We study the non-parametric estimation of a multidimensional unknown density f in a tomography problem based on independent and identically distributed observations, whose common density is proportional to the Radon transform of f. We identify the underlying statistical inverse problem and use a spectral cut-off regularisation to deduce an estimator. A fully data-driven choice of the cut-off parameter m in R+ is proposed and studied. To discuss the bias-variance trade off, we consider Sobolev spaces and show the minimax-optimality of the spectral cut-off density estimator. In a simulation study, we illustrate a reasonable behaviour of the studied fully data-driven estimator.

翻译：本文研究了基于独立同分布观测的断层扫描问题中多维未知密度f的非参数估计，这些观测的共同密度与f的Radon变换成正比。我们识别了潜在的统计逆问题，并采用谱截断正则化方法推导出估计量。提出并研究了R+中截断参数m的完全数据驱动选择。为探讨偏差-方差权衡，我们考虑Sobolev空间，证明了谱截断密度估计量的极小极大最优性。通过模拟研究，我们展示了所研究的完全数据驱动估计量的合理行为。