Analysis of resolution of tomographic-type reconstruction from discrete data for a class of distributions*This work was supported in part by NSF Grants DMS-1615124 and DMS-1906361. (3rd December 2020) - Katsevich, Alexander
Abstract: Let f ( x ), x ∈ R 2, be a piecewise smooth function with a jump discontinuity across a smooth surface S . Let f Λ ϵ denote the Lambda tomography (LT) reconstruction of f from its discrete Radon data f ̂ ( α k, p j ) . The sampling rate along each variable is ∼ ϵ . First, we compute the limit f 0 ( x ̌ ) = lim ϵ → 0 ϵ f Λ ϵ ( x 0 + ϵ x ̌ ) for a generic x 0 ∈ S . Once the limiting function f 0 ( x ̌ ) is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that straight segments of S lead to non-local artifacts in f Λ ϵ, and that these artifacts are of the same strength as the useful singularities of f Λ ϵ . We also show that f Λ ϵ ( x ) does not converge to its continuous analogue f Λ = (−Δ) 1/2 f as ϵ → 0 even if x ∉ S . Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators B with the same canonical relation as the classical Radon transform adjoint, and a class of distributions g ∈ E ′ ( Z n ), Z n ≔ S n − 1 × R, and obtain easy to use formulas for the DTB when B g is computed from discrete data g ( α k →, p j ) . Exact and LT reconstructions are particular cases of this more general theory.