Further information is available in the book by Joachim Frank Three-Dimensional Electron Microscopy of Macromolecular Assemblies Oxford University Press, (February 2006)

- Amplitude contrast ratio
- Amplitude correction
- Angular reconstitution
- Astigmatism
- Autocorrelation function (ACF)
- Back-projection
- Coherence
- Classification
- Common lines method
- Convolution product
- Contrast transfer function (CTF; also: phase contrast transfer function)
- Correspondence analysis (CA)
- Cross-correlation function (CCF)
- Defocus
- Defocus spread
- Differential phase residual (DPR)
- Drift
- Electron energy
- Eulerian angles
- Fourier ring correlation (FRC)
- Fourier shell correlation (FSC)
- Fourier space
- Fourier transform (2D)
- Gaussian envelope halfwidth
- High-pass filtration
- Lambda
- Low-pass filtration
- Padding
- Pixel size
- Point spread function
- Power spectrum
- Random-conical reconstruction
- Refinement
- Resolution
- Signal to noise ratio
- Source size
- Spatial frequency
- Spherical aberration
- SSNR
- Three-dimensional projection matching
- Thon rings
- Wiener fltration

**Angular reconstitution:**a common-lines method designed to determine the relative angles between three projections of an object with arbitrary symmetry [van Heel, 1987a]. The method works on the basis of comparing sinograms, which are plots of line projections as a function of angle.

**Astigmatism(axial):**an electron-optical lens aberration that causes the defocus to be a function of azimuth, and the contrast transfer function to deviate from circular symmetry about the optical axis. As a consequence, the Thon rings deform into elliptic or hyperbolic patterns, depending on the size of defocus and the size of the astigmatic defocus difference. See astigmatism as defined by SPIDER operations.

**Classification:**separation of an image set into subsets according to the similarity of features [Frank, 1990]. Automated classification can be done directly on the image set, based on [generalized] Euclidean distances, or on a representation of the image set in a coordinate system with reduced dimensionality, obtained by principal component analysis or correspondence analysis. There are two often-used classification methods: K-means and hierarchical ascendant classification (HAC). In K-means, the data set is split into K (a given number) subsets in such a way that each subset is maximally compact, as measured by the intra-subset variance.

**Differential phase residual (DPR):**a measure of statistical dependency between two averages, computed over rings in Fourier space as a function of ring radius (= spatial frequency, or resolution) [Frank et al., 1981]. Typically, the DPR rises from a low value (good average phase agreement) to a high value around 110 degrees (agreement between statistically unrelated images). A cutoff value of 45 degrees is commonly used as resolution measure.

**Eulerian angles:**a set of three angles that define an orientation, or direction, in space. It goes back to the mathematician Euler, and was used to describe motions in celestial mechanics.

**Fourier ring correlation (FRC):**a measure of statistical dependency between two averages, computed by comparison of rings in Fourier space [Saxton and Baumeister, 1982]. As the Differential Phase Residual, the FRC is used to establish the reproducible resolution in single-particle averaging. Typically, the curve falls off from a value of one (perfect agreement) toward zero (no correlation). The point where it falls below 0.5 is a realistic measure of resolution [Bottcher et al., 1997; Penczek, 1998 (=Appendix in Malhotra et al., 1998)].

**Fourier shell correlation (FSC):**same as Fourier ring correlation, except that values of the discrete Fourier transforms are compared within corresponding shells, instead of rings [van Heel, 1987b.

**Gaussian envelope halfwidth:**the halfwidth of the Gaussian in Fourier space, where the Gaussian is used to modify the envelope of the model CTF function. It is a composite parameter, used to account for a variety of effects, including drift, specimen charging effects, and multiple inelastic-elastic scattering. See TF, Note #2.

**High-pass filtration:**multiplication of the Fourier transform of an image with a rotationally symmetric 2D function ("the filter") that attenuates amplitudes at low spatial frequencies k and increases amplitudes for high spatial frequencies. This has the result of enhancing the edges in the image while suppressing all slow-moving variations. Note that in SPIDER, the FF operation operates on the Fourier transform, while the FQ ("Filter Quick") operation processes the image directly.

