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.^[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 areIf 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 satisfiesThis 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.