International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 319-320   | 1 | 2 |

## Section 2.5.6.8. Three-dimensional reconstruction in the general case

B. K. Vainshteinc

#### 2.5.6.8. Three-dimensional reconstruction in the general case

| top | pdf |

In the general case of 3D reconstruction from projections the projection vector τ occupies arbitrary positions on the projection sphere (Fig. 2.5.6.2). Then, as in (2.5.6.15), we can construct the three-dimensional spatial synthesis. To do this, let us transform the two-dimensional projections by extending them along τ as in (2.5.6.13) into three-dimensional projection functions .

Analogously to (2.5.6.15), such a three-dimensional synthesis is the integral over the hemisphere (Fig. 2.5.6.2) this is the convolution of the initial function with (Vainshtein, 1971b).

To obtain the exact reconstruction of we find, from each , the modified projection (Vainshtein & Orlov, 1974; Orlov, 1975)

By extending along τ we transform them into . Now the synthesis over the angles gives the three-dimensional function

The approximation for a discrete set of angles is written on the right. In this case we are not bound by the coaxial projection condition which endows the experiment with greater possibilities; the use of object symmetry also profits from this. To carry out the 3D reconstruction (2.5.6.25) or (2.5.6.27) one should know all three Euler's angles ψ, θ, α (Fig. 2.5.6.2).

The projection vectors should be distributed more or less uniformly over the sphere (Fig. 2.5.6.2). This can be achieved by using special goniometric devices.

Another possibility is the investigation of particles which, during the specimen preparation, are randomly oriented on the substrate. This, in particular, refers to asymmetric ribosomal particles. In this case the problem of determining these orientations arises.

The method of spatial correlation functions may be applied if a large number of projections with uniformly distributed projection directions is available (Kam, 1980). The space correlation function is the averaged characteristic of projections over all possible directions which is calculated from the initial projections or the corresponding sections of the Fourier transform. It can be used to find the coefficients of the object density function expansion over spherical harmonics, as well as to carry out the 3D reconstruction in spherical coordinates.

Another method (Van Heel, 1984) involves the statistical analysis of image types, subdivision of images into several classes and image averaging inside the classes. Then, if the object is rotated around some axis, the 3D reconstruction is carried out by the iteration method.

If such a specimen is inclined at a certain angle with respect to the beam, then the images of particles in the preferred orientation make a series of projections inclined at an angle β and having a random azimuth. The azimuthal rotation is determined from the image having zero inclination.

If particles on the substrate have a characteristic shape, they may acquire a preferable orientation with respect to the substrate, their azimuthal orientation α being random (Radermacher et al., 1987).

In the general case, the problem of determining the spatial orientations of randomly distributed identical three-dimensional particles with an unknown structure may be solved by measuring their two-dimensional projections (Fig. 2.5.6.1) if the number i of such projections is not less than three, (Vainshtein & Goncharov, 1986a,b; Goncharov et al. 1987; Goncharov, 1987). The direction of the vector along which the projection is obtained is set by the angle (Fig. 2.5.6.2).

The method is based on the analysis of one-dimensional projections of two-dimensional projections where α is the angle of the rotation about vector τ in the p plane.

Lemma 1. Any two projections and (Fig. 2.5.6.6) have common (identical) one-dimensional projections : Vectors and (Fig. 2.5.6.3) determine plane h in which they are both lying. Vector is normal to plane h and parallel to axis of the one-dimensional projection ; both and axes along which the projections and are constructed are perpendicular to .

 Figure 2.5.6.6 | top | pdf |Relative position of the particle and planes of projection.

The corresponding lemma in the Fourier space states:

Lemma 2. Any two plane transforms, and intersect along the straight line (Fig. 2.5.6.7); the one-dimensional transform is the transform of .

 Figure 2.5.6.7 | top | pdf |Section of a three-dimensional Fourier transform of the density of the particles, corresponding to plane projections of this density.

Thus in order to determine the orientations of a three-dimensional particle it is necessary either to use projections in real space or else to pass to the Fourier space (2.5.6.5).

Now consider real space. The projections are known and can be measured but angles of their rotation about vector (Fig. 2.5.6.8) are unknown and should be determined. Let us choose any two projections and and construct a set of one-dimensional projections and by varying angles and . In accordance with Lemma 1, there exists a one-dimensional projection, common for both and , which determines angles and along which and should be projected for obtaining the identical projection (Fig. 2.5.6.5). Comparing and and using the minimizing function it is possible to find such a common projection . (A similar consideration in Fourier space yields .)

 Figure 2.5.6.8 | top | pdf |Plane projections of a three-dimensional body. The systems of coordinates in planes (a) and (b) are chosen independently of one another.

The mutual spatial orientations of any three non-coplanar projection vectors , , can be found from three different two-dimensional projections , and by comparing the following pairs of projections: and , and , and and , and by determining the corresponding , and . The determination of angles , and reduces to the construction of a trihedral angle formed by planes , and . Then the projections with the known can be complemented with other projections and the corresponding values of ω can be determined. Having a sufficient number of projections and knowing the orientations , it is possible to carry out the 3D reconstruction of the object [see (2.5.6.27); Orlov, 1975; Vainshtein & Goncharov, 1986a; Goncharov et al., 1987].

### References

Goncharov, A. B. (1987). Integral geometry and 3D-reconstruction of arbitrarily oriented identical particles from their electron micrographs. Sov. Phys. Crystallogr. 32, 663–666.Google Scholar
Goncharov, A. B., Vainshtein, B. K., Ryskin, A. I. & Vagin, A. A. (1987). Three-dimensional reconstruction of arbitrarily oriented identical particles from their electron photomicrographs. Sov. Phys. Crystallogr. 32, 504–509.Google Scholar
Kam, Z. (1980). Three-dimensional reconstruction of aperiodic objects. J. Theor. Biol. 82, 15–32.Google Scholar
Orlov, S. S. (1975). Theory of three-dimensional reconstruction. II. The recovery operator. Sov. Phys. Crystallogr. 20, 429–433.Google Scholar
Radermacher, M., McEwen, B. & Frank, J. (1987). Three-dimensional reconstruction of asymmetrical object in standard and high voltage electron microscopy. Proc. Microscop. Soc. Canada, XII Annual Meet., pp. 4–5.Google Scholar
Vainshtein, B. K. (1971b). Finding the structure of objects from projections. Sov. Phys. Crystallogr. 15, 781–787.Google Scholar
Vainshtein, B. K. & Goncharov, A. B. (1986a). Determination of the spatial orientation of arbitrarily arranged identical particles of unknown structure from their projections. Sov. Phys. Dokl. 287, 278–283.Google Scholar
Vainshtein, B. K. & Goncharov, A. B. (1986b). Proceedings of the 11th International Congress on Electron Microscopy, Kyoto, Vol. 1, pp. 459–460.Google Scholar
Vainshtein, B. K. & Orlov, S. S. (1974). General theory of direct 3D reconstruction. Proceedings of International Workshop, Brookhaven National Laboratory, pp. 158–164.Google Scholar
Van Heel, M. (1984). Multivariate statistical classification of noisy images (randomly oriented biological macromolecules). Ultramicroscopy, 13, 165–184.Google Scholar