The symbol of the normal operator for the d-plane transform on the Euclidean space

TL;DR

This paper computes the symbol of the normal operator for the d-plane transform in Euclidean space, revealing its relation to the Laplacian's power and deriving the filtered backprojection formula.

Contribution

It provides a direct computation of the normal operator's symbol and links it to the Laplacian, offering an alternative derivation of the filtered backprojection formula.

Findings

The symbol of the normal operator is the product of the Laplacian's power symbol and a constant.

The derivation offers a new perspective on the filtered backprojection formula.

The approach simplifies understanding of the d-plane transform's normal operator.

Abstract

We directly compute the symbol of the normal operator for the d-plane transform on the Euclidean space. We show that this symbol is the product of the symbol of the power of the Laplacian of order -d/2 and a constant given by an invariant integral over excess-dimensional spaces. This leads to an alternative derivation of the filtered backprojection formula for the d-plane transform.