A breakdown of injectivity for weighted ray transforms in multidimensions

Abstract

We consider weighted ray-transforms PW (weighted Radon transforms along straight lines) in R d , d ≥ 2, with strictly positive weights W . We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on R d . In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on R d × S d−1 . In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of W is slightly relaxed. We also give examples of continous strictly positive W such that dim ker PW ≥ n in the space of infinitely smooth compactly supported functions on R d for arbitrary n ∈ N ∪ {∞}, where W are infinitely smooth for d = 2 and infinitely smooth almost everywhere for d ≥ 3.

An Example of Non-uniqueness for Radon Transforms with Continuous Positive Rotation Invariant Weights

“…We expect that the results of the present work admit generalizations to the weighted Radon transforms R d,n W along n-dimensional planes in R d for arbitrary d and n such that 1 ≤ n < d, d ≥ 2. For n = 1 such results are already obtained in [GonNov17].…”

Geodesic ray transform with matrix weights for piecewise constant functions

“…[PSUZ19] No. [GN17] Table 1. Is the X-ray transform injective on manifolds that admit a strictly convex function?…”

“…Our theorem holds if dim(M ) ≥ 2, W ∈ C(SM ; GL(k, C)), and functions are piecewise constant in comparison to [PSUZ19] where it is assumed that dim(M ) ≥ 3, W ∈ C ∞ (SM ; GL(k, C)) and functions are smooth. In the Euclidean space R n with n ≥ 3 injectivity is known for C 1,α weights [Ilm16] but there is an example of non-injectivity for W ∈ C α by Goncharov and Novikov [GN17]. Boman constructed an example of a smooth nonvanishing weight on the plane for which the weighted X-ray transform for smooth functions is non-injective [Bom93].…”