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On discrete X-ray transform

Roman Novikov, Basant Lal Sharma

TL;DR

This work analyzes a discrete version of the X-ray transform on $\mathbb{Z}^d$ with the goal of non-overdetermined reconstruction from limited data. It distinguishes irrational and rational ray directions, establishing direct pointwise inversions for irrational rays via $f(x)=\mathcal{P}f(\gamma_{x,\theta})$ and a slice-by-slice, Cormack-type reconstruction for rational rays on structured sets $\mathpzc{T}^*$ (and extensions $\mathpzc{T}^*_{\alpha,\beta}$) under $\mathrm{supp}\,f \subset B_r$, with results that extend to weighted versions $\mathcal{P}_W$. The paper also clarifies the relationship between discrete and continuous X-ray transforms, showing exact correspondences for lattice delta inputs and deriving regular-case relations that connect $\mathcal{P}^{\text{con}}$ and $\mathcal{P}^{\text{dis}}$ via residuals like $\delta f$; these bridges provide a framework to transfer insights between continuous tomography and its discrete sampling. Overall, the results deliver non-overdetermined, staircase-like reconstruction procedures for both irrational and rational ray families, include weighted generalizations, and illuminate the discrete-to-continuous transition with potential computational benefits in tomographic sensing.

Abstract

We consider a discrete version of X-ray transform going back, in particular, to Strichartz (1982). We suggest non-overdetermined reconstruction for this discrete transform. Extensions to weighted (attenuated) analogues are given. Connections to the continuous case are presented.

On discrete X-ray transform

TL;DR

This work analyzes a discrete version of the X-ray transform on with the goal of non-overdetermined reconstruction from limited data. It distinguishes irrational and rational ray directions, establishing direct pointwise inversions for irrational rays via and a slice-by-slice, Cormack-type reconstruction for rational rays on structured sets (and extensions ) under , with results that extend to weighted versions . The paper also clarifies the relationship between discrete and continuous X-ray transforms, showing exact correspondences for lattice delta inputs and deriving regular-case relations that connect and via residuals like ; these bridges provide a framework to transfer insights between continuous tomography and its discrete sampling. Overall, the results deliver non-overdetermined, staircase-like reconstruction procedures for both irrational and rational ray families, include weighted generalizations, and illuminate the discrete-to-continuous transition with potential computational benefits in tomographic sensing.

Abstract

We consider a discrete version of X-ray transform going back, in particular, to Strichartz (1982). We suggest non-overdetermined reconstruction for this discrete transform. Extensions to weighted (attenuated) analogues are given. Connections to the continuous case are presented.
Paper Structure (11 sections, 6 theorems, 114 equations)

This paper contains 11 sections, 6 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Non-overdetermined inversions for discrete X-ray transform
  3. Irrational rays
  4. Rational rays
  5. Weighted case
  6. Relations with the continuous case
  7. Case of delta functions
  8. Regular case
  9. Proofs of some estimates
  10. Proof of estimate \ref{['defT']}
  11. Proof of Remark \ref{['rem3p1']}

Key Result

Proposition 3.3

Let the assumptions of Theorem pro3p1 be satisfied. Then the projection $(\mathcal{P}^{con}f)_\theta$ on $X_{\theta,r,\epsilon}$ uniquely determines $(\mathcal{P}^{dis}f^{dis})_\theta$, for any fixed $\theta\in\mathbb{S}^{d-1}$ and any $\epsilon>0$. More precisely, for each $y\in X^{dis}_{\theta,r}$

Theorems & Definitions (10)

  • Conjecture 2.3
  • Conjecture 2.4
  • Remark 3.1
  • Proposition 3.3
  • Corollary 3.4
  • Corollary 3.5
  • Proposition 3.6
  • Proposition 3.7
  • Corollary 3.8
  • Remark 3.9