Resolution analysis of inverting the generalized N-dimensional Radon transform in ℝ^n from discrete data

02/17/2021

∙

by   Alexander Katsevich, et al.

∙

0

∙

share

Let ℛ denote the generalized Radon transform (GRT), which integrates over a family of N-dimensional smooth submanifolds 𝒮_ỹ⊂𝒰, 1≤ N≤ n-1, where an open set 𝒰⊂ℝ^n is the image domain. The submanifolds are parametrized by points ỹ⊂𝒱̃, where an open set 𝒱̃⊂ℝ^n is the data domain. The continuous data are g=ℛ f, and the reconstruction is f̌=ℛ^*ℬ g. Here ℛ^* is a weighted adjoint of ℛ, and ℬ is a pseudo-differential operator. We assume that f is a conormal distribution, supp(f)⊂𝒰, and its singular support is a smooth hypersurface 𝒮⊂𝒰. Discrete data consists of the values of g on a lattice ỹ^j with the step size O(ϵ). Let f̌_ϵ=ℛ^*ℬ g_ϵ denote the reconstruction obtained by applying the inversion formula to an interpolated discrete data g_ϵ(ỹ). Pick a generic pair (x_0,ỹ_0), where x_0∈𝒮, and 𝒮_ỹ_0 is tangent to 𝒮 at x_0. The main result of the paper is the computation of the limit f_0(x̌):=lim_ϵ→0ϵ^κf̌_ϵ(x_0+ϵx̌). Here κ≥ 0 is selected based on the strength of the reconstructed singularity, and x̌ is confined to a bounded set. The limiting function f_0(x̌), which we call the discrete transition behavior, allows computing the resolution of reconstruction.