2024/05/22 by Alexander Katsevich, Katsevich, Alexander · 2 citations
Computer Science · Medicine · #Digital Image Processing Techniques #FOS: Mathematics #Medical Image Segmentation Techniques #Medical Imaging Techniques and Applications #Numerical Analysis (math.NA) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2405.13269
openalex publication_date 2024/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider a wide class of generalized Radon transforms \mathcal R, which act in ℝn for any n≥ 2 and integrate over submanifolds of any codimension N, 1≤ N≤ n-1. Also, we allow for a fairly general reconstruction operator \mathcal A. The main requirement is that \mathcal A be a Fourier integral operator with a phase function, which is linear in the phase variable. We consider the task of image reconstruction from discrete data gj,k = (\mathcal R f)j,k + ηj,k. We show that the reconstruction error Nεrec=\mathcal A ηj,k satisfies Nrec(\check x;x0)=limε→0Nεrec(x0+ε\check x), \check x∈ D. Here x0 is a fixed point, D⊂ℝn is a bounded domain, and ηj,k are independent, but not necessarily identically distributed, random variables. Nrec and Nεrec are viewed as continuous random functions of the argument \check x (random fields), and the limit is understood in the sense of probability distributions. Under some conditions on the first three moments of ηj,k (and some other not very restrictive conditions on x0 and \mathcal A), we prove that Nrec is a zero mean Gaussian random field and explicitly compute its covariance. We also present a numerical experiment with a cone beam transform in ℝ3, which shows an excellent match between theoretical predictions and simulated reconstructions.