee low g unless their maps add distinct information.

MODEL BOUNDARY · SYNTHETIC IMAGE-DOMAIN SENSE, NOT A SCANNER PRESCRIPTION

The slice and loop positions are schematic. C_c(r) uses smooth analytic complex sensitivity proxies—not measured B₁⁻ maps, loaded-coil electromagnetic simulation, or patient-specific calibration. The 3D sensitivity and channel surfaces use magnitude as calibrated height plus redundant normalized lightness, while wrapped complex phase controls cyclic hue; folded planes, phasor rows, solved bars, and g terrain are explicitly mathematical insets rather than physical structures in a patient. The g terrain is linear in g−1 at 0.72 world unit per g unit and caps at g = 5 for framing; exact mean, peak, local values, and the number of capped samples remain printed. The √R bar has its own scale and is never folded into the g height. Ψ uses equal channel variance and one shared correlation value. The forward model is ideal Cartesian, single-direction uniform phase undersampling with known maps and a noiseless displayed solve; g reports the predicted statistical noise amplification. It omits map-estimation error, coupling, motion, partial Fourier, regularization, filtering, EPI distortion, simultaneous multi-slice, compressed sensing, and k-space methods such as GRAPPA. A one-channel R>1 system is deliberately shown as unsolvable.

PRIMARY PHYSICS SOURCES USED FOR THIS MODEL Roemer et al. · 1990The NMR phased array↗ Pruessmann et al. · 1999SENSE: Sensitivity encoding for fast MRI↗

10 / THE COMPLETE CHAIN

From current waveform to one image voxel.

01
Gradient current
Amplifiers drive X, Y, Z coil windings. Current and geometry create a controlled field slope in T/m.
I(t) → G(t)
02
Spin phase
The local field changes Larmor frequency. Accumulated phase records the gradient’s area through time.
φ(r,t) = γ r·∫G dt
03
k-space sample
The receiver sums every transverse spin at the current spatial-frequency coordinate.
k(t) = γ̄∫G dt
04
Image estimate
An inverse Fourier transform separates the superposed spatial frequencies into locations.
ρ̂(r) = ℱ⁻¹{S(k)}

REFERENCE DESK

Continue into the physics.

Clinical overviewK-space in the clinic↗ Interactive-teaching reviewK-space tutorial and image consequences↗ Open reviewCMR physics for clinicians↗ ISMRM courseSpatial encoding in MRI↗ Trajectory reviewEPI and spiral imaging↗ Signal derivationGradient phase and k-space↗ Cartesian acquisitionPrephasing, readout, and ADC trajectories↗ Oblique coordinatesLogical-to-physical gradient rotation↗ RF coil reviewBody transmit and local receive↗ Relaxation modelSpoiled GRE steady-state signal↗ Stationary-state theoryRF- and gradient-spoiled magnetization↗ Experimental spoiling limitsIdeal theory versus incomplete spoiling↗ Slice-selection modelRF bandwidth, Gz, and slice position↗ Artifact reviewTruncation, aliasing, and filtering↗

Choose a lesson.

Connect acquisition choices to voxel size, signal, folding, parallel imaging, blur, and ringing.

Shape the voxel
in all three axes.

Thicker slab

Thinner slab

Move the slice

Change k-space.
The image answers.

Truncate extent

Increase spacing

Apodize

When slices overlap, the neighbours change what this slice can signal.

Skip phase lines.
Let the coil array help unfold them.

A physical coil array supplies spatially distinct complex measurements.

The selected image location is a vector of coil measurements—not one brightness.

From current waveform to one image voxel.

Gradient current

Spin phase

k-space sample

Image estimate

Continue into the physics.

Formula explanation

Separate commands, physical quantities, and calculated results

Symbols & units

If one quantity changes

Every abbreviation, prefix, and conversion

Change one number

What this view shows

Controls that reveal the relationship

Decode every symbol.

Readout

Explain it as if I am 10

Start with a field.

Connect acquisition choices to voxel size, signal, folding, parallel imaging, blur, and ringing.

Shape the voxelin all three axes.

Thicker slab

Thinner slab

Move the slice

Change k-space.The image answers.

Truncate extent

Increase spacing

Apodize

Skip phase lines.Let the coil array help unfold them.

From current waveform to one image voxel.

Gradient current

Spin phase

k-space sample

Image estimate

Continue into the physics.

Shape the voxel
in all three axes.

Change k-space.
The image answers.

Skip phase lines.
Let the coil array help unfold them.