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Two-Dimensional Unknown View Tomography from Unknown Angle Distributions

Kaishva Chintan Shah, Karthik S. Gurumoorthy, Ajit Rajwade

TL;DR

The paper tackles 2D unknown-view tomography (UVT) where both projection angles $\{\theta_i\}$ and their distribution $F$ are unknown. It introduces a cross-validation-based alternating optimization to jointly estimate the image $g$ and the angle distribution $\Gamma$, using order statistics and Laplacian eigenmaps for angular ordering and two distribution models: a semi-parametric mixture of von Mises (MVF) and a PMF. The method reconstructs images from a reconstruction set and evaluates fit on a held-out validation set via a cross-validation error $J$, updating $\Gamma$ by gradient-based steps (softmax for MVF weights or mirror-descent for PMF) and then performing final reconstruction with the updated parameters. Experiments on four 512×512 images demonstrate improved reconstruction quality over a uniform-angle graph Laplacian baseline and competitive results versus an oracle angle reconstruction, highlighting the method’s potential for cryo-EM and CT calibration when angle distributions are unknown.

Abstract

This study presents a technique for 2D tomography under unknown viewing angles when the distribution of the viewing angles is also unknown. Unknown view tomography (UVT) is a problem encountered in cryo-electron microscopy and in the geometric calibration of CT systems. There exists a moderate-sized literature on the 2D UVT problem, but most existing 2D UVT algorithms assume knowledge of the angle distribution which is not available usually. Our proposed methodology formulates the problem as an optimization task based on cross-validation error, to estimate the angle distribution jointly with the underlying 2D structure in an alternating fashion. We explore the algorithm's capabilities for the case of two probability distribution models: a semi-parametric mixture of von Mises densities and a probability mass function model. We evaluate our algorithm's performance under noisy projections using a PCA-based denoising technique and Graph Laplacian Tomography (GLT) driven by order statistics of the estimated distribution, to ensure near-perfect ordering, and compare our algorithm to intuitive baselines.

Two-Dimensional Unknown View Tomography from Unknown Angle Distributions

TL;DR

The paper tackles 2D unknown-view tomography (UVT) where both projection angles and their distribution are unknown. It introduces a cross-validation-based alternating optimization to jointly estimate the image and the angle distribution , using order statistics and Laplacian eigenmaps for angular ordering and two distribution models: a semi-parametric mixture of von Mises (MVF) and a PMF. The method reconstructs images from a reconstruction set and evaluates fit on a held-out validation set via a cross-validation error , updating by gradient-based steps (softmax for MVF weights or mirror-descent for PMF) and then performing final reconstruction with the updated parameters. Experiments on four 512×512 images demonstrate improved reconstruction quality over a uniform-angle graph Laplacian baseline and competitive results versus an oracle angle reconstruction, highlighting the method’s potential for cryo-EM and CT calibration when angle distributions are unknown.

Abstract

This study presents a technique for 2D tomography under unknown viewing angles when the distribution of the viewing angles is also unknown. Unknown view tomography (UVT) is a problem encountered in cryo-electron microscopy and in the geometric calibration of CT systems. There exists a moderate-sized literature on the 2D UVT problem, but most existing 2D UVT algorithms assume knowledge of the angle distribution which is not available usually. Our proposed methodology formulates the problem as an optimization task based on cross-validation error, to estimate the angle distribution jointly with the underlying 2D structure in an alternating fashion. We explore the algorithm's capabilities for the case of two probability distribution models: a semi-parametric mixture of von Mises densities and a probability mass function model. We evaluate our algorithm's performance under noisy projections using a PCA-based denoising technique and Graph Laplacian Tomography (GLT) driven by order statistics of the estimated distribution, to ensure near-perfect ordering, and compare our algorithm to intuitive baselines.
Paper Structure (4 sections, 2 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 4 sections, 2 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Method
  3. Experiments
  4. Discussion and Conclusion

Figures (3)

  • Figure 1: Visual comparison between the ground truth and reconstructions from 5000 noisy projections, obtained from our algorithm, ORP, and graph Laplacian assuming uniform distribution (GLTU) as in Coifman2008. Our technique where we learn the distribution, is able to reconstruct image details much better than GLTU.
  • Figure 2: Comparison of the estimated and original distributions for MVF (col. 1) and PMF models (col. 3), along with the histogram of absolute angle differences (cols. 2 and 4 resp.). The estimated and original MVF both contain 5 components. The estimated PMF was computed with 25 bins, whereas the original PMF, which was generated using 50 bins, was combined into 25 bins for representation purposes.
  • Figure 3: The cross-validation error is shown against iterations for images 1 and 4 and both distribution models (PMF and MVF).