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Full reconstruction of a vector field from restricted Doppler and first integral moment transforms in ℝ n

Abstract We show that a vector field in ℝ n {\mathbb{R}^{n}} can be reconstructed uniquely from the knowledge of restricted Doppler and first integral moment transforms. The line complex we consider consists of all lines passing through a fixed curve γ ⊂ ℝ n {\gamma\subset\mathbb{R}^{n}} . The question of reconstruction of a symmetric m-tensor field from the knowledge of the first m + 1 {m+1} integral moments was posed by Sharafutdinov [Integral Geometry of Tensor Fields, Inverse Ill-posed Probl. Ser. 1, De Gruyter, Berlin, 1994, p. 78]. In this work, we provide an answer to Sharafutdinov’s question for the case of vector fields from restricted data comprising of the first two integral moment transforms.