MICROLOCAL ANALYSIS OF \bfitd-PLANE TRANSFORM ON THE EUCLIDEAN SPACE*

We study the basic properties of the 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 computed tomography (CT) scanners. In this paper we obtain a concrete expression of the canonical relation between the d-plane transform and the quantitative properties of the filtered back-projection of the product of the images of the d-plane transform. The FBP of the product is related to the metal streaking artifacts of CT images. Our result is a generalization of recent results of Park, Choi, and Seo [Comm. Pure Appl. Math., 70 (2017), pp. 2191--2217] and Palacios, Uhlmann, and Wang [SIAM J. Math. Anal., 50 (2018), pp. 4914--4936] for the X-ray transform on the plane.