Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.5, pp. 315-316   | 1 | 2 |

Section The object and its projection

B. K. Vainshteinc The object and its projection

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In electron microscopy we obtain a two-dimensional image [\varphi_{2} ({\bf x}_{\tau})] – a projection of a three-dimensional object [\varphi_{3} ({\bf r})] (Fig.[link]): [\varphi_{2} ({\bf x}_{\tau}) = {\textstyle\int} \varphi_{3} ({\bf r})\ {\rm d}\tau \quad {\boldtau} \perp {\bf x.} \eqno(] The projection direction is defined by a unit vector [{\boldtau} (\theta, \psi)] and the projection is formed on the plane x perpendicular to [{\boldtau.}] The set of various projections [\varphi_{2} ({\bf x}_{\tau_{i}}) = \varphi_{2i} ({\bf x}_{i})] may be assigned by a discrete or continuous set of points [{\boldtau}_{i} (\theta_{i}, \psi_{i})] on a unit sphere [| {\boldtau} | = 1] (Fig.[link]). The function [\varphi ({\bf x}_{\tau})] reflects the structure of an object, but gives information only on [{\bf x}_{\tau}] coordinates of points of its projected density. However, a set of projections makes it possible to reconstruct from them the three-dimensional (3D) distribution [\varphi_{3} (xyz)] (Radon, 1917[link]; DeRosier & Klug, 1968[link]; Vainshtein et al., 1968[link]; Crowther, DeRosier & Klug, 1970[link]; Gordon et al., 1970[link]; Vainshtein, 1971a[link]; Ramachandran & Lakshminarayanan, 1971[link]; Vainshtein & Orlov, 1972[link], 1974[link]; Gilbert, 1972a[link]; Herman, 1980[link]). This is the task of the three-dimensional reconstruction of the structure of an object: [\hbox{set } \varphi_{2} ({\bf x}_{i}) \rightarrow \varphi_{3} ({\bf r}). \eqno(]


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A three-dimensional object [\varphi_{3}] and its two-dimensional projection [\varphi_{2}].


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The projection sphere and projection [\varphi_{2}] of [\varphi_{3}] along τ onto the plane [{\bf x} \perp {\bf \boldtau}]. The case [{\bf \boldtau} \perp z] represents orthoaxial projection. Points indicate a random distribution of τ.

Besides electron microscopy, the methods of reconstruction of a structure from its projections are also widely used in various fields, e.g. in X-ray and NMR tomography, in radioastronomy, and in various other investigations of objects with the aid of penetrating, back-scattered or their own radiations (Bracewell, 1956[link]; Deans, 1983[link]; Mersereau & Oppenheim, 1974[link]).

In the general case, the function [\varphi_{3} ({\bf r})] ([link] (the subscript indicates dimension) means the distribution of a certain scattering density in the object. The function [\varphi_{2} ({\bf x})] is the two-dimensional projection density; one can also consider one-dimensional projections [\varphi_{1} (x)] of two- (or three-) dimensional distributions. In electron microscopy, under certain experimental conditions, by functions [\varphi_{3} ({\bf r})] and [\varphi_{2} ({\bf x})] we mean the potential and the projection of the potential, respectively [the electron absorption function μ (see Section 2.5.4[link]) may also be considered as `density']. Owing to a very large depth of focus and practical parallelism of the electron beam passing through an object, in electron microscopy the vector τ is the same over the whole area of the irradiated specimen – this is the case of parallel projection.

The 3D reconstruction ([link] can be made in the real space of an object – the corresponding methods are called the methods of direct three-dimensional reconstruction (Radon, 1917[link]; Vainshtein et al., 1968[link]; Gordon et al., 1970[link]; Vainshtein, 1971a[link]; Ramachandran & Lakshminarayanan, 1971[link]; Vainshtein & Orlov, 1972[link], 1974[link]; Gilbert, 1972a[link]).

