Weighted Divergent Beam Ray Transform: Reconstruction, Unique continuation and Stability

TL;DR

This paper proves that symmetric tensor fields can be reconstructed from partial weighted divergent beam ray data, establishes unique continuation properties using fractional Laplacian techniques, and provides explicit formulas and stability results for these transforms.

Contribution

It introduces new reconstruction formulas and stability estimates for the weighted divergent beam ray transform, extending understanding of tensor field recovery and unique continuation.

Findings

Tensor fields can be recovered pointwise from partial data.

Unique continuation holds for the fractional divergent beam ray transform.

Explicit reconstruction formulas and stability estimates are derived.

Abstract

In this article, we establish that any symmetric $m$-tensor field can be recovered pointwise from partial data of the $k$-th weighted divergent ray transform for any $k \in \mathbb{Z}^{+} \cup\{0\}$. Using the unique continuation property of the fractional Laplacian, we further prove the unique continuation of the fractional divergent beam ray transform for both vector fields and symmetric 2-tensor fields. Additionally, we derive explicit reconstruction formulas and stability results for vector fields and symmetric 2-tensor fields in terms of fractional divergent beam ray transform data. Finally, we conclude by proving a unique continuation result for the divergent beam ray transform for functions.