From 9a884e7c428115244335cc949efb674b6202986f Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Tue, 28 Jul 2026 11:19:24 -0500 Subject: [PATCH 1/2] :memo: Add a theory note on phase offset ambiguities Derives what the unknown turn count left by spatial unwrapping does to the MCPC-3D-S offset estimate, why resolving it per frame is unstable when the bulk field approaches the half-wrap, and how a mismatched offset/field pair becomes detectable. Documents the two-stage selector MEDIC uses and the requirements on the field estimator. Theory only; no measured results. --- notes/phase-offset-ambiguities.pdf | Bin 0 -> 136871 bytes notes/phase-offset-ambiguities.typ | 428 +++++++++++++++++++++++++++++ 2 files changed, 428 insertions(+) create mode 100644 notes/phase-offset-ambiguities.pdf create mode 100644 notes/phase-offset-ambiguities.typ diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d1e6af361ef3ae6d9f47089dc86f96c7014d4a26 GIT binary patch literal 136871 zcmeFa2UrwK(>5&TgkZvivMOd-HtzxkqKJwjVgyu#6;YA|5p&KtVFEEDCREHhXGKLZ zqnL9SVIrht4)v4z~pdaC^?BxEa(yz8W|KWh=>YNhqDAR2O5It8D4^< zsXj64Ap*=WOIsrF2p$k6h>Z>k4vP$BY2%wTw9(^+Xk&ufjg<;~)xl9i)GHdBc`<}b757%ufF!U7R;@Eo3I!Fld#Ys2>v9Dk-oN*GHeNa1_Z@2 zRSN+Bu!tb7&cKgpTA|cy1*w^Is1$m@4>*a54_6C(J9&F_5(Gs@hliWEf8aE zAi#1tl?l2JNW{H{dF>w*79J~ziWI~~MW_Y+V}c?$EaT@i6=IS`YQ*ACY9#yt7BpXaZxOJ9SeY;*FlWib|Uel z?Mk5px1Ne}R18a0#|mKXb%=75Urf4z2%MPIfF+JHLq$$PMNTR3pt?hqxqp~CJcO-V 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Every voxel also carries a receive-coil phase offset, +present at zero echo time, common to all echoes, and set by coil geometry rather +than by the field. This offset varies rapidly in space, steeply near coil +elements and discontinuously between receive channels, so measured phase is not +spatially smooth even where the field is. Spatial unwrapping assumes that +neighbouring voxels differ by less than half a turn, and raw phase violates that +assumption. Removing the offset restores the smoothness later processing needs. + +The offset is never measured. It must be inferred from the same wrapped phase +that it corrupts, and the inference leaves one integer undetermined. This note +sets out the consequences of that integer, the rule conventionally used to fix +it, the conditions under which the rule fails, and the procedure MEDIC uses to +identify and repair those cases. + += Phase offset estimation + +#defn[ +*Notation.* A quantity is *wrapped* when confined to $(-pi, pi]$. + +#table( + columns: (auto, auto, 1fr), + stroke: none, + inset: (x: 4pt, y: 2.5pt), + [$theta$, $f$], [], [coil offset (rad) and field offset (Hz)], + [$phi_e$], [wrapped], [the recorded phase at echo $e$], + [$Delta phi$], [wrapped], [the wrapped phase difference $W(phi_1 - phi_0)$], + [#phiuw], [unwrapped], [that difference with its turn count restored], + [$M$], [integer], [the unwrapper's error in whole turns], + [$hat(theta)$], [wrapped], [the offset estimate], + [$psi_e$], [neither], [recorded phase less an estimated offset], +) +] + +== Signal model + +Acquire $E$ echoes at times $t_0 < t_1 < ... < t_(E-1)$. Field evolution is +linear in time, so the phase at echo $e$ before wrapping is +$theta + 2 pi f t_e$. What is recorded is that value wrapped into one turn. With +$W(x) = arg(e^(i x))$, + +$ + phi_e = W(theta + 2 pi f t_e). +$ + +Without $W$, two echoes would determine $theta$ and $f$ by a straight-line fit. +With it the line is known only modulo $2 pi$, and infinitely many $(theta, f)$ +pairs remain compatible with the data. + +Two properties of $W$ are used below. It is invariant under whole turns, +$W(x + 2 pi m) = W(x)$ for integer $m$, so every real $x$ decomposes uniquely as +$x = W(x) + 2 pi m$. Wrapping discards $m$ and unwrapping recovers it. Second, a +difference of wrapped values returns the wrapped difference only after a second +wrap, + +$ + W(W(x) - W(y)) = W(x - y), +$ + +because $W(x) - W(y)$ spans $(-2 pi, 2 pi)$ and coincides with $W(x - y)$ only +when it happens to lie within one turn. + +== Echo spacing + +An echo-planar train produces echoes at constant spacing, + +$ + t_e = "TE"_0 + e dot #dte, quad e = 0, 1, ..., E-1, +$ + +with $"TE"_0$ the first echo time and #dte the spacing. Two field scales follow +from #dte alone. The *wrap spacing* $1 slash #dte$ separates neighbouring +candidate solutions for the field. The *half-wrap* $1 slash (2 #dte)$ lies +midway between them, and serves as the decision boundary for any rule that +prefers the smallest field. + +== Recovering the offset + +The construction below is MCPC-3D-S. Because $theta$ is common to every echo, it +cancels in a difference. For the first two echoes, + +$ + Delta phi = W(phi_1 - phi_0) + = W((theta + 2 pi f t_1) - (theta + 2 pi f t_0)) + = W(2 pi f #dte). +$ + +The middle step follows from @eq:diffid, since each $phi_e$ differs from +$theta + 2 pi f t_e$ by whole turns. The wrap remains outside the subtraction +because $phi_1 - phi_0$ is a difference of wrapped values. + +$Delta phi$ is itself wrapped, but a difference of neighbouring echoes varies +slowly in space, so a region-growing algorithm can restore its missing turns. +Spatial unwrapping recovers phase differences between voxels rather than +absolute turn counts, so what it returns differs from the true unwrapped +difference by an unknown whole number of them: + +$ + #phiuw = 2 pi f #dte + 2 pi M, +$ + +with $M = 0$ when the unwrapper is correct. Extrapolating to $t = 0$ then +recovers the offset. Writing $k = "TE"_0 slash #dte$, + +$ + hat(theta) = W(phi_0 - k dot #phiuw). +$ + +The outer $W$ is required because #phiuw is unbounded and $k dot #phiuw$ with +it. $M$ is the only quantity in @eq:est that the data leaves open. + += The half-wrap limit + +Substituting @eq:uw into @eq:est, and reading the field from the same +difference, gives the solution implied by any $M$: + +$ + hat(theta)_M = W(theta - 2 pi k M), + quad quad + hat(f)_M = f + M / (#dte). +$ + +The two errors are locked together. The field moves by $M$ whole wrap spacings +and the offset by $-2 pi k M$, and neither occurs without the other. The +candidate solutions form a discrete family, one per integer, separated by +$1 slash #dte$ in field. + +#defn[ +*Proposition 1.* For evenly spaced echoes and any integer $M$, the pair +$(hat(theta)_M, hat(f)_M)$ reproduces @eq:model identically: +$W(hat(theta)_M + 2 pi hat(f)_M t_e) = phi_e$ at every echo. +] + +#proof[ +*Proof.