**Lambda:**[1] the electron wavelength.*Lambda*is calculated from the relativistic electron energy*kV*:

Lambda = 12.398 / sqrt[kV * (1022 + kV)]kV Angstroms 100 0.03701 120 0.03349 140 0.03074 160 0.02851 180 0.02665 200 0.02508 300 0.01969 400 0.01644

[2] in back projection,*Lambda*is a parameter used to control the speed of convergence. See BP RP, Note #3.

**Low-pass filtration:**multiplication of the Fourier transform of an image with a 2D function ("the filter") that attenuates amplitudes at high spatial frequencies k. This has the result of blurring the image, and of eliminating sharp edges and noise. Low-pass filtration is routinely used to limit the information in a reconstruction to the reproducible resolution, as found by resolution criteria (see Fourier Shell Correlation, Differential Phase Residual). [See High-pass filtration, regarding SPIDER operations FF and FQ].

**Power spectrum:**intensity [= squared amplitude] of the Fourier transform, presented either in the form of an image (= the outcome of the PW operation) or as a profile (= the outcome of averaging the 2D power spectrum azimuthally). [Note that the SPIDER operation PW furnishes the amplitude, not the squared amplitude! This makes it easier to use the distribution in displays and other operations, because of the high dynamic range]. The complete length of the abscissa in such a plot corresponds to the highest resolution represented in the digitization: 1/(2 x sampling distance). For example, if the scanning is done with 20 microns, and the electron optical magnification was 50,000 x, then the highest resolution represented in the digitized image is 5 x 10**4/(2 x 20 x 10**4) A**(-1) = 1/8 A**(-1). This means that the total length of the abscissa corresponds to 1/8 A**(-1) in this case.

**Random-conical reconstruction:**a method of data collection and reconstruction used for single particles, typically used initially in a project, to obtain a first low-resolution reconstruction of the macromolecule [Radermacher et al., 1987]. Two images of the same specimen field are collected, one with untilted grid, the other with the grid tilted by 50 to 60 degrees. Any set of particles presenting the same view in the untilted-specimen image form a random-conical projection set in the associated tilted-specimen image.

**Resolution:**the extent of meaningful information in Fourier space, given by a spatial frequency radius. In 2D averaging or 3D reconstruction of macromolecules in single-particle form, the resolution is estimated by splitting the dataset into halves, and comparing averages or 3D maps from the independently processed halfsets by using a measure of reproducibility. [See also Fourier ring correlation, Fourier shell correlation, Differential phase residual.]

**Three-dimensional projection matching:**a method of refinement in reconstructing single particles from their projections [Penczek et al., 1994]. A preliminary 3D map, obtained by random-conical reconstruction or angular reconstitution, is used to compute "predicted projections" on a grid in angular space, which are held in the computer memory. One by one, the experimental projections are now compared (by cross-correlation, See CCF) with the set of predicted projections, to find the best Eulerian angles representing their projection direction. Next, a refined 3D map is computed, and again this is used to compute "predicted projections," etc. For self-consistent, homogeneous data sets, this procedure converges in a higher-resolution 3D map.

Frank, J. (1990) *Quart. Rev. Biophys.*
23, 281-329.

Frank, J. (1996)
*Three-dimensional Electron Microscopy of Macromolecular
Assemblies.* Academic Press, San Diego.

Frank, J., Verschoor, A., and Boublik, M.
(1981) *Science* 214, 1353-1355.

Malhotra, A., Penczek, P., Agrawal, R.K.,
Gabashvili, I.S., Grassucci, R.A., Juenemann, R., Burkhardt, N.,
Nierhaus, K.H., and Frank, J. (1998) *J. Mol. Biol.* 280,
103-116.

Penczek, P., Grassucci, R.A. and Frank, J.
(1994) *J. Ultramicroscopy* 53, 251-270.

Radermacher, M., Wagenknecht, T.,
Verschoor, A., and Frank, J. (1987) *J. Microsc.* 146,
113-136.

Saxton, W.O. and Baumeister, W. (1982)
*J. Microsc.* 127, 127-138.

van Heel, M. (1987a) *Ultramicroscopy*
21, 111-124.

Unser, M., Trus, B.L., and Steven, A.C.
(1987) *Ultramicroscopy* 30, 429-434.

van Heel, M. (1987b) *Ultramicroscopy*
21, 95-100.

Zhu, J., Penczek, P., Schroder, R., and Frank, J.
(1997) *J Struct Biol* 118, 197-219.

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