Glossary

Glossary of terms used in image processing and single-particle methods

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

• Amplitude contrast ratio: For a weak phase, weak amplitude object, the ratio Qo between the contrast transferred by cos(gamma) and sin(gamma) is called amplitude contrast ratio. It is different for different atoms, and strictly speaking also different for different spatial frequencies. Overlooking this dependency, the ratio is often given as a constant. For a specimen composed of different atomic species, Qo can be empirically determined. It is in the range of 10% for cryo-EM specimens, and almost double for specimens negatively stained by heavy-metal salts.
  

• Amplitude correction: correction of the 3D Fourier amplitudes of the EM density map after angular refinement. This correction can be based on an analytical formula, namely by application of a temperature factor with negative B, or by reference to empirical measurements obtained by low-angle X-ray solution scattering (Gabashvili et al., 2000).

• 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.
  
  

• Autocorrelation function (ACF): the cross-correlation function of an image with itself. The ACF is centrosymmetric, and its highest peak is always in the origin, a shift position where both signal and noise contents of the image overlap.
  

• Back-projection: a method of 3D reconstruction from 2D projections. It is based on superposing 3D functions ("back-projection bodies") obtained by translating the 2D projections along the directions of projection. To obtain the correct result, the projections must be weighted in Fourier space prior to weighted back-projection.
  

• 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.
  

• Coherence (spatial): the degree to which separate parts of the electron wave have defined phase relationships, and are able to interfere with each other in the image formation process. There exists a gradation from full coherence (all parts of the wave have defined phase relationships) over partial coherence (intermediate situation) to incoherence (no defined phase relationships). In bright field electron microscopy, the contrast transfer function is normally given for the fully coherent situation, and the influence of partial coherence is expressed by an envelope function.
  

• Common lines method: a method of relating different projections of the same object to one another. In Fourier space, 2D projections are represented by central planes, which intersect each other along lines that go through the origin: the common lines.
  

• Contrast transfer function (CTF; also: phase contrast transfer function): function that describes the transfer of information from the object (i.e., the object's projection) to the contrast observed in the image for bright field electron microscopy (the usual mode of imaging). If both object and image contrast are expressed in terms of their Fourier transforms O(k) and I(k), then I(k) = CTF(k) * O(k). The two-dimensional variable k = (kx , ky ) is called spatial frequency. In the absence of astigmatism, the contrast transfer function is a function of the radius k = sqrt(kx**2 + ky**2 ) only - transfer of information is not direction-dependent.
  

• Convolution product: a 2D or 3D function obtained as the result of a superposition integral. The integrand is an image or volume, and the superposed kernel is a typically short-range "point spread function" that expresses the effect of a linear optical system in the formation of the image of an "object." As the cross-correlation function, the convolution product can be computed via FFT by taking advantage of a Fourier theorem. Images should be padded prior to FFT computation of the convolution, to avoid circular self-overlap.
  

• Correspondence analysis (CA): a multivariate statistical analysis technique, used to analyze patterns of variation in an image data set. Together with principal component analysis, CA belongs to the eigenvector methods of ordination. In electron microscopy, the technique is used to (A) reduce the dimensionality of the data set, so that automated methods of classification are facilitated; or (B) obtain an immediate inventory of the different components of variability (e.g., depth of staining, rocking of a molecule, presence or absence of a domain, conformational variability).
  

• Cross-correlation function (CCF): a 2D or 3D function that is obtained by forming the scalar cross-product of two images or volumes(i.e., the sum of products of equivalent pixels) as a function of a 2D or 3D shift vector. The CCF is often used to achieve alignment between two imagesor volumes, since it displays a high value (a peak) at the place where a motif contained in both images come into register. Note that for expediency, the CCF is computed by taking advantage of a Fourier theorem. Because of that, images or volumes should be padded prior to CCF computation, to avoid circular self-overlap.

• Defocus: defocus setting (with reference to the Gaussian focus) in the electron microscope, normally given in Angstrom units, which determines the shape of the CTF. For different settings of the defocus, different bands of the object spectrum are emphasized. Note that in the electron optical convention, overfocus means negative defocus values, underfocus (the only useful range) means positive defocus values.

