Estimating alignment errors in sets of 2-D images.

We describe a robust and accurate method for the estimation of alignment errors for a set of two-dimensional images, in the case where the true pattern is unknown. The intended application of the proposed method is cryo-electron microscopy, where two-dimensional views of individual proteins in random orientations are observed in the electron microscope at low signal-to-noise ratio. By representing images in the basis of Fourier-harmonic coordinates and constructing averages and average intensities, we demonstrate that the variances of translation and rotational errors as well as of the Gaussian noise can be recovered. This machinery therefore allows one to isolate the various categories of errors that impede the quality of results in single particle reconstructions into constituent parts: translational errors, rotational errors, and additive noise.

[1]  JEFFREY WOOD,et al.  Invariant pattern recognition: A review , 1996, Pattern Recognit..

[2]  G J Jensen,et al.  Alignment error envelopes for single particle analysis. , 2001, Journal of structural biology.

[3]  Padmaja Natarajan,et al.  A statistical approach to computer processing of cryo-electron microscope images: virion classification and 3-D reconstruction. , 2003, Journal of structural biology.

[4]  Z Yin,et al.  An ab initio algorithm for low-resolution 3-D reconstructions from cryoelectron microscopy images. , 2001, Journal of structural biology.

[5]  Laurent Joyeux,et al.  Efficiency of 2D alignment methods. , 2002, Ultramicroscopy.

[6]  R. Muller,et al.  Spectrum of 100-kyr glacial cycle: orbital inclination, not eccentricity. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[7]  J Frank.  Three-dimensional reconstruction of single molecules. , 1981, Methods in cell biology.

[8]  Peter C. Doerschuk,et al.  Explicit orthonormal fixed bases for spaces of functions that are totally symmetric under the rotational symmetries of a platonic solid , 1996 .

[9]  S. Provencher,et al.  Three-dimensional reconstruction from electron micrographs of disordered specimens. I. Method. , 1988, Ultramicroscopy.

[10]  J. Frank.  Three-Dimensional Electron Microscopy of Macromolecular Assemblies , 2006 .

[11]  S. Provencher,et al.  Three-dimensional reconstruction from electron micrographs of disordered specimens. II. Implementation and results. , 1988, Ultramicroscopy.

[12]  J Frank,et al.  Three-dimensional reconstruction of single particles embedded in ice. , 1992, Ultramicroscopy.

[13]  W. Wriggers,et al.  Fast rotational matching. , 2002, Acta crystallographica. Section D, Biological crystallography.

[14]  J Frank,et al.  The structure of the stalk surface layer of a brine pond microorganism: correlation averaging applied to a double layered lattice structure , 1985, Journal of microscopy.

[15]  M Schatz,et al.  Invariant classification of molecular views in electron micrographs. , 1990, Ultramicroscopy.

[16]  Howard Georgi,et al.  Lie Algebras in Particle Physics , 1982 .

[17]  M. Heel,et al.  The portal protein of bacteriophage SPP1: a DNA pump with 13‐fold symmetry. , 1993, The EMBO journal.

[18]  Brian M. Sadler,et al.  Shift- and rotation-invariant object reconstruction using the bispectrum , 1992 .

[19]  H. Ahmedov,et al.  Unitary representations of the two-dimensional Euclidean group in the Heisenberg algebra , 2000 .

[20]  Joachim Frank,et al.  chapter 16 Three-Dimensional Reconstruction of Single Molecules , 1981 .

[21]  J. C. Dainty,et al.  Knox–Thompson and triple-correlation imaging through atmospheric turbulence , 1988 .

[22]  Song Jin,et al.  Reconstruction of speckled images using bispectra , 1992 .

[23]  Z. Kam.  The reconstruction of structure from electron micrographs of randomly oriented particles. , 1980, Journal of theoretical biology.

[24]  Pablo Chacón,et al.  Fast rotational matching of rigid bodies by fast Fourier transform acceleration of five degrees of freedom. , 2003, Acta crystallographica. Section D, Biological crystallography.

[25]  Georgios B. Giannakis,et al.  Signal reconstruction from multiple correlations: frequency- and time-domain approaches , 1989 .

[26]  J Frank.  Quantitative three-dimensional reconstruction. Introduction. , 1981, Methods in cell biology.

[27]  Joachim Frank,et al.  chapter 12 Introduction , 1981 .

[28]  Pawel A Penczek,et al.  Three-dimensional spectral signal-to-noise ratio for a class of reconstruction algorithms. , 2002, Journal of structural biology.

[29]  R. Wade,et al.  Electron microscope transfer functions for partially coherent axial illumination and chromatic defocus spread , 1977 .

[30]  W. O. Saxton,et al.  The correlation averaging of a regularly arranged bacterial cell envelope protein , 1982, Journal of microscopy.

[31]  Willy Wriggers,et al.  2D fast rotational matching for image processing of biophysical data. , 2003, Journal of structural biology.

[32]  M. Heel,et al.  On the Stretching of Distorted Images of Two-Dimensional Crystals , 1980 .

[33]  M. Unser,et al.  A new resolution criterion based on spectral signal-to-noise ratios. , 1987, Ultramicroscopy.

[34]  Donald L. Snyder,et al.  Image recovery from correlations , 1992 .

[35]  J. Frank,et al.  The ribosome at improved resolution: new techniques for merging and orientation refinement in 3D cryo-electron microscopy of biological particles. , 1994, Ultramicroscopy.