Transform-based backprojection for volume reconstruction of large format electron microscope tilt series
Section snippets
Motivation for this study
In recent years, the format of digital image detectors for the electron microscope has been progressively increasing. These resulting high-resolution large format images are playing an important role in determining the three-dimensional (3D) structure and function of cells and sub-cellular organelles at the scale of proteins and protein complexes (Martone et al., 2002a, Martone et al., 2002b). Many of these cellular structures can extend for tens to hundreds of microns. Large format digital
Image formation in an electron microscope
To understand how the distortions affecting tilt-series images arise, we must first consider image formation in a uniform magnetic field B. An electron moving in a magnetic field can generally be understood to experience a force according towhich acts perpendicular to the plane defined by the velocity vector and the magnetic field lines, where q is the charge and v is the instantaneous velocity. Electrons moving in a magnetic field will generally orbit the magnetic field lines, moving
Objectives of the study
Given the problems outlined above, the task of tomographic reconstruction is somewhat akin to the job of accurately mapping the organs of a jellyfish swimming at a distance in the ocean and photographed through the bottom of a soft drink bottle. This is, of course, somewhat of an exaggeration, but given the requirement for high precision in our reconstructions, the simile is apt. To continue with the analogy, the various movements of the animal, the bending of the light rays in the seawater,
Tomographic reconstruction
After image acquisition, the reconstruction process is divided into three phases. The first phase is to identify the precise location of a set of image features consistent across the image series. The second phase of the reconstruction is to develop geometric correspondences between the configurations of features in the images, and if necessary to transform the images geometrically to bring the features into better alignment. This is the alignment process, and can be done automatically once
Development of projection models and application to the alignment problem
To align a series of exposures, one must define a suitable model for the optics. The general projective model, which models the optics of straight-line trajectories, and is a generalization of pinhole camera models, is given by the following formula:where P is a projection, G is a nonsingular linear transform, t is a translation, and λ is a scaling function on the (X, Y, Z) coordinates. In this case, the projection map gives the destination in the imaging plane of a trajectory γ,
Development environment
Because of the depth of capabilities of the IMOD software system our system has been developed, to a large degree, as an extension of the IMOD packages. Wherever possible we have used the same file formats as used by IMOD. In particular image data files, fiducial data files, xyz coordinate models, and re-projection error data are the same as those in IMOD. This allows us to use the 3dmod program and its plugin modules to view data and reconstructions, refine fiducial models, and use the bead
Alignment
Re-projection error is a common measure of alignment accuracy, and gives a good indication of the quality of the subsequent reconstruction (Baldwin et al., 2005, Brandt et al., 2001a, Brandt et al., 2001b, Penczek et al., 1995). To assess the effectiveness of our alignment models, we compared the mean square error produced by the projective, quadratic, and cubic models. Initial tracking was performed on a seed model, in which a number of gold particles were located in the zero-tilt image of the
Bundle adjustment
The conjugate gradient method for computing the XYZ model along with the projective transforms has proved to be quite stable, with rapid convergence to a local minimum. We have observed problems only when the fiducial coordinates used as input to the process contain gross errors such as duplicate or missing points. Use of the projective model, on the other hand, carries some well-known ambiguities, because the bundle adjustment is well defined only up to a projective map. We have some control
Discussion and conclusions
An objective of this work has been the development of a method, which could adapt to the complexity of the electron microscope as an optical instrument without having to untangle each separate phenomenon and estimate its effect on the process of image formation. The model of image formation as a geometrically nonlinear integral transform from a three-dimensional object to a two-dimensional image has proved to be a simple, yet robust vehicle for the inclusion of numerous optical effects. In
Acknowledgments
The authors thank Professor Jose Maria Carazo of Autonoma University, Madrid, for his encouragement of this work and his careful reading and criticism of early drafts of the manuscript. The authors further thank Drs. Kyoshi Hama, Tatsuo Arii, and David Mastronarde for helping to inspire this work. We are especially grateful to Dr. Mastronarde for his detailed explanations of the modules of the IMOD package that were foundational to this effort. We also express our appreciation to Ben Kopek and
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