* Working modulo $2 pi$, so that $W$ of either side agrees, +$ + hat(theta)_M + 2 pi hat(f)_M t_e + &equiv theta - 2 pi k M + 2 pi (f + M slash #dte) t_e \ + &= theta + 2 pi f t_e + 2 pi M (t_e slash #dte - k). +$ +By @eq:spacing, $t_e slash #dte = k + e$, so the parenthesis is the integer $e$, +the final term is a whole number of turns, and both sides wrap alike. +#h(1fr) $qed$ +] + +The cancellation depends on the even spacing. Written as +$t_e slash #dte = k + e$, the ratio $k$ locates the echo train on the #dte +lattice, and the $-2 pi k M$ carried by the offset is what cancels it. Since the +cancellation holds at every echo simultaneously, a longer echo train places no +further constraint on $M$. Unequal spacing would leave $t_e slash #dte - k$ +non-integral at some echo and so produce a detectable residue, but an EPI train +is evenly spaced by construction. + +== The rule ROMEO applies + +No measurement selects $M$, so the choice must come from elsewhere. ROMEO's +global correction, which MEDIC calls, subtracts a single multiple of $2 pi$ +from the unwrapped phase so that the median rounded turn count over the mask +becomes zero. The effect is to select the candidate whose field is smallest, + +$ + hat(M) = op("argmin")_M abs(f + M slash #dte), +$ + +which is equivalent to requiring $abs(hat(f)) < 1 slash (2 #dte)$. + +The rule encodes a physical expectation rather than a measurement. Before +imaging, the scanner sets its demodulation frequency to the centre of the water +resonance, so the residual field is a small offset rather than an arbitrary one. +Where that expectation holds, the rule returns the correct solution. Where it +does not, the rule returns an alias, and by +#link()[Prop. 1] nothing in the data distinguishes the two. + += Why the rule fails on some frames + +A wrong value of $M$, chosen once and applied to a whole acquisition, would +matter very little. Every frame would be displaced by the same $1 slash #dte$. +The field offset is already referenced to an arbitrary demodulation frequency, +so a fixed additive constant changes neither temporal contrast nor spatial +structure, and by #link()[Prop. 1] it could not be detected in any +case. + +The difficulty is that $M$ is not chosen once. Three properties of the procedure +combine to produce a failure. + +The first is that the choice is repeated on every frame. A functional +acquisition is processed frame by frame, and each frame resolves $M$ from its +own data, without reference to the frames around it. + +The second is that the quantity compared against the half-wrap is an estimate. A +median rounded turn count is a location statistic over a mask of voxels, each +carrying its own field, and it absorbs whatever varies between frames, including +physiological change, subject motion, and noise in the voxels the mask admits. + +The third is that the comparison has a hard boundary. Far from the half-wrap the +margin is wide, and no plausible perturbation of the estimate alters the +outcome. Close to the half-wrap the margin approaches zero. + +For a subject whose field lies near the half-wrap, $M$ is therefore decided on +each frame independently, by a noisy statistic sitting close to a threshold, and +different frames fall on different sides of it. Because $M$ is an integer, a +frame that falls on the wrong side is displaced by a full wrap rather than by a +small amount. The resulting time series is stable to a fraction of a hertz +across most frames and displaced by $1 slash #dte$ on a scattered subset, with +no intermediate values. + +The damage is temporal. A uniformly wrong field is harmless, but a field that +takes two values within a single acquisition corrupts the temporal structure the +acquisition was performed to measure. + += Detecting a failed choice + +#link()[Prop. 1] appears to rule out any correction. It applies, +however, to a *matched* pair, meaning an offset $hat(theta)_M$ together with the +field $hat(f)_M$ that accompanies it. The pipeline does not produce the pair in +a single step. The offset is formed first, from #phiuw by @eq:est. The field is +derived afterwards, by removing that offset and unwrapping in time, and the +second unwrapping applies a global correction of its own without reference to +the $M$ the offset assumed. + +When the two stages disagree, the result falls outside the family in @eq:pair. +The offset belongs to one candidate and the field to another, so +#link()[Prop. 1] does not apply to it. This is the situation on a +frame where the rule has tipped, because the two stages use different masks and +different statistics and can land on opposite sides of the boundary. + +Removing the offset from the recorded phase leaves + +$ + psi_e = phi_e - hat(theta)_M, +$ + +a difference of two wrapped values, and therefore not itself a phase. +Unwrapping it in time and fitting + +$ + psi_e approx c + s dot t_e +$ + +gives a slope $s$, which is the field, and an intercept $c$, the value the line +extrapolates to at $t = 0$. A self-consistent reconstruction has had its offset +removed, so its line passes through the origin and $c = 0$. + +#defn[ +*Proposition 2.* If the fit @eq:fit is taken on the solution whose slope is the +true field $f$, then +$ + psi_e equiv 2 pi f t_e + c mod 2 pi, + quad quad c = W(2 pi k M), +$ +with $c$ identical at every echo. +] + +#proof[ +*Proof.* $psi_e equiv (theta + 2 pi f t_e) - (theta - 2 pi k M) += 2 pi f t_e + 2 pi k M$, and $2 pi k M equiv c mod 2 pi$. The constant does not +depend on $e$. #h(1fr) $qed$ +] + +Two properties make $c$ usable. Being identical at every echo, it displaces the +whole fitted line rather than averaging away across echoes. Being a single +global constant rather than a per-voxel effect, it can be aggregated over a mask +without dilution. + +The intercept measures neither the error in the offset nor the error in the +field. It measures whether the two are mutually consistent, which is the +property the data can check. Its limitation follows from +#link()[Prop. 1]: a matched pair gives $c = 0$, so the intercept +cannot rank it against the correct solution. Where more than one candidate +remains consistent, the data has been exhausted. + += The selection algorithm + +The procedure runs once per frame. Its output is an integer correction $N$ +applied to #phiuw before the offset of @eq:est is formed, leaving a residual +error of $M + N$, and the aim is $N = -M$. + +#defn[ +*Step 1.* Consider $N in {-1, 0, +1}$. A single step is a full wrap of global +field, and the global correction has already brought the field inside the +half-wrap, so a correction of more than one wrap is not reachable. + +*Step 2.* For each $N$, shift #phiuw by $2 pi N$, form the implied offset by +@eq:est, remove it from both echoes, unwrap in time, fit @eq:fit, and record +$abs(c)$ aggregated over a conservative brain mask. + +*Step 3.* Discard any candidate whose $abs(c)$ exceeds a cutoff. By +#link()[Prop. 2], neighbouring candidates are separated in +intercept by $abs(W(2 pi k))$, so the cutoff is placed at half that separation, +which makes the test a nearest-neighbour assignment between two possibilities +rather than a tuned threshold. The implementation also measures the largest +intercept observed among the candidates and adopts the stricter of the two +scales, which keeps the test honest when the second unwrapping has already +absorbed part of the residue. + +*Step 4.