  We model the shape and spin characteristics of an object population when
there are not enough data to model its single members. The data are random
projection areas of the members. We construct a mapping $f(x)\rightarrow C(y)$,
$x\in\mathbb{R}^2$, $y\in\mathbb{R}$, where $f(x)$ is the distribution function
of the shape elongation and spin vector obliquity, and $C(y)$ is the cumulative
distribution function of an observable $y$ describing the variation of the
observed projection areas of one member, and show that the mapping is
invertible. Using the projected area of an ellipsoid as our model, we obtain
analytical basis functions for a function series of $C(y)$ and prove uniqueness
and stability properties of the inverse problem. Even though the model error is
considerably larger than the measurement noise for realistic cases of arbitrary
shapes (such as asteroids), the main characteristics of $f(x)$ (such as the
locations of peaks) are robustly recovered from the data.
