Analysis of resolution of tomographic-type reconstruction from discrete data for a class of conormal distributions
share
Let f(x), x∈R^2, be a piecewise smooth function with a jump discontinuity across a smooth surface S. Let f_Λϵ denote the Lambda tomography (LT) reconstruction of f from its discrete Radon data f̂(α_k,p_j). The sampling rate along each variable is ∼ϵ. First, we compute the limit f_0(x̌)=lim_ϵ→0ϵ f_Λϵ(x_0+ϵx̌) for a generic x_0∈ S. Once the limiting function f_0(x̌) 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_Λϵ, and that these artifacts are of the same strength as the useful singularities of f_Λϵ. We also show that f_Λϵ(x) does not converge to its continuous analogue f_Λ=(-Δ)^1/2f 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 conormal 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_k⃗,p_j). Exact and LT reconstructions are particlular cases of this more general theory.
READ FULL TEXT