* If one candidate survives, take it, whatever its field. If none +survives, change nothing, since a failed fit is not evidence for any particular +candidate. If several survive, they are indistinguishable to the data, and +@eq:argmin selects the one with the smallest estimated $abs(hat(f))$. +] + +The two stages answer different questions, and neither suffices alone. Step 3 +uses evidence and rejects candidates the data contradicts. Step 4 uses an +assumption, and applies it only among candidates the data has already accepted, +which prevents the assumption from overriding the evidence. + +Step 4 applies the same rule as @eq:argmin, so it is worth stating what it +changes. The global correction takes an unweighted median of *rounded* turn +counts over a dilated mask. Rounding discards the sub-wrap information that +indicates how close the decision is, and dilation admits edge voxels whose +fields are extreme. Step 4 estimates the field itself, weighted, over a +conservative mask. The rule is unchanged; the statistic it is applied to is not. + +One degenerate case deserves mention. If $k$ is a whole number then +$W(2 pi k M) = 0$ for every $M$, so the intercept carries no information. +@eq:pair then gives $hat(theta)_M = W(theta)$ regardless of $M$, so no choice +alters the output. The implementation detects the zero separation and makes no +change, which is the correct behaviour. + += Estimating the field + +Step 4 compares $abs(hat(f))$ against the half-wrap, so the reliability of the +comparison depends on what that estimate measures and on how steady it is +between frames. + +== What the rule refers to + +@eq:argmin is a statement about *bulk tissue*, since bulk tissue is what the +scanner's frequency adjustment centres. It is not a statement about the mean +field, nor about the median field over whatever mask happens to be available. + +The field distribution across a brain is asymmetric. It has a narrow peak at the +bulk-tissue value together with long tails from air-tissue interfaces, sinuses, +signal dropout and mask edges, where susceptibility gradients are large. Any +quantile of the whole distribution is displaced toward whichever tail is +heavier. Writing the distribution as a mixture, + +$ + p(f) = (1 - epsilon) dot p_"bulk" (f) + epsilon dot p_"tail" (f), +$ + +the rule refers to the centre of $p_"bulk"$, and the estimator should recover it +with $epsilon$ as small as possible. Two consequences follow. + +Enlarging the mask makes the estimate worse. Dilating a brain mask admits +precisely the voxels that populate $p_"tail"$, which raises $epsilon$ and +displaces any quantile of $p$ further from the bulk centre. This reverses the +usual expectation that more data improves an estimate. + +A bias of a fraction of a wrap is enough to change the decision. The comparison +is against the half-wrap, so estimator bias adds directly to the apparent +distance from the boundary. A subject whose bulk field already lies near the +half-wrap can be carried across it by bias alone, at which point the decision +reflects the statistic rather than the data. + +== The estimator + +MEDIC uses a magnitude-weighted median over an eroded brain mask. Erosion +lowers $epsilon$ geometrically by removing edge voxels, and the weighting lowers +it by suppressing low-signal voxels, which are over-represented in the tails. +The weight is the magnitude of the second echo, because the field is read from a +difference whose noise is dominated by the weaker of the two. + +A median is used rather than a mean or an M-estimator. The bulk distribution is +broad and right-skewed, and the half-wrap boundary falls within its body rather +than out in a tail, so a statistic pulled toward the mean reads systematically +closer to the boundary without describing bulk tissue any more accurately. + +Continuity is the binding requirement. The estimate decides a threshold +comparison that has to give the same answer on consecutive frames of one +subject, and an unstable estimator would reintroduce at this stage exactly the +inconsistency described in section 4. A median moves smoothly as the samples +move. Estimators defined by an $arg max$, including modes in any of their usual +forms, do not, since field distributions are frequently multimodal and a +near-tie between peaks makes the estimate jump between frames. Stability across +frames matters more here than freedom from bias. + From d94ad0b7d90abec243727bd2723f3b03497b8ecb Mon Sep 17 00:00:00 2001 From: Andrew Van Date: Tue, 28 Jul 2026 13:41:40 -0500 Subject: [PATCH 2/2] :memo: Clarify and correct the phase offset ambiguities note - Replace the appendix's "plot phase against echo time" prose with a drawn figure of the candidate pencil, and state the pivot at TE0 as a consequence of the proof rather than something the reader must picture. - Add section 8.1 giving what the unwrapper, the two masks, and the weighted median must provide, as properties rather than settings, so the spec is self-contained without describing any particular implementation. - Correct the delta = 0 guard: an exact float comparison never fires, so specify a tolerance test. - Drop the even-sized-median qualification, which was opaque and whose closing claim about the detector was wrong (for k near 0.58-0.74 it accepts a single wrong candidate rather than rejecting all of them). - Remove the test-fixture reference and its two mistranscribed voxel counts so the note stands on its own. - Add a units statement, sharpen the second unwrap's description, condense the recap that restated sections 6 and 7, and shorten the reference-data block. - Layout: let the appendix flow instead of forcing a pagebreak, and keep the reference-data box unbreakable; 11 pages down to 10. --- notes/phase-offset-ambiguities.pdf | Bin 136871 -> 219014 bytes notes/phase-offset-ambiguities.typ | 730 +++++++++++++++++++++-------- 2 files changed, 524 insertions(+), 206 deletions(-) diff --git a/notes/phase-offset-ambiguities.pdf b/notes/phase-offset-ambiguities.pdf index d1e6af361ef3ae6d9f47089dc86f96c7014d4a26..41fd7bc6f7287d9c6514ca66a7e1cc618e4421de 100644 GIT binary patch literal 219014 zcmeEv2b@$z)-ORpSVctv18S=vC^9qM_jVXmBnOo=L`BeHXkcU#Gd&6_qJUx+F<}HT z0qVMp`~BYg-s5 zO;5~3L@&96iZdnYw$VuMvP@<9=<1?Oae3K^x!uxLcpxDjvEdPXH#`YLfD=eCxL8g! 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z1jJRC2C@O#b6~(om$E(lytBkROu1dE1!0gHk8 z!y@a@uu$v)9Wp3a0W=7QK-Ubm7c3FNT_7qbt|B-T6n_yM=xWi*6trX@`@;cGhsI%{ zoCb73pqvI-r+0_|wL!LlBSN*oS}O?h4^ZzzJmb+2?t&~E_I*4Sif?P?0`Gx`Vkk)e z;IrZ3{1kMRVO@ea4cQu~*&&${Km?|hGXY2eqQQ9+NJXLA03qOEof8PKOhK6k$$$X& zt`Q;%wgX_25PzUm1>phM0@53RQ^*d8z))$=ON47Wgb3F{$U5~o$kt#yAd;Z`3L&AO zz5sv&I39tv1#~tN0^2hQ3&%zhj!2u+fG#lXgCqis8zjIWtz1AVMJp>Ti2Li%AbZAw zlz~=npb~<5H=ydq(dr1aPa%5&83S|=0iYA=DPU1>ZHh&apj;2oAY8+O8k^RJu;8AA z*0(@*1?60T2Gx=P!PcPxfT58yXbnUDi(Tg~0fYr87sY}an^xx_xU4dKxvKI{OM_7<3LgPSkMr*rR0MF3A zfI&#VU_sg#j|FWmNTwjxLUxJ;f3&&;uSGo&eA{NwJ&^bWM9Fi3YNS~G$5+3#i5}bp8ZX%R-;!r3E$8n&J zqSXh8T2PG$u3ezq8?;a$-vXWr)f%941K|+p%0syT=o~_KkZUa&S{uiqaS#>*8YEL3 zxCVpg;-Op_jDydCCO~!tu9=`*8Ha{@WPk?c@;Ee{j{_PMYjL2lOlt>#2IWKGh6;-3 zfClwO01fgZ9B5=hItM*aT3Z9@Gb{tp6olu(m<`%yunmD`8ZF-Az|{_XUi>;d!*_-_ zP`}ao5lHW$d&O6)gXMxJLU@1!SFX?=pqEA)Q$V!<`xa<#K<6dE{T$FOf$$I9B0{+eph0;F4qUs? z?g4I&p}GeT!YZv?z)cp!GXV?Vy%RtkN~=q7GziCuK+d#wPXgr!4Q=g77uwtrbaOxv z_HTR!SsfIkz+;>+4fL~Vb7hcNAhh@lXargu2Q(6`?XGo<{;6Lo#oo<>Oko9&E*ZIn zkk_6^lKuTG)bFP because $W(x) - W(y)$ spans $(-2 pi, 2 pi)$ and coincides with $W(x - y)$ only -when it happens to lie within one turn. +when it happens to land in $(-pi, pi]$ already. == Echo spacing @@ -121,9 +128,10 @@ $ with $"TE"_0$ the first echo time and #dte the spacing. Two field scales follow from #dte alone. The *wrap spacing* $1 slash #dte$ separates neighbouring -candidate solutions for the field. The *half-wrap* $1 slash (2 #dte)$ lies -midway between them, and serves as the decision boundary for any rule that -prefers the smallest field. +candidate solutions for the field. The *half-wrap* $1 slash (2 #dte)$ is half +that spacing, the distance from any candidate to the midpoint between it and its +neighbour, and serves as the decision boundary for any rule that prefers the +smallest field. == Recovering the offset @@ -134,7 +142,7 @@ $ Delta phi = W(phi_1 - phi_0) = W((theta + 2 pi f t_1) - (theta + 2 pi f t_0)) = W(2 pi f #dte). -$ +$ The middle step follows from @eq:diffid, since each $phi_e$ differs from $theta + 2 pi f t_e$ by whole turns. The wrap remains outside the subtraction @@ -143,286 +151,596 @@ because $phi_1 - phi_0$ is a difference of wrapped values. $Delta phi$ is itself wrapped, but a difference of neighbouring echoes varies slowly in space, so a region-growing algorithm can restore its missing turns. Spatial unwrapping recovers phase differences between voxels rather than -absolute turn counts, so what it returns differs from the true unwrapped -difference by an unknown whole number of them: - -$ - #phiuw = 2 pi f #dte + 2 pi M, -$ - -with $M = 0$ when the unwrapper is correct. Extrapolating to $t = 0$ then -recovers the offset. Writing $k = "TE"_0 slash #dte$, +absolute turn counts, so what it returns may differ from the true unwrapped +difference by a whole number of turns. Write #phiuw for what it returns. +Extrapolating to $t = 0$ then recovers the offset, with +$k = "TE"_0 slash #dte$, $ hat(theta) = W(phi_0 - k dot #phiuw). $ The outer $W$ is required because #phiuw is unbounded and $k dot #phiuw$ with -it. $M$ is the only quantity in @eq:est that the data leaves open. +it. That unrecovered number of turns is the only thing @eq:est leaves open. -= The half-wrap limit += The candidate ladder -Substituting @eq:uw into @eq:est, and reading the field from the same -difference, gives the solution implied by any $M$: +The turn count in #phiuw is unknown, so consider every value it could take. +Shifting what the unwrapper returned by whole turns, $ - hat(theta)_M = W(theta - 2 pi k M), - quad quad - hat(f)_M = f + M / (#dte). -$ + #phiuw^((N)) = #phiuw + 2 pi N, quad N in ZZ, +$ -The two errors are locked together. The field moves by $M$ whole wrap spacings -and the offset by $-2 pi k M$, and neither occurs without the other. The -candidate solutions form a discrete family, one per integer, separated by -$1 slash #dte$ in field. +produces one *candidate* per integer, with $N = 0$ the value the unwrapper +(ROMEO) actually returned. A candidate is not a partial answer. Each one fixes an +offset and a field together, and both follow from what has already been +established. -#defn[ -*Proposition 1.* For evenly spaced echoes and any integer $M$, the pair -$(hat(theta)_M, hat(f)_M)$ reproduces @eq:model identically: -$W(hat(theta)_M + 2 pi hat(f)_M t_e) = phi_e$ at every echo. -] +The offset comes from putting @eq:shift into @eq:est. The shift enters multiplied +by $k$, so -#proof[ -*Proof.* Working modulo $2 pi$, so that $W$ of either side agrees, $ - hat(theta)_M + 2 pi hat(f)_M t_e - &equiv theta - 2 pi k M + 2 pi (f + M slash #dte) t_e \ - &= theta + 2 pi f t_e + 2 pi M (t_e slash #dte - k). + hat(theta)_N = W(phi_0 - k(#phiuw + 2 pi N)) = W(hat(theta)_0 - 2 pi k N). +$ + +The field is the slope of the unwrapped difference, phase gained per unit time +over the interval #dte, so + $ -By @eq:spacing, $t_e slash #dte = k + e$, so the parenthesis is the integer $e$, -the final term is a whole number of turns, and both sides wrap alike. -#h(1fr) $qed$ -] + hat(f)_N = (#phiuw^((N))) / (2 pi #dte) = hat(f)_0 + N / (#dte). +$ + +One candidate is correct. Write $N^*$ for its index, so that +$hat(theta)_(N^*) equiv theta$ and $hat(f)_(N^*) = f$. Nothing observable +identifies $N^*$, which is the whole difficulty. + +Two features of @eq:offn and @eq:fldn matter later. The first is that a single +integer fixes both quantities. There is no candidate carrying the correct field +alongside an incorrect offset, or the reverse: they move together, the field in +steps of $1 slash #dte$ and the offset in steps of $2 pi k$. The second is that +the candidates are *discrete*. The field is not uncertain within an interval; it +is determined up to a set of isolated values spaced $1 slash #dte$ apart, with +nothing in between available as an answer. Whatever goes wrong later is therefore +wrong by a whole wrap, never by a little. -The cancellation depends on the even spacing. Written as -$t_e slash #dte = k + e$, the ratio $k$ locates the echo train on the #dte -lattice, and the $-2 pi k M$ carried by the offset is what cancels it. Since the -cancellation holds at every echo simultaneously, a longer echo train places no -further constraint on $M$. Unequal spacing would leave $t_e slash #dte - k$ -non-integral at some echo and so produce a detectable residue, but an EPI train -is evenly spaced by construction. +== The rule ROMEO applies -== The rule ROMEO applies +One candidate has to be chosen, and ROMEO's global correction is what chooses it. +After spatially unwrapping, it subtracts $2 pi$ times the median rounded turn +count over the mask, which leaves that median at zero, -No measurement selects $M$, so the choice must come from elsewhere. ROMEO's -global correction, which MEDIC calls, subtracts a single multiple of $2 pi$ -from the unwrapped phase so that the median rounded turn count over the mask -becomes zero. The effect is to select the candidate whose field is smallest, +$ + "median" round(#phiuw slash 2 pi) = 0. +$ + +That condition reads directly as a statement about the field. By @eq:fldn a turn +of the difference is $1 slash #dte$ of field, so rounding the turn count to zero +is the same as requiring $ - hat(M) = op("argmin")_M abs(f + M slash #dte), -$ + abs(hat(f)_0) < 1 / (2 #dte), +$ + +the field lying inside the *half-wrap*. Because neighbouring candidates are a +full wrap apart, exactly one of them satisfies @eq:half, so the rule selects the +candidate of smallest field magnitude: the rung of the ladder nearest zero. +@eq:cg tests that condition through one particular statistic, an unweighted +median of rounded turn counts over its own mask; which summary of the field is +the right one is taken up in @sec:field. + +@eq:half is an assumption rather than a measurement, and its justification is +physical. Before imaging, the scanner sets its demodulation frequency to the +centre of the water resonance, so the residual field is expected to be a small +offset rather than an arbitrary one. -which is equivalent to requiring $abs(hat(f)) < 1 slash (2 #dte)$. +Where that expectation holds the rule returns $N^*$. Where it does not, it returns +a neighbouring rung instead, and nothing in the data marks the difference. -The rule encodes a physical expectation rather than a measurement. Before -imaging, the scanner sets its demodulation frequency to the centre of the water -resonance, so the residual field is a small offset rather than an arbitrary one. -Where that expectation holds, the rule returns the correct solution. Where it -does not, the rule returns an alias, and by -#link()[Prop. 1] nothing in the data distinguishes the two. +#defn[ +*Proposition 1.