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Signal Reconstruction from Samples at Unknown Locations with Application to 2D Unknown View Tomography

Sheel Shah, Kaishva Shah, Karthik S. Gurumoorthy, Ajit Rajwade

TL;DR

This work tackles the challenge of reconstructing signals and images when sample locations and projection angles are unknown. It extends 1D theory from fully known sampling to quasi-bandlimited signals with arbitrary, known sampling distributions and imperfect sample ordering, deriving explicit error bounds that decay with the number of samples $N$ and depend on noise $\sigma$, distribution density lower bound $\zeta$, and ordering error parameters. Building on this, the authors cast 2D unknown-view tomography (UVT) as a special case of reconstructing a QBL signal from samples at unknown locations, and they derive asymptotic bounds for 2D QBL image reconstruction from 1D Radon projections under unknown angles drawn from a known distribution, supported by comprehensive simulations. The results provide the first analytical performance guarantees for 2D UVT in this sampling-theory framework, demonstrating that accurate reconstruction is achievable under mild conditions and offering a principled basis for ordering algorithms (e.g., Laplacian Eigenmaps) and denoising in high-noise or nonuniform settings. The findings have practical implications for applications such as cryo-electron microscopy and remote sensing, where projection parameters are often imperfect or unknown, and they open avenues for extending the theory to 3D and adaptive angle-distribution estimation.

Abstract

It is well known that a band-limited signal can be reconstructed from its uniformly spaced samples if the sampling rate is sufficiently high. More recently, it has been proved that one can reconstruct a 1D band-limited signal even if the exact sample locations are unknown, but given a uniform distribution of the sample locations and their ordering in 1D. In this work, we extend the analytical error bounds in such scenarios for quasi-bandlimited (QBL) signals, and for the case of arbitrary but known sampling distributions. We also prove that such reconstruction methods are resilient to a certain proportion of errors in the specification of the sample location ordering. We then express the problem of tomographic reconstruction of 2D images from 1D Radon projections under unknown angles (2D UVT) with known angle distribution, as a special case for reconstruction of QBL signals from samples at unknown locations with known distribution. Building upon our theoretical background, we present asymptotic bounds for 2D QBL image reconstruction from 1D Radon projections in the unknown angles setting, and present an extensive set of simulations to verify these bounds in varied parameter regimes. To the best of our knowledge, this is the first piece of work to perform such an analysis for 2D UVT and explicitly relate it to advances in sampling theory, even though the associated reconstruction algorithms have been known for a long time.

Signal Reconstruction from Samples at Unknown Locations with Application to 2D Unknown View Tomography

TL;DR

This work tackles the challenge of reconstructing signals and images when sample locations and projection angles are unknown. It extends 1D theory from fully known sampling to quasi-bandlimited signals with arbitrary, known sampling distributions and imperfect sample ordering, deriving explicit error bounds that decay with the number of samples and depend on noise , distribution density lower bound , and ordering error parameters. Building on this, the authors cast 2D unknown-view tomography (UVT) as a special case of reconstructing a QBL signal from samples at unknown locations, and they derive asymptotic bounds for 2D QBL image reconstruction from 1D Radon projections under unknown angles drawn from a known distribution, supported by comprehensive simulations. The results provide the first analytical performance guarantees for 2D UVT in this sampling-theory framework, demonstrating that accurate reconstruction is achievable under mild conditions and offering a principled basis for ordering algorithms (e.g., Laplacian Eigenmaps) and denoising in high-noise or nonuniform settings. The findings have practical implications for applications such as cryo-electron microscopy and remote sensing, where projection parameters are often imperfect or unknown, and they open avenues for extending the theory to 3D and adaptive angle-distribution estimation.

