X-rays of currents and projections of forms

TL;DR

This paper introduces a new Radon-like transform for differential forms in R^n, provides an explicit inversion formula, and demonstrates how to reconstruct forms and currents from their projections onto subspaces.

Contribution

It presents a novel Radon-like transform for differential forms, along with an explicit inversion formula and applications to reconstructing forms and currents from projections.

Findings

Derived an explicit inversion formula for the new transform.

Showed how to synthesize p-forms from projections onto k-planes.

Provided a method to reconstruct currents from their projections.

Abstract

We introduce and study a new Radon-like transform that averages projected differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit inversion formula for our transform on the space of rapidly-decaying smooth p-forms. Our transform differs from the one in Gelfand-Graev-Shapiro. Moreover, if it can be extended to a somewhat larger space of p-forms, our inversion formula will allow the synthesis of any rapidly-decaying smooth p-form on R^n as a (continuous) superposition of pullbacks from p-forms on k-dimensional subspaces. In turn, such synthesis implies an explicit formula (which we derive) for reconstructing compactly supported currents in R^n (e.g., compact oriented k-dimensional subvarieties) from their oriented projections onto k-planes.