* For evenly spaced echoes, every candidate predicts the same +recorded phase. That is, $W(hat(theta)_N + 2 pi hat(f)_N t_e)$ does not depend on +$N$, at any echo. +] -= Why the rule fails on some frames +The rule is therefore forced rather than merely convenient: no statistic computed +from the phase can rank the candidates. Proved in #link()[Appendix A], +which draws out the consequences and sets out what the result does not cover. -A wrong value of $M$, chosen once and applied to a whole acquisition, would -matter very little. Every frame would be displaced by the same $1 slash #dte$. -The field offset is already referenced to an arbitrary demodulation frequency, -so a fixed additive constant changes neither temporal contrast nor spatial -structure, and by #link()[Prop. 1] it could not be detected in any -case. += Why the rule fails on some frames -The difficulty is that $M$ is not chosen once. Three properties of the procedure +A wrong candidate, chosen once and applied to a whole acquisition, would matter +very little. Every frame would be displaced by the same $1 slash #dte$ of field, and the +field offset is already referenced to an arbitrary demodulation frequency, so a +fixed additive constant changes neither temporal contrast nor spatial structure. + +The difficulty is that the choice is not made once. Three properties of @eq:cg combine to produce a failure. -The first is that the choice is repeated on every frame. A functional -acquisition is processed frame by frame, and each frame resolves $M$ from its -own data, without reference to the frames around it. - -The second is that the quantity compared against the half-wrap is an estimate. A -median rounded turn count is a location statistic over a mask of voxels, each -carrying its own field, and it absorbs whatever varies between frames, including -physiological change, subject motion, and noise in the voxels the mask admits. - -The third is that the comparison has a hard boundary. Far from the half-wrap the -margin is wide, and no plausible perturbation of the estimate alters the -outcome. Close to the half-wrap the margin approaches zero. - -For a subject whose field lies near the half-wrap, $M$ is therefore decided on -each frame independently, by a noisy statistic sitting close to a threshold, and -different frames fall on different sides of it. Because $M$ is an integer, a -frame that falls on the wrong side is displaced by a full wrap rather than by a -small amount. The resulting time series is stable to a fraction of a hertz -across most frames and displaced by $1 slash #dte$ on a scattered subset, with -no intermediate values. - -The damage is temporal. A uniformly wrong field is harmless, but a field that -takes two values within a single acquisition corrupts the temporal structure the +The first is that it is evaluated on every frame. A functional acquisition is +processed frame by frame, and each frame applies @eq:cg to its own data, without +reference to the frames around it. + +The second is that the left side of @eq:cg is a sample median over a mask of +voxels, each carrying its own field. It absorbs whatever varies between frames, +including physiological change, subject motion, and noise in the voxels the mask +admits. + +The third is that @eq:half is a comparison against a fixed boundary. When +$abs(hat(f)_0)$ is far below $1 slash (2 #dte)$ the margin is wide and no +plausible perturbation of the median changes which candidate satisfies the +inequality. As the field approaches the half-wrap that margin approaches zero. + +The practical consequence is that ROMEO's global correction is occasionally off by +a single turn. For a subject whose field lies near the half-wrap the candidate is +chosen on each frame independently, by a noisy median sitting close to a +threshold, so different frames fall on different sides of it. Because the +candidates are a full wrap apart, a frame that falls on the wrong side is +displaced by $1 slash #dte$ rather than by a small amount. The resulting time +series is stable to a fraction of a hertz across most frames and displaced by a +full wrap on a scattered subset, with no intermediate values. + +The damage is temporal. A uniformly wrong field is harmless, but a field taking +two values within a single acquisition corrupts the temporal structure the acquisition was performed to measure. -= Detecting a failed choice += Detecting a failed choice + +By #link()[Prop. 1] no test can rank one candidate against another. +The test in this section asks a different question: whether what the pipeline has +produced is a candidate at all. -#link()[Prop. 1] appears to rule out any correction. It applies, -however, to a *matched* pair, meaning an offset $hat(theta)_M$ together with the -field $hat(f)_M$ that accompanies it. The pipeline does not produce the pair in -a single step. The offset is formed first, from #phiuw by @eq:est. The field is -derived afterwards, by removing that offset and unwrapping in time, and the -second unwrapping applies a global correction of its own without reference to -the $M$ the offset assumed. +The question has content because the offset and the field come from two separate +steps. The first uses #phiuw to form the offset, by @eq:offn. The second removes +that offset from every echo, spatially unwraps the first echo — applying @eq:cg a +second time, now to offset-removed phase — and unwraps the remaining echoes +against it to recover the field. Nothing carries the first step's choice of shift +into the second. -When the two stages disagree, the result falls outside the family in @eq:pair. -The offset belongs to one candidate and the field to another, so -#link()[Prop. 1] does not apply to it. This is the situation on a -frame where the rule has tipped, because the two stages use different masks and -different statistics and can land on opposite sides of the boundary. +Consider a frame on which the first unwrapping has tipped, so the offset removed +belongs to a neighbouring rung rather than the correct one. The echoes still carry +the correct field, and the second unwrapping recovers it, because @eq:cg prefers +the small field and in offset-removed data the correct field is the small one. What +comes out therefore has an offset from one rung and a field from another. That +combination is not a rung of the ladder, so +#link()[Prop. 1] does not cover it, and it leaves a signature in the +data. -Removing the offset from the recorded phase leaves +== The signature +Removing an offset $hat(theta)_N$ from the recorded phase leaves + +$ + psi_e = phi_e - hat(theta)_N, $ - psi_e = phi_e - hat(theta)_M, -$ -a difference of two wrapped values, and therefore not itself a phase. -Unwrapping it in time and fitting +a difference of two wrapped values, and therefore not itself a phase. Unwrapping +it in the echo direction and fitting a line, $ - psi_e approx c + s dot t_e + psi_e approx c + s dot t_e, $ gives a slope $s$, which is the field, and an intercept $c$, the value the line -extrapolates to at $t = 0$. A self-consistent reconstruction has had its offset -removed, so its line passes through the origin and $c = 0$. +extrapolates to at $t = 0$. + +The intercept is the signature. Removing the offset that belongs to a given field +leaves phase that starts at zero, so a matched offset and field give a line +through the origin, $c = 0$. A mismatched pair does not, and by how much is fixed. #defn[ -*Proposition 2.* If the fit @eq:fit is taken on the solution whose slope is the -true field $f$, then +*Proposition 2.* Let the echo unwrap return the correct field, while the offset +removed was $hat(theta)_N$ for some $N != N^*$. Then $ psi_e equiv 2 pi f t_e + c mod 2 pi, - quad quad c = W(2 pi k M), + quad quad c = W(2 pi k (N - N^*)), $ with $c$ identical at every echo. ] #proof[ -*Proof.* $psi_e equiv (theta + 2 pi f t_e) - (theta - 2 pi k M) -= 2 pi f t_e + 2 pi k M$, and $2 pi k M equiv c mod 2 pi$. The constant does not -depend on $e$. #h(1fr) $qed$ +*Proof.