Microlocal analysis of d-plane transform on the Euclidean space

We study the basic properties of d-plane transform on the Euclidean space as a Fourier integral operator, and its application to the microlocal analysis of streaking artifacts in its filtered back-projection. The d-plane transform is defined by integrals of functions on the n-dimensional Euclidean space over all the d-dimensional planes, where 0<d<n. This maps functions on the Euclidean space to those on the affine Grassmannian G(d,n). This is said to be X-ray transform if d=1 and Radon transform if d=n-1. When n=2 the X-ray transform is thought to be measurements of CT scanners. In this paper we obtain concrete expression of the canonical relation of the d-plane transform and quantitative properties of the filtered back-projection of the product of the images of the d-plane transform. The latter one is related to the metal streaking artifacts of CT images, and some generalization of recent results of Park-Choi-Seo (2017) and Palacios-Uhlmann-Wang (2018) for the X-ray transform on the plane.