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

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In the general case of 3D reconstruction [\varphi_{3} ({\bf r})] from projections [\varphi_{2} ({\bf x}_{\tau})] the projection vector τ occupies arbitrary positions on the projection sphere (Fig. 2.5.6.2[link]). Then, as in (2.5.6.15)[link], we can construct the three-dimensional spatial synthesis. To do this, let us transform the two-dimensional projections [\varphi_{2i} [{\bf x}, {\boldtau} (\theta, \psi)_{i}]] by extending them along τ as in (2.5.6.13)[link] into three-dimensional projection functions [\varphi_{3} ({\bf r}_{\tau_{i}})].

Analogously to (2.5.6.15)[link], such a three-dimensional synthesis is the integral over the hemisphere (Fig. 2.5.6.2[link]) [\eqalignno{ \Sigma_{3} ({\bf r}) &= {\textstyle\int\limits_{\omega}} \varphi_{3} ({\bf r},{ \boldtau}_{i}) \ \hbox{d}\omega_{\tau} = \varphi ({\bf r}) * |{\bf r}|^{-2} &\cr &\simeq \Sigma \varphi_{3i} [{\bf r}_{\tau (\theta, \, \psi)_{i}}] \simeq \varphi_{3} ({\bf r}) + B\hbox{;} &(2.5.6.25)\cr}] this is the convolution of the initial function with [|{\bf r}|^{-2}] (Vainshtein, 1971b[link]).

To obtain the exact reconstruction of [\varphi_{3} ({\bf r})] we find, from each [\varphi_{2} ({\bf x}_{{\boldtau}})], the modified projection (Vainshtein & Orlov, 1974[link]; Orlov, 1975[link]) [\tilde{\varphi}_{2} ({\bf x}_{\tau}) = \int {\varphi_{2} ({\bf x}_{\boldtau}) - \varphi_{2} ({\bf x}'_{\boldtau}) \over |{\bf x}_{\boldtau} - {\bf x}'_{\boldtau}|^{3}} \ \hbox{d} s_{{\bf x}^\prime}. \eqno(2.5.6.26)]

By extending [\varphi_{2} ({\bf x}_{{\boldtau}})] along τ we transform them into [\tilde{\varphi}_{3} ({\bf r}_{{\boldtau}})]. Now the synthesis over the angles [\omega_{\tau} = (\theta, \psi, \alpha)_{\tau}] gives the three-dimensional function [\varphi_{3} ({\bf r}) = {1 \over 4 \pi^{3}} \int \tilde{\varphi}_{3} ({\bf r}_{{\boldtau}}) \ \hbox{d}\omega_{{\boldtau}} \simeq \sum\limits_{i} \tilde{\varphi}_{3i} [{\bf r}_{\tau (\theta, \, \psi, \, \alpha)_{i}}]. \eqno(2.5.6.27)]

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)[link] or (2.5.6.27)[link] one should know all three Euler's angles ψ, θ, α (Fig. 2.5.6.2[link]).

The projection vectors [{\boldtau}_{i}] should be distributed more or less uniformly over the sphere (Fig. 2.5.6.2[link]). 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[link]). 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[link]) 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[link]).

In the general case, the problem of determining the spatial orientations of randomly distributed identical three-dimensional particles [\varphi_{3} ({\bf r})] with an unknown structure may be solved by measuring their two-dimensional projections [p ({\bf x}_{\tau})] (Fig. 2.5.6.1[link]) [p ({\bf x}_{\tau_{i}}) \equiv \varphi_{2} ({\bf x}_{\tau_{i}}) \simeq \textstyle\int \varphi_{3} ({\bf r}) \ \hbox{d}\tau_{i}\quad {\bf x} \perp {\boldtau}_{i}\hbox{;} \eqno(2.5.6.1a)] if the number i of such projections is not less than three, [i \geq 3] (Vainshtein & Goncharov, 1986a[link],b[link]; Goncharov et al. 1987[link]; Goncharov, 1987[link]). The direction of the vector [{\boldtau}_{i}] along which the projection [p ({\boldtau}_{i})] is obtained is set by the angle [\omega_{i} (\theta_{i}, \psi_{i})] (Fig. 2.5.6.2[link]).

The method is based on the analysis of one-dimensional projections [q_{\alpha}] of two-dimensional projections [p ({\bf x}_{\tau_{i}})] [q (x_{\perp \alpha}) = \textstyle\int p ({\bf x}_{\tau_{i}}) \ \hbox{d}x_{\parallel \alpha}, \eqno(2.5.6.28)] where α is the angle of the rotation about vector τ in the p plane.