* By @eq:offn, $hat(theta)_N equiv hat(theta)_(N^*) - 2 pi k (N - N^*) +equiv theta - 2 pi k (N - N^*)$. Then +$psi_e equiv (theta + 2 pi f t_e) - (theta - 2 pi k (N - N^*)) += 2 pi f t_e + 2 pi k (N - N^*)$, and the constant does not depend on $e$. +#h(1fr) $qed$ ] -Two properties make $c$ usable. Being identical at every echo, it displaces the -whole fitted line rather than averaging away across echoes. Being a single +Two properties make $c$ usable in practice. Being identical at every echo, it +displaces the whole fitted line rather than averaging away across echoes. Being one global constant rather than a per-voxel effect, it can be aggregated over a mask without dilution. -The intercept measures neither the error in the offset nor the error in the -field. It measures whether the two are mutually consistent, which is the -property the data can check. Its limitation follows from -#link()[Prop. 1]: a matched pair gives $c = 0$, so the intercept -cannot rank it against the correct solution. Where more than one candidate -remains consistent, the data has been exhausted. +== What the intercept can and cannot settle + +The intercept measures neither an error in the offset nor an error in the field. It +measures whether the two are mutually consistent, and consistency between them is +the one property the data can check. + +That makes it a filter rather than a ranking. Among the shifts of @eq:shift, those +with a large intercept are not candidates and can be discarded. Those with +$c approx 0$ are candidates, and #link()[Prop. 1] then applies in +full: they predict identical data and the intercept cannot separate them. Choosing +among the survivors still requires @eq:half. + +Both steps are needed, for different reasons. The filter is needed because a shift +can report a reasonable-looking field while carrying an offset that does not +generate it, and since that offset is subsequently removed from every echo, the +error propagates into everything downstream. The rule is needed because the filter +leaves more than one survivor. What the filter contributes is that when the first +unwrapping tips, the surviving set is not the same set, and applying the rule to +the shifted set recovers the frame. = The selection algorithm -The procedure runs once per frame. Its output is an integer correction $N$ -applied to #phiuw before the offset of @eq:est is formed, leaving a residual -error of $M + N$, and the aim is $N = -M$. +The procedure runs once per frame and returns one shift $N$ to apply to #phiuw. #defn[ -*Step 1.* Consider $N in {-1, 0, +1}$. A single step is a full wrap of global -field, and the global correction has already brought the field inside the -half-wrap, so a correction of more than one wrap is not reachable. +*Step 1.* Consider $N in {-1, 0, +1}$. One step is a full wrap of global field, +and @eq:cg has already brought the field inside the half-wrap, so a correction of +more than one wrap is not reachable. -*Step 2.* For each $N$, shift #phiuw by $2 pi N$, form the implied offset by -@eq:est, remove it from both echoes, unwrap in time, fit @eq:fit, and record -$abs(c)$ aggregated over a conservative brain mask. +*Step 2.* For each $N$, form $hat(theta)_N$ by @eq:offn, remove it from both +echoes, unwrap in time, fit @eq:fit, and record $abs(c)$ aggregated over a +conservative brain mask. *Step 3.* Discard any candidate whose $abs(c)$ exceeds a cutoff. By -#link()[Prop. 2], neighbouring candidates are separated in -intercept by $abs(W(2 pi k))$, so the cutoff is placed at half that separation, -which makes the test a nearest-neighbour assignment between two possibilities -rather than a tuned threshold. The implementation also measures the largest -intercept observed among the candidates and adopts the stricter of the two -scales, which keeps the test honest when the second unwrapping has already -absorbed part of the residue. +#link()[Prop. 2] the candidates of Step 1 have intercepts $0$ and +$plus.minus W(2 pi k)$, so the cutoff is placed at half that separation, making +the test a nearest-neighbour assignment between two possibilities rather than a +tuned threshold. The implementation also measures the largest intercept observed +among the candidates and adopts the stricter of the two scales, which keeps the +test honest when the echo unwrap has already absorbed part of the residue. *Step 4.* If one candidate survives, take it, whatever its field. If none -survives, change nothing, since a failed fit is not evidence for any particular -candidate. If several survive, they are indistinguishable to the data, and -@eq:argmin selects the one with the smallest estimated $abs(hat(f))$. +survives, change nothing, since a failed fit is not evidence for any candidate. +If several survive, they are indistinguishable to the data, and @eq:half decides +among them by smallest $abs(tilde(f)_N)$, the field the reconstruction actually +returned. ] -The two stages answer different questions, and neither suffices alone. Step 3 -uses evidence and rejects candidates the data contradicts. Step 4 uses an -assumption, and applies it only among candidates the data has already accepted, -which prevents the assumption from overriding the evidence. +Steps 3 and 4 answer different questions and neither suffices alone. Step 3 uses +evidence, rejecting candidates the data contradicts. Step 4 uses the assumption +of @eq:half, and applies it only among candidates the data has already accepted, +which is what stops the assumption from overriding the evidence. -Step 4 applies the same rule as @eq:argmin, so it is worth stating what it -changes. The global correction takes an unweighted median of *rounded* turn -counts over a dilated mask. Rounding discards the sub-wrap information that -indicates how close the decision is, and dilation admits edge voxels whose -fields are extreme. Step 4 estimates the field itself, weighted, over a -conservative mask. The rule is unchanged; the statistic it is applied to is not. +Step 4 restates @eq:half, so it is worth being clear about what it changes. +@eq:cg evaluates the condition as an unweighted median of *rounded* turn counts +over a dilated mask. Rounding discards the sub-wrap information that says how +close the decision was, and dilation admits edge voxels whose fields are extreme. +Step 4 evaluates $abs(tilde(f)_N)$ directly, weighted, over a conservative mask. +The condition is the same; the statistic is not, and @sec:field is about why that +matters. One degenerate case deserves mention. If $k$ is a whole number then -$W(2 pi k M) = 0$ for every $M$, so the intercept carries no information. -@eq:pair then gives $hat(theta)_M = W(theta)$ regardless of $M$, so no choice -alters the output. The implementation detects the zero separation and makes no -change, which is the correct behaviour. - -= Estimating the field - -Step 4 compares $abs(hat(f))$ against the half-wrap, so the reliability of the +$W(2 pi k (N - N^*)) = 0$ for every $N$, so the intercept carries no +information. By @eq:offn the offset $hat(theta)_N = W(hat(theta)_0)$ is then also +independent of $N$, so no choice alters the output, and leaving $N = 0$ is +correct. A guard on $delta = 0$ states that intent, but it need not be what +enforces it: in floating point $W(2 pi k)$ evaluates to a rounding residue near +$10^(-16)$ rather than to zero, and the cutoff that residue sets is then small +enough that every candidate fails it, so the empty-survivor case of Step 4 returns +$N = 0$ regardless. Either route gives the same answer. + += Estimating the field + +Step 4 compares $abs(tilde(f))$ against the half-wrap, so the reliability of the comparison depends on what that estimate measures and on how steady it is between frames. -== What the rule refers to - -@eq:argmin is a statement about *bulk tissue*, since bulk tissue is what the +@eq:half is a statement about *bulk tissue*, since bulk tissue is what