Estimation of length density Lv from vertical slices of unknown thickness

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Abstract
SUMMARY: Length density of lineal features, Lv, is an important stereological parameter. The efficient stereological procedure for the estimation of Lv from the counting measurement performed on the projected images of the vertical slices (foils) is modified and improved: Lv can be now estimated from vertical slices of unknown thickness, and the slices need not be of the same thickness. The required assumption‐free stereological relationship is image is the average number of intersections of straight test lines parallel to the vertical axis with the projected images of the lineal features in the vertical slices, per unit test line length. is the average number of intersections of the lineal features with the parallel planes of the vertical slices, per unit area. Note that there are two parallel planes in a slice, and therefore their total area is twice the area of the observed projected image frame. is the average number of intersections of orientated cycloid‐shaped test lines (minor axis perpendicular to vertical axis) with the projected images of the lineal features in the vertical slices, per unit length. For practical applications of this result, it is necessary uniquely to identify the points of intersections of the lineal features with the parallel planes of the slices in the projected images, so that can be estimated unambiguously. However, in practice, this is not a problem in biological, as well as materials microstructures.