Momentum ray transforms

Abstract

The momentum ray transform $I^k$ integrates a rank $m$ symmetric tensor field $f$ over lines with the weight $t^k$: $(I^{k}f)(x,\xi )={\int }_{-\infty }^{\infty }{t}^k⟨f(x+t\xi ),{\xi }^m⟩dt.$ 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 $\Vert f{\Vert }_{H_t^s({\mathbb{R}}^n)}$ through some norm of $(I^{0}f,I^{1}f,\dots ,I^{m}f)$. The $H_t^s$-norm is a modification of the Sobolev norm weighted differently at high and low frequencies. Using the Reshetnyak formula, we obtain a stability estimate.