Abstract

It is well known that a band-limited signal can be reconstructed from its uniformly spaced samples if the sampling rate is sufficiently high. More recently, it has been proved that one can reconstruct a 1D band-limited signal even if the exact sample locations are unknown, but given a uniform distribution of the sample locations and their ordering in 1D. In this work, we extend the analytical error bounds in such scenarios for quasi-bandlimited (QBL) signals, and for the case of arbitrary but known sampling distributions. We also prove that such reconstruction methods are resilient to a certain proportion of errors in the specification of the sample location ordering. We then express the problem of tomographic reconstruction of 2D images from 1D Radon projections under unknown angles (2D UVT) with known angle distribution, as a special case for reconstruction of QBL signals from samples at unknown locations with known distribution. Building upon our theoretical background, we present asymptotic bounds for 2D QBL image reconstruction from 1D Radon projections in the unknown angles setting, and present an extensive set of simulations to verify these bounds in varied parameter regimes. To the best of our knowledge, this is the first piece of work to perform such an analysis for 2D UVT and explicitly relate it to advances in sampling theory, even though the associated reconstruction algorithms have been known for a long time.
Paper Structure (25 sections, 23 theorems, 52 equations, 7 figures, 1 table)

This paper contains 25 sections, 23 theorems, 52 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work and Preliminaries
  3. Signal Reconstruction from Samples at Unknown Locations
  4. Classical Tomographic Reconstruction
  5. Tomographic Reconstruction from Unknown Angles of Projection
  6. Overview of Approach in this Paper
  7. Signal Reconstruction in 1D from Samples at Unknown Locations
  8. Original Problem
  9. Quasi-bandlimited signals
  10. Model for Errors in Specification of Sample Ordering
  11. Model for Noise in Samples
  12. 1D signal reconstruction with ordering errors
  13. Comparison with Compressed Sensing
  14. Tomographic Image Reconstruction from Unknown Projection Angles
  15. Experiments
  16. ...and 10 more sections

Key Result

Theorem 1

Consider the setting in (P3). Choose $k_0 = \left\lceil \frac{\log(N)}{\gamma} \right\rceil$ as the bandwidth of a band-limited approximation, $g^{k_0}(t)$, of the original QBL signal $g(t)$ (i.e., $g^{k_0}(t) := \sum_{|k| \leq k_0} a_k e^{j2\pi kt}$). Define our reconstructed signal as follows: where $\{t'_i\}_{i=1}^N$ are as defined in (P3), $\{\Bar{a}_k\}_{k=1}^N$ are discrete approximations o

Figures (7)

  • Figure 1: Diagram showing a ring for a constant $\rho$ but varying $\theta$ in the Fourier domain.
  • Figure 2: An example of shuffles (left two rows) and shifts (right two rows).
  • Figure 3: In the first two rows, the plots are presented for settings S1, S2, S3 (defined in the text) for angles drawn from $\textrm{Uniform}(0,2\pi)$ (left) and $\textrm{VM}(\mu=\pi,\kappa=1)$ (right). First row: Squared relative reconstruction error $E$ (averaged over 10 runs) versus the number of projections $N$ keeping $f_n = 0.05$. A plot of $\mathcal{O}{(1/N)}$ and $\mathcal{O}((\log N)^3/N)$ are shown for comparison. See SM supplemental material for plots for $\kappa = 2$. Second row: Plot of $E$ versus squared noise percentage $f_n ^ 2$ for $N = 5000$. Third row: Plot of $E$ versus $N_{\bar{\delta}}$ for $f_n = 0.05$ and $N = 5000$ for setting S2.
  • Figure 4: Leftmost: ground truth image; Sample reconstructions and relative reconstruction error ($E$) values with $N \in \{2000,4000,6000\}$ projections and $f_n \in \{0.05,0.08,0.1\}$ for setting S3, i.e. via Laplacian eigenmaps, for angles drawn from $\textrm{Uniform}(0,2\pi)$ (left) and $\textrm{VM}(\pi,1)$ (right). Results with other $\kappa$ values are presented in supplemental material.
  • Figure 5: Variation of $N_{\bar{\delta}}$ averaged over 10 instances with noise percentage $f_n$ (first row). Histograms of $N_{\bar{\delta}}$ values over the different instances for $f_n = 0.05$ (second row). In both cases, the angles are drawn from $\textrm{Uniform}(0,2\pi)$ (left) and $\textrm{VM}(\pi,1)$ (right). All results are for $N = 5000$ projections.
  • ...and 2 more figures

Theorems & Definitions (42)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Corollary 1
  • Lemma 3
  • Theorem 2
  • Corollary 2
  • Lemma 4
  • Theorem 3
  • Lemma 5
  • ...and 32 more