Summary: Knowledge about the foliage angle density g(a) of the leaves in the canopy of trees is crucial in foresty management, modelling canopy reflectance, and environmental monitoring. It is usually determined from observations of the contact frequency \(f(\beta)\) by solving a version of the first kind Fredholm integral equation derived by J. E. Reeve [Appendix in J. Warren Wilson, Inclined point quadrats. New Phytologist 59, 1-8 (1960)].
However, for inference purposes, the practitioner uses functionals defined on \(g(\alpha)\), such as the leaf area index F, rather than \(g(\alpha)\) itself. J. B. Miller [A formula for average foliage density. Aust. J. Botanics 15, 141-144 (1967)] has shown that F can be computed directly from \(f(\beta)\) without solving the integral equation. In this paper, we show that his result is a special case of a general transformation for linear functionals defined on \(g(\alpha)\). The key is the existence of an alternative inversion formula for the integral equation to that derived by J. B. Miller [J. Aust. Math. Soc. 4, 397-402 (1964; Zbl 0195.121)].