Least-Squares Fitting of Two 3-D Point Sets
IEEE Transactions on Pattern Analysis and Machine Intelligence
A method of general moments for orienting 2D projections of unknown 3D objects
Computer Vision, Graphics, and Image Processing
Elements of information theory
Elements of information theory
SIAM Review
The mathematics of computerized tomography
The mathematics of computerized tomography
Information Theory and Reliable Communication
Information Theory and Reliable Communication
Structure and View Estimation for Tomographic Reconstruction: A Bayesian Approach
CVPR '06 Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition - Volume 2
A Framework for Discrete Integral Transformations I—The Pseudopolar Fourier Transform
SIAM Journal on Scientific Computing
Uniqueness of tomography with unknown view angles
IEEE Transactions on Image Processing
Feasibility of tomography with unknown view angles
IEEE Transactions on Image Processing
Graph Laplacian Tomography From Unknown Random Projections
IEEE Transactions on Image Processing
Viewing Angle Classification of Cryo-Electron Microscopy Images Using Eigenvectors
SIAM Journal on Imaging Sciences
Sensor network localization by eigenvector synchronization over the euclidean group
ACM Transactions on Sensor Networks (TOSN)
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The cryo-electron microscopy reconstruction problem is to find the three-dimensional (3D) structure of a macromolecule given noisy samples of its two-dimensional projection images at unknown random directions. Present algorithms for finding an initial 3D structure model are based on the “angular reconstitution” method in which a coordinate system is established from three projections, and the orientation of the particle giving rise to each image is deduced from common lines among the images. However, a reliable detection of common lines is difficult due to the low signal-to-noise ratio of the images. In this paper we describe two algorithms for finding the unknown imaging directions of all projections by minimizing global self-consistency errors. In the first algorithm, the minimizer is obtained by computing the three largest eigenvectors of a specially designed symmetric matrix derived from the common lines, while the second algorithm is based on semidefinite programming (SDP). Compared with existing algorithms, the advantages of our algorithms are five-fold: first, they accurately estimate all orientations at very low common-line detection rates; second, they are extremely fast, as they involve only the computation of a few top eigenvectors or a sparse SDP; third, they are nonsequential and use the information in all common lines at once; fourth, they are amenable to a rigorous mathematical analysis using spectral analysis and random matrix theory; and finally, the algorithms are optimal in the sense that they reach the information theoretic Shannon bound up to a constant for an idealized probabilistic model.