Momentum ray transforms

TL;DR

This paper introduces an algorithm to reconstruct symmetric tensor fields from their momentum ray transforms, extending previous work on the classical ray transform with new formulas and stability estimates for tensor data.

Contribution

The paper develops a reconstruction algorithm for tensor fields from multiple momentum ray transforms and derives Reshetnyak formulas for stability analysis in specific cases.

Findings

Reconstruction algorithm for tensor fields from momentum ray transforms.

Reshetnyak formulas relating Sobolev norms of tensors to transform data.

Stability estimates for tensor field recovery using the derived formulas.

Abstract

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $ (I^k\!f)(x,\xi)=\int_{-\infty}^\infty t^k\langle f(x+t\xi),\xi^m\rangle\,dt. $ In particular, the ray transform $I=I^0$ was studied by several authors since it had many tomographic applications. We present an algorithm for recovering $f$ from the data $(I^0\!f,I^1\!f,\dots, I^m\!f)$. In the cases of $m=1$ and $m=2$, we derive the Reshetnyak formula that expresses $\|f\|_{H^s_t({\mathbb{R}}^n)}$ through some norm of $(I^0\!f,I^1\!f,\dots, I^m\!f)$. The $H^{s}_{t}$-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.