= Missing-section ambiguity in volume reconstruction
<Missing-section ambiguity in volume reconstruction> is nonuniqueness caused by unobserved material between acquired sections. Even perfect <image registration> cannot determine every structure inside that material. It is an <inverse problem>: different three-dimensional specimens can produce exactly the same retained <image stack>. The serial-section acquisition sequence makes the missing observation explicit: cut, collect and image sections, then align and stack the surviving observations.<Serial Section TEM by JEOL>[Figure1(a)–(e), physical sections, alignment and stacking]{ref}
For an original exact counterexample, consider a binary column of three adjacent <voxels>, ordered along the cutting direction. A value 1 denotes a marked material and 0 denotes its absence. Two possible columns are
$$
A=(1,1,1),\qquad B=(1,0,1).
$$
If the middle section is missing, both acquired records are $(1,1)$. Yet $A$ is one connected column and $B$ consists of two separated parts. The acquisition map $P$ that retains only the first and last entries satisfies
$$
P(A)=P(B),\qquad A\ne B.
$$
This is an ambiguity in the observations, not a failure of a particular reconstruction program. <Linear interpolation> inserts the middle value 1 and selects $A$, but that choice uses a smoothness assumption rather than an additional measurement.
With expected section spacing $40\,\mathrm{nm}$, losing the section centered at $z=40\,\mathrm{nm}$ leaves acquired centers at $0$ and $80\,\mathrm{nm}$. Keep their physical positions: silently relabeling them as adjacent $40\,\mathrm{nm}$ observations also distorts depth. Additional physical observations or justified constraints may reduce the possible reconstructions. A visually smooth surface alone cannot distinguish the two columns above.
\Image[/missing-section-ambiguity.png]
{title=Original binary-volume counterexample: the two possible middle layers give identical retained sections but different connectivity}
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