X-ray Tomography of One-forms with Partial Data

Abstract

If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions.

Partial data problems in scalar and vector field tomography

“…The next theorem generalizes the result in [21] where the authors assume that dF | V = 0 instead of (dF ) ij ∈ A V .…”

“…We prove that if there are some constant coefficient partial differential operators P ij (D) such that P ij (D)(dF ) ij | V = 0 and the integrals of F over all lines intersecting V vanish, then F must be a potential field (F is the gradient of some scalar field). This is a generalization of a recent result in [21]. The partial data result is proved by using a relation between the normal operator of the X-ray transform of scalar fields and the normal operator of the X-ray transform of vector fields (see lemma 4.4).…”

“…The partial data problem of vector fields is known to be uniquely solvable up to potential fields, if dF | V = 0 [21]. Similarly, the normal operator of the X-ray transform of vector fields has a unique continuation property under the assumptions [21].…”

“…The unique continuation of (−∆) s implies a unique continuation result for the normal operator N 0 of the X-ray transform X 0 , and the uniqueness for the partial data problem follows directly from the unique continuation of N 0 . This approach which uses the unique continuation of the normal operator in proving uniqueness for partial data problems was also used in [20,21].…”

“…[41]). Therefore theorem 1.7 can be seen as a solenoidal injectivity result in terms of partial data (see [21] and [33,41]). Theorem 1.7 holds also for vector fields F ∈ (S (R n )) n which components are Schwartz functions since in that case (dF…”