Normal operators for momentum ray transforms, II: Saint Venant operator

TL;DR

This paper investigates how symmetric tensor fields can be reconstructed from momentum ray transform data using the Saint Venant operator, providing explicit formulas linking the data to local tensor information.

Contribution

It derives explicit formulas expressing the Saint Venant operator in terms of the normal operators of the momentum ray transform, revealing the local information obtainable from the data.

Findings

Explicit formula for the Saint Venant operator in terms of normal operators

Characterization of local tensor information recoverable from ray transform data

Enhanced understanding of tensor field determination from momentum ray transforms

Abstract

The momentum ray transform $I_m^k$ integrates a rank $m$ symmetric tensor field $f$ on ${\mathbb R}^n$ over lines with the weight $t^k$, $I_m^kf(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,\mathrm{d}t$. Let $N^k_m=(I^k_m)^*I^k_m$ be the normal operator of $I_m^k$. To what extent is a symmetric $m$-tensor field $f$ determined by the data $(N_m^0f,\dots,N_m^rf)$ given for some $0\le r\le m$? The Saint Venant operator $W^r_m$ is a linear differential operator of order $m-r$ with constant coefficients on the space of symmetric $m$-tensor fields. We derive an explicit formula expressing $W^r_mf$ in terms of $(N_m^0f,\dots,N_m^rf)$. The tensor field $W^r_mf$ represents the full local information on $f$ that can be extracted from the data $(N_m^0f,\dots,N_m^rf)$.