Shape and spin distributions of large object populations from random projection areas

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.