= Sholl profile
{c}
{title2=$N(r)$}
A <Sholl profile> records intersection counts as a function of distance from a chosen center. For a thin geometric tracing it is an exact counting function; a profile obtained from sampled images also depends on the software's sampling and segmentation conventions.
In an ideal planar model, let $\Gamma$ be a finite embedded <rooted tree> of <line segments> in the <Euclidean plane> with chosen root $c$. Its <Sholl profile> counts distinct intersections:
$$
N(r)=\#\bigl\{x\in\Gamma:\lVert x-c\rVert=r\bigr\},\qquad r>0.
$$
This is a geometric model for thin tracings; the software documentation separately addresses sampling <binary images> and handling <image segmentation>.<Sholl Analysis in the SNT documentation>[Overview; Segmentation requirements; sampled profile metrics]{ref}
For an original simple example, take three straight radial segments from the origin, pointing in different directions, with lengths 1, 2 and 4. For radii $r=0.5,1.5,3,5$, the counts are respectively $3,2,1,0$. The segment of length 1 contributes at $r=1$ but not beyond it; record the boundary convention when defining the count. No branch points are needed for this decreasing profile: different lengths of terminal segments already produce it.
The profile retains radial occupancy but discards the angles and the identities of the intersected paths. In particular, <a Sholl profile does not determine rooted tree topology>. Sampling only a finite list of radii loses still more information; it should not be silently equated with knowledge of the complete continuous-radius counting function.
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