Chapter 8
Tomographic Reconstruction
from Electron Micrographs
J. Bernard Heymann
Abstract Reconstruction from a tilt series of electron micrographs is based on the
assumption that each image represents a projection through the specimen, and that the
3D information can be recovered by “back-projecting” all the images in the correct
geometry. We use algorithms that are integrative (back-projection or Fourier inversion) or iterative (algebraic or maximum entropy methods). In practical tomography,
we can only record a finite set of images at a dose low enough to avoid radiation
damage, yielding noisy tomograms with missing information. The quality of the
tomograms depends on the algorithmic details, but also on the pre-processing of the
tilt series images, and post-processing of the tomographic volume.
8.1 Principles of Reconstruction
Any modality of imaging yields by nature a two-dimensional (2D) image of a
three-dimensional (3D) object or scene. In the electron microscope (as in X-ray
imaging), we treat the images as projections because of the large depth-of-focus [1].
The image is approximated to a good degree as an integration of the density along
the beam direction. Radon [2, 3] showed that it is in principle possible to recover
the 3D information from such images taken with different projection directions.
Herman [4] reminded us that implementing such a mathematical operation is
non-trivial, given that (i) we only have a finite number of images, (ii) there are
inaccuracies in the measurements, and (iii) an efficient approach is required. de
Rosier and Klug [1] formulated the first practical implementation of reconstruction
from micrographs in frequency space (FSR or frequency space reconstruction).
Around the same time, real space reconstruction algorithms were developed,
including weighted back-projection (WBP) [5] and the algebraic reconstruction
technique (ART) [6]. These algorithms form the basis of most subsequent
J. Bernard Heymann (&)
National Institute of Arthritis, Musculoskeletal and Skin Diseases,
National Institutes of Health, 50 South Dr, Bethesda, MD 20892, USA
e-mail: heymannb@mail.nih.gov
© Springer International Publishing AG 2018
E. Hanssen (ed.), Cellular Imaging, Biological and Medical Physics,
Biomedical Engineering, https://doi.org/10.1007/978-3-319-68997-5_8
209
Tomographic Reconstruction
from Electron Micrographs
J. Bernard Heymann
Abstract Reconstruction from a tilt series of electron micrographs is based on the
assumption that each image represents a projection through the specimen, and that the
3D information can be recovered by “back-projecting” all the images in the correct
geometry. We use algorithms that are integrative (back-projection or Fourier inversion) or iterative (algebraic or maximum entropy methods). In practical tomography,
we can only record a finite set of images at a dose low enough to avoid radiation
damage, yielding noisy tomograms with missing information. The quality of the
tomograms depends on the algorithmic details, but also on the pre-processing of the
tilt series images, and post-processing of the tomographic volume.
8.1 Principles of Reconstruction
Any modality of imaging yields by nature a two-dimensional (2D) image of a
three-dimensional (3D) object or scene. In the electron microscope (as in X-ray
imaging), we treat the images as projections because of the large depth-of-focus [1].
The image is approximated to a good degree as an integration of the density along
the beam direction. Radon [2, 3] showed that it is in principle possible to recover
the 3D information from such images taken with different projection directions.
Herman [4] reminded us that implementing such a mathematical operation is
non-trivial, given that (i) we only have a finite number of images, (ii) there are
inaccuracies in the measurements, and (iii) an efficient approach is required. de
Rosier and Klug [1] formulated the first practical implementation of reconstruction
from micrographs in frequency space (FSR or frequency space reconstruction).
Around the same time, real space reconstruction algorithms were developed,
including weighted back-projection (WBP) [5] and the algebraic reconstruction
technique (ART) [6]. These algorithms form the basis of most subsequent
J. Bernard Heymann (&)
National Institute of Arthritis, Musculoskeletal and Skin Diseases,
National Institutes of Health, 50 South Dr, Bethesda, MD 20892, USA
e-mail: heymannb@mail.nih.gov
© Springer International Publishing AG 2018
E. Hanssen (ed.), Cellular Imaging, Biological and Medical Physics,
Biomedical Engineering, https://doi.org/10.1007/978-3-319-68997-5_8
209