### Function analysis

### Daghestan Electronic Mathematical Reports, Issue 2 (2014)

# Inversion formulas for tensor imaging on incomplete data

### UDK: 517.587

### Pages: 75 - 86

### DOI: 10.31029/demr.2.6

In this article new inverse formulas for a ray transform of symmetric tensor fields with incomplete data are given. The Radon transform of Saint-Venant operator of tensor field is uniquely determined when the line integrals of the field along lines which form an $n$-dimensional complexes in the space $\mathbb{R}^n$ are given. Three most common complex are considered: the family of lines intersecting a given curve, intersecting a given curve at infinity and tangent a given surface. In the case of the complex of lines intersecting a curve at infinite the formula containing only two-fold integration is obtained.

**Keywords: **
symmetric tensor field, ray transform, reconstruction formula, complex of lines, Radon transform, solenoidal part, Saint-Venant operator.