X-rays of forms and projections of currents

Abstract:

We study a new Radon-like transform that averages projected $p$-forms in $\mathbf R^{n}$ over affine $(n-k)$-spaces. 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 the work by Gelfand, Graev and Shapiro (1969). 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 $\mathbf R^{n}$ as a (continuous) superposition of pullbacks from $p$-forms on $k$-dimensional subspaces. In turn, such a synthesis implies an explicit formula (which we derive) for reconstructing compactly supported currents in $\mathbf R^{n}$ (e.g., compact oriented $k$-dimensional subvarieties) from their oriented projections onto $k$-planes.