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A binary selection mask records which pixels belong to a chosen subset of an image. On a finite pixel grid $\Omega$, it is the indicator functionIt can represent an area region of interest, the result of image thresholding or a combination of selection rules. The mask describes membership; its entries need not equal the underlying image intensities.

For a selected pixel count $N>0$, the count and selected arithmetic mean areConsider the original arraysThe selected values are 0, 20 and 40, so $N=3$ and ${\bar{I}}_M=20$. The selected zero-valued pixel still contributes to the denominator. Replacing excluded pixels by zeros and averaging all six entries would instead give $60/6=10$: that asks a different question. An empty mask has $N=0$, so this selected-mean formula is undefined.

For rectangular pixels with calibrated dimensions $\Delta x=0.5\mathrm{mm}$ and $\Delta y=0.2\mathrm{mm}$, the complete-pixel area from a calibrated pixel count is $A=N\Delta x\Delta y=0.30\mathrm{m}{\mathrm{m}}^2$. These are stipulated array values and cell dimensions, not a measurement of an object.

A stored mask can use another two-value encoding. ImageJ's Create Mask command stores 255 inside an area selection and 0 outside, corresponding to $B_{ij}=255M_{ij}$ in this model.^[ref] Summing $B$ gives 765 in the example, rather than the selected count 3. Convert the encoding or count selected entries before interpreting a sum as an area.

Two normalized masks $M$ and $Q$ combine by set intersection through $M_{ij}{Q}_{ij}$. Their complement within the specified grid is $1-M_{ij}$. Record the foreground-background polarity, coordinate alignment and grid extent when transferring a mask between images.

An image lookup table maps a stored pixel value or index to a displayed color. For an 8-bit grayscale image with stored values $I_{ij}\in \{0,\dots ,255\}$, a table can be writtenusing an RGB color model triple. Changing $C$ changes the appearance without necessarily changing $I$. ImageJ explicitly distinguishes Invert LUT, which changes the display, from inversion of the pixel values.^[ref]

For an original example, store the rowThe identity grayscale table is $C(v)=(v,v,v)$; an inverted table is $C_{\mathrm{inv}}(v)=(255-v,255-v,255-v)$. The same four stored values appear in opposite brightness order under these tables, but their arithmetic mean remains $447/4=111.75$. An image thresholding rule selecting stored values above 100 still gives the binary selection mask $(0,0,1,1)$.

Numerically replacing the pixels by $255-I$ instead produces $(255,191,127,0)$. Its arithmetic mean is $573/4=143.25$, and the same strict threshold gives $(1,1,1,0)$. Thus an inverted-looking image does not by itself establish that its numerical values or selected subset were inverted.

The distinction is especially visible with binary selection masks: the documented Create Mask command stores 255 inside the selection and 0 outside, while an inverting lookup table can display the selected class as black.^[ref] Record the foreground-background polarity from the stored encoding and algorithm, rather than inferring it from screen brightness.

A program can also apply a lookup mapping to the data. ImageJ's Apply LUT command changes gray values and restores the identity table so that the displayed appearance is retained.^[ref] Distinguish that operation from assigning a display table. A table mapping several input values to the same color can conceal numerical distinctions in the display even while those distinctions remain in the original stored image.

Sholl analysis summarizes the radial arrangement of a traced structure by counting its intersections with concentric Euclidean circles around a chosen center. The resulting Sholl profile reports how the count changes with radius. A computational workflow must specify the center, spatial calibration, sampled radii and the convention used to identify distinct intersections. The software documentation distinguishes analysis of segmented images from analysis of reconstructed tracings.^[ref]

For an original center-choice example, take the straight segment $\Gamma =\{(x,0):0\le x\le 2\}$. With center $(0,0)$, a circle of radius $r$ meets the segment once for $0<r\le 2$. With center $(1,0)$, it meets the same segment twice for $0<r\le 1$ and not at all for $r>1$. At $r=0.75$, therefore, the two center choices give counts 1 and 2, although the traced object has not changed. The center is part of the measurement definition.

For binary images, the foreground also needs an explicit meaning. The SNT documentation treats zero-valued pixels as background independently of the displayed lookup table and illustrates how an inverted segmentation can make the program sample background instead of structure.^[ref] Record the foreground-background polarity and radial sampling interval with the result.

The radial summary is useful for comparing where intersections occur, but a Sholl profile does not determine rooted tree topology. A fitted curve, sampled count sequence and complete ideal Sholl profile are different data objects; none should be silently substituted for another.