= Area from a calibrated pixel count
{title2=$A=N\Delta x\Delta y$}
<Area from a calibrated pixel count> assigns an <area> to a selected collection of complete image pixels. For $N$ selected pixels of calibrated widths $\Delta x$ and $\Delta y$,
$$
\boxed{A=N\Delta x\Delta y.}
$$
This rectangular-cell model permits unequal horizontal and vertical spacing. In <ImageJ>, area is reported in square pixels or calibrated square units according to the spatial calibration and measurement settings.<ImageJ User Guide by Tiago Ferreira and Wayne Rasband>[IJ1.46r, section30.7 Set Measurements, Area; section30.8 Set Scale]{ref}
For an original threshold example, let four stored gray values be 40, 90, 130, 200. A rule selecting values at least 128 retains two pixels. If $\Delta x=\Delta y=0.5\,\mathrm{\mu m}$,
$$
A=2(0.5)(0.5)\,\mathrm{\mu m^2}=0.50\,\mathrm{\mu m^2}.
$$
At $\Delta x=0.5\,\mathrm{\mu m}$ and $\Delta y=0.8\,\mathrm{\mu m}$, the same pixel selection instead represents $0.80\,\mathrm{\mu m^2}$. Pixel count alone is not an area in physical units.
For a <binary selection mask> $M_{ij}\in\{0,1\}$, the count is $N=\sum_{ij}M_{ij}$. A fractional-coverage model would instead use fractions $0\le f_{ij}\le1$ and $A=\Delta x\Delta y\sum_{ij}f_{ij}$; the two rules need not agree at boundaries. A <region of interest> and an <image thresholding> rule also select pixels for different reasons. State the selection and calibration before interpreting the result as an estimate of a specimen's geometric area. The numbers above are stipulated image data, not a physical specimen measurement.
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