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We model the shape and spin characteristics of an object population whenthere are not enough data to model its single members. The data are randomprojection 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 functionof the shape elongation and spin vector obliquity, and $C(y)$ is the cumulativedistribution function of an observable $y$ describing the variation of theobserved projection areas of one member, and show that the mapping isinvertible. Using the projected area of an ellipsoid as our model, we obtainanalytical basis functions for a function series of $C(y)$ and prove uniquenessand stability properties of the inverse problem. Even though the model error isconsiderably larger than the measurement noise for realistic cases of arbitraryshapes (such as asteroids), the main characteristics of $f(x)$ (such as thelocations of peaks) are robustly recovered from the data.

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