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 :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, . . . , I m f ). In the cases of m = 1 and m = 2, we derive the Reshetnyak formula that expresses f H s t (R n ) through some norm of (I 0 f, I 1 f, . . . , 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.

Optimal recovery of a radiating source with multiple frequencies along one line

“…A different way of using several weights is to study the momentum ray transform [29,16]. In that problem, one defines the momentum ray transform for points…”

Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

“…Suppose that each component F k (x), k = 1, ..., d of a vector field F (x) is a function from the Schwartz class S(R d ). Then potential ϕ and fields F p and F s given by equations ( 8)- (10) have the following decay rates at infinity…”

“…Corollary 5. Suppose F (x) is a C ∞ vector field defined on R d and decaying at infinity at rates given by equation ( 1), and F p + F s are defined by equations ( 8)- (10). Then…”

Momentum ray transforms, II: range characterization in the Schwartz space

“…We emphasize that such a tensor field is not unique, it is determined by (3.10) up to an arbitrary potential field (the definition of a potential tensor field is presented in [5], as well as the the definition of the inner derivative d which is used in the next paragraph). Let us fix some tensor field g ∈ S(R 2 ; S m R 2 ) satisfying (3.10).…”

“…, I m f ). The inversion algorithm is presented in [5]. Quite similarly to (1.6), we introduce the operators…”