the scanner's frequency adjustment centres. It is not a statement about the mean field, nor about the median field over whatever mask happens to be available. -The field distribution across a brain is asymmetric. It has a narrow peak at the -bulk-tissue value together with long tails from air-tissue interfaces, sinuses, -signal dropout and mask edges, where susceptibility gradients are large. Any -quantile of the whole distribution is displaced toward whichever tail is -heavier. Writing the distribution as a mixture, +A median is used rather than a mean or an M-estimator. The bulk distribution is +broad and right-skewed, with long tails from air-tissue interfaces, sinuses and +mask edges, and the half-wrap boundary falls within its body rather than out in a +tail. A statistic pulled toward the mean therefore reads systematically closer to +the boundary without describing bulk tissue any more accurately. + +MEDIC uses that median over an eroded brain mask, weighted by squared magnitude. +Both choices work against the tails: erosion removes edge voxels geometrically, +and the weighting suppresses low-signal voxels, which the tails over-represent. +The weight is taken from the second echo rather than the first, because the field +is read from a difference whose noise is dominated by the weaker of the two. + + += The algorithm in full + +Collected here as a specification. All quantities are per-voxel unless a +reduction over a mask is written explicitly, and the whole procedure is applied +independently to each frame. + +#defn[ +*Input.* Magnitudes $m_0, m_1$ and wrapped phases $phi_0, phi_1$ at the first two +echo times $"TE"_0$ and $"TE"_1$; an unwrapping mask $Omega$; a conservative mask +$Omega_"c" subset.eq Omega$ for reductions. + +*Output.* The phase offset $hat(theta)$. + +*Constants.* $#dte = "TE"_1 - "TE"_0$, $quad k = "TE"_0 slash #dte$, $quad +delta = abs(W(2 pi k))$. + +#v(0.5em) +*1. Unwrap the phase difference.* +$ + S = m_0 m_1 e^(i (phi_1 - phi_0)), + quad + Delta phi = arg S, + quad + #phiuw = "ROMEO"(Delta phi; abs(S), Omega). +$ +ROMEO returns a spatially unwrapped field with @eq:cg imposed. + +*2. Reconstruct each candidate.* For $N in {-1, 0, +1}$: +$ + hat(theta)_N &= W(phi_0 - k (#phiuw + 2 pi N)), \ + (psi_0^((N)), psi_1^((N))) &= "ROMEO"_"4D" ((phi_0, phi_1) - hat(theta)_N; + (m_0, m_1), Omega, ("TE"_0, "TE"_1)), \ + s^((N)) &= (psi_1^((N)) - psi_0^((N))) slash #dte, + quad quad + tilde(f)_N = s^((N)) slash 2 pi, \ + c_N &= abs(op("med")_(Omega_"c") [psi_0^((N)) - s^((N)) "TE"_0]). +$ +$"ROMEO"_"4D"$ imposes @eq:cg a second time, which is why $tilde(f)_N$ need not +equal the ladder field $hat(f)_N$ of @eq:fldn, and so why $c_N$ can be nonzero at +all. +*3. Set the rejection scale.* $ - p(f) = (1 - epsilon) dot p_"bulk" (f) + epsilon dot p_"tail" (f), + sigma = min(max_N c_N, delta), + quad quad + cal(C) = {N : c_N < sigma slash 2}. $ +If $delta$ is zero, return $hat(theta)_0$: $k$ is then integral, so by @eq:offn the +offset does not depend on $N$ and no choice alters the output. Test this within a +tolerance rather than against exact zero, since $W(2 pi k)$ evaluates to a small +rounding residue in floating point. -the rule refers to the centre of $p_"bulk"$, and the estimator should recover it -with $epsilon$ as small as possible. Two consequences follow. +*4. Select.* +$ + hat(N) = cases( + 0 & "if" cal(C) = nothing, + N & "if" cal(C) = {N}, + display(op("arg min")_(N in cal(C)) abs(op("med")_(Omega_"c")^(w) tilde(f)_N)) + & "otherwise," quad w = m_1^2. + ) +$ -Enlarging the mask makes the estimate worse. Dilating a brain mask admits -precisely the voxels that populate $p_"tail"$, which raises $epsilon$ and -displaces any quantile of $p$ further from the bulk centre. This reverses the -usual expectation that more data improves an estimate. +*5. Return* $hat(theta)_(hat(N))$. +] -A bias of a fraction of a wrap is enough to change the decision. The comparison -is against the half-wrap, so estimator bias adds directly to the apparent -distance from the boundary. A subject whose bulk field already lies near the -half-wrap can be carried across it by bias alone, at which point the decision -reflects the statistic rather than the data. +Three details of the specification carry the reasoning of the earlier sections. +The two reductions differ deliberately: step 2 takes a plain median, since it +estimates one global constant, while step 4 takes a median weighted by $m_1^2$ of +a distribution whose shape matters, for the reasons in @sec:field. The scale +$sigma$ in step 3 takes the smaller of the analytic separation $delta$ and the +largest intercept observed, which makes the consistency test harder to pass and so +biases the procedure toward leaving $hat(N) = 0$. And the cases in step 4 are +ordered, not interchangeable: an empty $cal(C)$ returns $0$ rather than consulting +@eq:half, a singleton is taken whatever its field, and only a tie reaches +@eq:half. + +== What the components must provide + +Three things are deliberately left open: the spatial unwrapper, the two masks, and +the weighted median. What matters is the property each must satisfy. + +*The unwrapper.* Steps 1 and 2 need one that imposes @eq:cg on what it returns, +step 2 on the echo it unwraps first. Nothing else about it is load-bearing: the +selection logic reads only the returned phase, and @sec:detect needs only that the +two unwrapping steps choose their shifts independently, since that is what pairs an +offset from one rung with a field from another and leaves a signature to detect. + +*The masks.* $Omega$ is generous, brain plus a margin, because the unwrapper must +grow through the whole region of interest. $Omega_"c"$ is interior, excluding edge +and low-signal voxels, because both reductions summarise a distribution whose tails +come from exactly those voxels (@sec:field). Nothing depends on how either is +built, only on their differing in that way, with $Omega_"c"$ well inside $Omega$. + +*The weighted median.* $op("med")^w$ is the value at which cumulative weight, over +voxels sorted by value, first reaches half the total. The convention matters only +at an exact tie, where any deterministic choice will do. + +== Reference data + +The failure of @sec:fail can be reproduced on public data. +#link("https://openneuro.org/datasets/ds006131")[`ds006131`] (v2.0.1, "PAFIN") +acquires five echoes at $"TE"_0 = 14.2$ ms and $#dte = 24.73$ ms, so +$delta = 2.68$ rad and the half-wrap is $20.22$ Hz. Two of its 51 subjects carry a +bulk field close enough to that boundary for @eq:cg to tip on individual frames: + +#defn(breakable: false)[ +#set par(justify: false) +#set text(size: 9.5pt) +`ds006131/sub-/ses-1/func/` \ +`sub-_ses-1_task-bao_dir-AP_run-01_echo-{1..5}_part-{mag,phase}_bold.nii.gz` \ +#v(0.35em) +$hat(N) != 0$ on 19 of 243 frames for sub-20828 and on 158 of 243 for sub-24630. +] -== The estimator +Afterwards neither field-map time series contains a wrap-sized step, where before +each carried one on every affected frame. The counts depend on the masks in use, so +treat them as a guide: what is robust is that these two subjects need a nonzero +$hat(N)$ on a scattered subset of frames and the other 49 do not. An implementation +returning $hat(N) = 0$ throughout on them has not reproduced the correction; one +returning nonzero shifts across many subjects is over-correcting. -MEDIC uses a magnitude-weighted median over an eroded brain mask. Erosion -lowers $epsilon$ geometrically by removing edge voxels, and the weighting lowers -it by suppressing low-signal voxels, which are over-represented in the tails. -The weight is the magnitude of the second echo, because the field is read from a -difference whose noise is dominated by the weaker of the two. +#counter(heading).update(0) +#set heading(numbering: "A.1") -A median is used rather than a mean or an M-estimator. The bulk distribution is -broad and right-skewed, and the half-wrap boundary falls within its body rather -than out in a tail, so a statistic pulled toward the mean reads systematically -closer to the boundary without describing bulk tissue any more accurately. - -Continuity is the binding requirement. The estimate decides a threshold -comparison that has to give the same answer on consecutive frames of one -subject, and an unstable estimator would reintroduce at this stage exactly the -inconsistency described in section 4. A median moves smoothly as the samples -move. Estimators defined by an $arg max$, including modes in any of their usual -forms, do not, since field distributions are frequently multimodal and a -near-tie between peaks makes the estimate jump between frames. Stability across -frames matters more here than freedom from bias. += Proof of Proposition 1 + +#proof[ +*Proof of #link()[Prop. 1].* Working modulo $2 pi$, so that $W$ of either side agrees, @eq:offn and +@eq:fldn give +$ + hat(theta)_N + 2 pi hat(f)_N t_e + &equiv hat(theta)_0 - 2 pi k N + 2 pi (hat(f)_0 + N slash #dte) t_e \ + &= hat(theta)_0 + 2 pi hat(f)_0 t_e + 2 pi N (t_e slash #dte - k). +$ +By @eq:spacing, $t_e slash #dte = k + e$, so the parenthesis is the integer $e$ +and the final term is $2 pi N e$, a whole number of turns. Both sides therefore +wrap alike. #h(1fr) $qed$ +] +== The geometry + +// Drawing geometry in points. `y` is measured upward from the baseline and +// flipped at draw time. One whole turn is `turn` tall and candidate 0 gains +// `rise` per echo spacing, so the drawn k is te0 / dte-x. +#let pencil = { + let te0 = 56.0 + let dte-x = 93.0 + let turn = 34.0 + let rise = 20.0 + let ypiv = 62.0 + let tend = 265.0 + let ox = 34.0 + let oy = 24.0 + let hbox = 224.0 + + let echoes = (te0, te0 + dte-x, te0 + 2.0 * dte-x) + + let px(t) = (ox + t) * 1pt + let py(y) = (hbox - oy - y) * 1pt + let val(n, t) = ypiv + (rise + n * turn) / dte-x * (t - te0) + + let thin = (paint: luma(50), thickness: 0.9pt) + let faint = (paint: luma(170), thickness: 0.4pt, dash: "dashed") + let hair = (paint: luma(105), thickness: 0.4pt) + + let dot(t, y, filled) = place( + dx: px(t) - 1.8pt, + dy: py(y) - 1.8pt, + circle( + radius: 1.8pt, + fill: if filled { black } else { white }, + stroke: 0.4pt + black, + ), + ) + + // dimension line with end caps; `side` puts the label left (-1) or right (+1) + let dim(t, ya, yb, label, side) = { + place(line(start: (px(t), py(ya)), end: (px(t), py(yb)), stroke: hair)) + for y in (ya, yb) { + place(line( + start: (px(t) - 2pt, py(y)), + end: (px(t) + 2pt, py(y)), + stroke: hair, + )) + } + place( + dx: px(t) + (if side > 0 { 4pt } else { -26pt }), + dy: py((ya + yb) / 2.0) - 5.5pt, + box( + width: 22pt, + align(if side > 0 { left } else { right }, text(size: 8.5pt, label)), + ), + ) + } + + box(width: (ox + tend + 46.0) * 1pt, height: hbox * 1pt, { + // baseline, and time guides drawn only up to the lowest candidate so they + // never collide with the dimension lines + place(line( + start: (px(-8.0), py(6.0)), + end: (px(tend + 6.0), py(6.0)), + stroke: hair, + )) + for (t, lab) in ( + (0.0, $0$), + (echoes.at(0), $"TE"_0$), + (echoes.at(1), $"TE"_1$), + (echoes.at(2), $"TE"_2$), + ) { + let low = calc.min(val(1, t), val(0, t), val(-1, t)) + place(line(start: (px(t), py(6.0)), end: (px(t), py(low - 6.0)), stroke: faint)) + place( + dx: px(t) - 11pt, + dy: py(-4.0), + box(width: 22pt, align(center, text(size: 8.5pt, lab))), + ) + } + + // the three candidate lines, labelled at the right + for n in (1, 0, -1) { + place(line( + start: (px(0.0), py(val(n, 0.0))), + end: (px(tend), py(val(n, tend))), + stroke: thin, + )) + place( + dx: px(tend) + 4pt, + dy: py(val(n, tend)) - 5.5pt, + text(size: 8.5pt, $N = #if n > 0 [$+1$] else if n == 0 [$0$] else [$-1$]$), + ) + } + + // offsets at t = 0, the shared pivot, and the samples at later echoes + for n in (1, 0, -1) { + dot(0.0, val(n, 0.0), false) + dot(echoes.at(1), val(n, echoes.at(1)), false) + dot(echoes.at(2), val(n, echoes.at(2)), false) + } + dot(te0, ypiv, true) + + // the two step sizes, read straight off the picture + dim(0.0, val(0, 0.0), val(1, 0.0), $2 pi k$, -1) + dim(echoes.at(1), val(0, echoes.at(1)), val(1, echoes.at(1)), $2 pi$, 1) + dim(echoes.at(2), val(0, echoes.at(2)), val(1, echoes.at(2)), $4 pi$, 1) + + // leader from an annotation in the empty upper left down to the pivot + place(line( + start: (px(te0 - 9.0), py(140.0)), + end: (px(te0 + 1.0), py(68.0)), + stroke: hair, + )) + place(dx: px(-6.0), dy: py(156.0), text(size: 8.5pt)[every candidate meets here]) + place(dx: px(-6.0), dy: py(202.0), text(size: 8.5pt)[unwrapped phase]) + }) +} + +#figure(pencil, caption: [ + Candidates as lines in phase against echo time, drawn for $k = 0.6$. All of + them meet at $"TE"_0$, so the offset step $2 pi k$ is exactly the phase a + one-wrap steeper slope gains by the first echo. Past $"TE"_0$ the lines + separate by one further turn per echo spacing, so at echo $e$ they stand whole + turns apart (open circles) and wrap to identical samples. +]) + +#v(0.6em) + +@fig:pencil is the same statement drawn. Against echo time each candidate is a +straight line, intercept $hat(theta)_N$ and slope $2 pi hat(f)_N$. Setting +$t = "TE"_0$ in the last line of the proof kills the parenthesis, so every +candidate takes the value $hat(theta)_0 + 2 pi hat(f)_0 "TE"_0$ at the first echo: +the candidates form a pencil of lines through one point. Past that point they +separate at one turn per echo spacing, standing $2 pi N e$ apart at $t_e$. +Wrapped phase cannot see a whole turn, so every line reproduces the same measured +samples. + +The two step sizes are consequences rather than choices. Requiring two candidates +to differ by whole turns at every echo forces their slopes to differ by one turn +per echo spacing, which is the field step $1 slash #dte$; that fixed, their +offsets must differ by the $2 pi k$ such a slope has already gained at $"TE"_0$, +which is exactly what makes the two lines meet there. + +== Consequences + +Three consequences follow, and the rest of the note rests on all of them. + +*The rule is unavoidable.* Since the candidates predict identical data, no +statistic computed from the phase can rank them. @eq:half is the only kind of +answer available, and any implementation must supply something like it. + +*A longer echo train does not help.* The cancellation holds at every echo at once, +so a five-echo acquisition constrains $N$ no better than a two-echo one. The later +echoes are not independent measurements of the ambiguity. This depends on the even +spacing of @eq:spacing rather than on phase imaging in general, but an EPI train is +evenly spaced by construction. + +*The undetectable errors are exactly enumerated.* #link()[Prop. 1] applies to a +candidate as defined in @eq:offn and @eq:fldn, an offset paired with the field of +the same $N$. Nothing outside that set is protected, which is what makes any +detection possible at all, and @sec:detect is about exploiting it.