Lemma 1. Any two projections [p_{1} ({\bf x}_{\tau_{i}})] and [p_{2} ({\bf x}_{\tau_{2}})] (Fig. 2.5.6.6[link]) have common (identical) one-dimensional projections [q_{12} (x_{12})]: [q_{12} (x_{12}) = q_{1, \, \alpha_{1} j} (x_{\perp \alpha_{1} j}) = q_{2, \, \alpha_{2} k} (x_{\perp \alpha_{2} k}). \eqno(2.5.6.29)] Vectors [{\boldtau}_{1}] and [{\boldtau}_{2}] (Fig. 2.5.6.3[link]) determine plane h in which they are both lying. Vector [m_{12} = \langle \tau_{1} \tau_{2}\rangle] is normal to plane h and parallel to axis [x_{12}] of the one-dimensional projection [q_{12}]; both [x_{\perp \alpha_{1} j}] and [x_{\perp \alpha_{2} j}] axes along which the projections [q_{1}] and [q_{2}] are constructed are perpendicular to [x_{12}].

[Figure 2.5.6.6]

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, [ \Phi_{2} ({\bf u}_{\tau_{1}}) = {\scr F}_{2} p_{1}] and [ \Phi_{2} ({\bf u}_{\tau_{2}}) = {\scr F}_{2} p_{2}] intersect along the straight line [v_{12}] (Fig. 2.5.6.7[link]); the one-dimensional transform [Q(v_{12})] is the transform of [ q_{12}: Q(v_{12}) = {\scr F}_{1} g_{12}].

[Figure 2.5.6.7]

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 [\omega_{i} (\theta_{i}, \psi_{i}, \alpha_{i})] of a three-dimensional particle [\varphi_{3, \, \omega_{i}} ({\bf r})] it is necessary either to use projections [p_{i}] in real space or else to pass to the Fourier space (2.5.6.5)[link].

Now consider real space. The projections [p_{i}] are known and can be measured but angles [\alpha_{ij}] of their rotation about vector [{\boldtau}_{i}] (Fig. 2.5.6.8[link]) are unknown and should be determined. Let us choose any two projections [p_{1}] and [p_{2}] and construct a set of one-dimensional projections [q_{1, \, \alpha_{1} j}] and [q_{2, \, \alpha_{2} k}] by varying angles [\alpha_{1j}] and [\alpha_{2k}]. In accordance with Lemma 1[link], there exists a one-dimensional projection, common for both [p_{1}] and [p_{2}], which determines angles [\alpha_{1j}] and [\alpha_{2k}] along which [p_{1}] and [p_{2}] should be projected for obtaining the identical projection [q_{12}] (Fig. 2.5.6.5[link]). Comparing [q_{1, \, \alpha_{1} j}] and [q_{2, \, \alpha_{2} k}] and using the minimizing function [D(1, 2) = |q_{1, \, \alpha_{1} j} - q_{2, \, \alpha_{2} k}|^{2} \eqno(2.5.6.30)] it is possible to find such a common projection [q_{12}]. (A similar consideration in Fourier space yields [Q_{12}].)

[Figure 2.5.6.8]

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 [{\boldtau}_{1}], [{\boldtau}_{2}], [{\boldtau}_{3}] can be found from three different two-dimensional projections [p_{1}], [p_{2}] and [p_{3}] by comparing the following pairs of projections: [p_{1}] and [p_{2}], [p_{1}] and [p_{3}], and [p_{2}] and [p_{3}], and by determining the corresponding [q_{12}], [q_{13}] and [q_{23}]. The determination of angles [\omega_{1}], [\omega_{2}] and [\omega_{3}] reduces to the construction of a trihedral angle formed by planes [h_{12}], [h_{13}] and [h_{23}]. Then the projections [p_{i} (\omega_{i})] with the known [\omega_{i}\ (i = 1, 2, 3)] can be complemented with other projections [(i = 4, 5, \ldots)] and the corresponding values of ω can be determined. Having a sufficient number of projections and knowing the orientations [\omega_{i}], it is possible to carry out the 3D reconstruction of the object [see (2.5.6.27)[link]; Orlov, 1975[link]; Vainshtein & Goncharov, 1986a[link]; Goncharov et al., 1987[link]].

References

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