On locality of Radon to Riesz transform

In this paper we present a novel approach to locally compute the Riesz transform from the knowledge of the Radon transform. Previous implementations of the Riesz transform are based on the Fourier or the Radon transforms and their inversion formulae, and therefore needs for the knowledge of the function or its Radon data on the whole domain. More recent attempts on rectangular domains involves convolutions with the Poisson kernel and local derivatives. On the other hand, starting from the links between the Riesz and the Radon transforms, we address in this paper a new local Radon based Riesz formula in the general n-dimensional case, for even n. The advantage of this formula, local in the Radon space, is pointed out in the bidimensional case, where we provide a new local Radon based Riesz algorithm, and conduct numerical tests for the estimation of the Riesz transform on convex sets, from truncated Radon data. Finally we study the robustness to noise of the current approach.