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

TL;DR

This paper develops a method to infer the shape and spin distribution of large object populations from random projection data, demonstrating the invertibility and robustness of the approach despite model errors.

Contribution

It introduces an invertible mapping from shape and spin distributions to observable projection area data, with analytical basis functions for ellipsoids and proofs of uniqueness and stability.

Findings

Main characteristics of distributions are robustly recovered

Analytical basis functions derived for ellipsoids

Invertibility of the mapping established

Abstract

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.