• Defocus spread: the spread of defocus due to the spread of electron energies or to the fluctuation of lens current.

• 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.

• Drift: a movement of the specimen during the exposure time, which causes a blurring of the image. It leaves a characteristic signature in the diffraction pattern, from which the type, the direction, and the extent of the movement can be inferred (Frank, 1969).

• Electron energy: the energy of an electron accelerated from its rest energy by a voltage U is given by eU. For example, electrons impinging on the specimen in an electron microscope operated at 100 kV have the energy 100 keV (kilo electron Volts).

• 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.

• Fourier space: the argument space of the Fourier transform, spanned by the components of the spatial frequency vector.

• Fourier transform (2D): mathematical representation of an image as a series of two-dimensional sine waves. The smaller the wavelength, the higher the spatial frequency, the higher the resolution of features in the image represented. Each Fourier component is indexed by its spatial frequency (kx, ky) and is given as a pair of numbers: amplidude (describing the strength of the sine wave) and phase (describing the position of the sine wave).
  

• 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].
  

• Padding: an operation that puts the given image into a larger-sized array, surrounding it with pixels equal to the mean value of the pixels within the image. Padding is used to avoid self-overlap in the Fourier-based computation of the ACF of an image, or the CCF of two images.
  

• Pixel size: the number of Angstroms per pixel in the digitized micrograph. Pixel size is computed as
  

  ps (A/p) = [10,000(A/u) * SR(u) * DF] / M

  where ps = pixelsize, SR = scanning resolution, DF = decimation factor, M = magnification
  (units: A = Angstroms, p = pixels, u = microns). Thus, if the images were acquired at 50,000 magnification, scanned at 7 microns, then reduced in size by 2, the pixel size would be (10000 * 7 * 2)/50000 or 2.8 Angstroms/pixel.
  

• Point-spread function: the kernel in the superposition integral (convolution product) that expresses the effect of a linear optical system in the formation of an image of an object. Point spread function and transfer function form Fourier pairs. In bright-field electron microscopy, the point-spread function is the Fourier transform of the contrast transfer function.
  

• 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.
  

• Refinement: generally, an iterative way of improving the accuracy, or resolution of a map by making use of constraints, see three-dimensional projection matching.
  

• 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.]
  

• Signal-to-noise ratio: ratio of signal variance to noise variance, computed for a given band limit.
  

• Source size: the size of the virtual electron source in the electron microscope, resulting in illumination divergence, which determines the degree of spatial coherence. The source size is given as an angle, in degrees or in terms of spatial frequency (i.e., mapped into the back focal plane). For details, see Frank (1973).
  

• Spatial frequency: (a) a vector giving the frequency components (number of cycles per unit length, kx and ky ) of a general 2D sine wave in x- and y-direction; or (b) the length of a spatial frequency vector. In the 2D Fourier transform plane, the spatial frequency vector gives the location of any point. Points that have the same [absolute] spatial frequency lie on a circle with radius sqrt(kx**2 + ky**2). For representation of 3D objects or images, the spatial frequency vector is three-dimensional, with components (kx , ky , kz), corresponding to 3D sine waves, and points with the same [absolute] spatial frequency lie on the surface of a sphere with radius sqrt(kx**2 + ky**2 + kz**2).
  

• Spectral signal-to-noise ratio (SSNR): measure of resolution proposed by Michael Unser and coworkers [Unser et al., 1987] [119-121].
  

• Spherical aberration (3rd order): electron optical lens aberration resulting in decreased focal length for rays that are farther from the optical axis.

• Thon rings: rings visible in the power spectrum of micrographs obtained by bright-field electron microscopy. These rings can be explained as the effect of the contrast transfer function, which modulates the Fourier transform of the object in a defocus-dependent way. Presence of axial astigmatism is evident from deviations of the Thon rings from perfect circular symmetry, defocus is gauged from the placement of the rings, and resolution can be gauged from the range of the rings. For these reasons, electron micrographs are routinely inspected by optical diffraction, prior to scanning and digital analysis.
  

• 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.
  

• Wiener filtration: application of the Wiener filter, designed to restore a signal in the presence of noise in an optimal way, in the least-square sense.
  

References

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.