Rate of Convergence of the Empirical Radon Transform

Motivated by recent work on exploratory projection pursuit, we define the empirical Radon transform. Its precise convergence rate at a fixed centre, uniform over all orientations, and its precise convergence rate uniformly over all centres and all orientations are derived. We show that in the sense of optimizing the uniform convergence rate, kernel-based empirical Radon transforms are optimal. We also describe convergence rates in mean square. An estimator of the original density can be reconstructed from the empirical Radon transform by direct inversion, provided that we restrict ourselves to sufficiently well-behaved functions. In that restricted class the rate of convergence in mean square of the latter estimators appears to be of somewhat smaller order than that of empirical Radon transforms.