On the other hand, three-dimensional reconstruction can be carried out using the Fourier transformation, i.e. by transition to reciprocal space. The Fourier reconstruction is based on the well known theorem: the Fourier transformation of projection [\varphi_{2}] of a three-dimensional object [\varphi_{3}] is the central (i.e. passing through the origin of reciprocal space) two-dimensional plane cross section of a three-dimensional transform perpendicular to the projection vector (DeRosier & Klug, 1968[link]; Crowther, DeRosier & Klug, 1970[link]; Bracewell, 1956[link]). In Cartesian coordinates a three-dimensional transform is [ \eqalignno{ {\scr F}_{3} [\varphi_{3} ({\bf r})] &= \Phi_{3} (uvw) = {\textstyle\int\!\!\int\!\!\int} \varphi_{3} (xyz) &\cr &\quad \times \exp \{2 \pi i (ux + vy + wz)\} \ \hbox{d}x \ \hbox{d}y \ \hbox{d}z. &(\cr}] The transform of projection [\varphi_{2} (xy)] along z is [ \eqalignno{ {\scr F}_{2} [\varphi_{2} (xy)] &= \Phi_{3} (uv0) = {\textstyle\int\!\!\int\!\!\int} \varphi_{3} (xyz) &\cr &\quad \times \exp \{2 \pi i (ux + vy + 0z)\} \ \hbox{d}x \ \hbox{d}y \ \hbox{d}z &\cr &= {\textstyle\int\!\!\int\!\!\int} \varphi_{3} (xyz) \ \hbox{d}z \exp \{2 \pi i (ux + vy)\} \ \hbox{d}x \ \hbox{d}y &\cr &= {\textstyle\int\!\!\int} \varphi_{2} (xy) \exp \{2 \pi i (ux + vy)\} \ \hbox{d}x \ \hbox{d}y &\cr &= \Phi_{2} (uv). &(\cr}] In the general case ([link] of projecting the plane [{\bf x} (xy) \| {\bf u} (uv) \perp {\boldtau}] along the vector τ [ {\scr F}_{2} [\varphi_{2} ({\bf x}_{\tau})] = \Phi_{2} ({\bf u}_{\tau}). \eqno(] Reconstruction with Fourier transformation involves transition from projections [\varphi_{2i}] at various [{\boldtau}_{i}] to cross sections [\Phi_{2i}], then to construction of the three-dimensional transform [\Phi_{3} ({\bf u})] by means of interpolation between [\varphi_{2i}] in reciprocal space, and transition by the inverse Fourier transformation to the three-dimensional distribution [\varphi_{3} ({\bf r})]: [ \eqalignno{ \hbox{set }& \varphi_{2i} ({\bf x}_{\tau i}) \rightarrow \hbox{ set } {\scr F}_{2} (\varphi_{2}) \cr &\equiv \hbox{ set } \Phi_{2i} \rightarrow \Phi_{3} \rightarrow {\scr F}_{3}^{-1} (\Phi_{3}) \equiv \varphi_{3} ({\bf r}). &(\cr}] Transition ([link] or ([link] from two-dimensional electron-microscope images (projections) to a three-dimensional structure allows one to consider the complex of methods of 3D reconstruction as three-dimensional electron microscopy. In this sense, electron microscopy is an analogue of methods of structure analysis of crystals and molecules providing their three-dimensional spatial structure. But in structure analysis with the use of X-rays, electrons, or neutrons the initial data are the data in reciprocal space [| \Phi_{2i} |] in ([link], while in electron microscopy this role is played by two-dimensional images [\varphi_{2i} ({\bf x})] [([link], ([link])] in real space.

In electron microscopy the 3D reconstruction methods are, mainly, used for studying biological structures (symmetric or asymmetric associations of biomacromolecules), the quaternary structure of proteins, the structures of muscles, spherical and rod-like viruses, bacteriophages, and ribosomes.

An exact reconstruction is possible if there is a continuous set of projections [\varphi_{\tau}] corresponding to the motion of the vector [{\boldtau}(\theta, \psi)] over any continuous line connecting the opposite points on the unit sphere (Fig.[link]). This is evidenced by the fact that, in this case, the cross sections [ {\scr F}_{2}] which are perpendicular to τ in Fourier space ([link] continuously fill the whole of its volume, i.e. give [ {\scr F}_{3}(\varphi_{3})] ([link] and thereby determine [ \varphi_{3} ({\bf r}) = {\scr F}^{-1} (\Phi_{3})].

In reality, we always have a discrete (but not continuous) set of projections [\varphi_{2i}]. The set of [\varphi_{2i}] is, practically, obtained by the rotation of the specimen under the beam through various angles (Hoppe & Typke, 1979[link]) or by imaging of the objects which are randomly oriented on the substrate at different angles (Kam, 1980[link]; Van Heel, 1984[link]). If the object has symmetry, one of its projections is equivalent to a certain number of different projections.

The object [\varphi_{3} ({\bf r})] is finite in space. For function [\varphi_{3} ({\bf r})] and any of its projections there holds the normalization condition [\Omega = {\textstyle\int} \varphi_{3} ({\bf r}) \ \hbox{d}v_{{\bf r}} = {\textstyle\int} \varphi_{2} ({\bf x}) \ \hbox{d}{\bf x} = {\textstyle\int} \varphi_{1} (x) \ \hbox{d}x, \eqno(] where Ω is the total `weight' of the object described by the density distribution [\varphi_{3}]. If one assumes that the density of an object is constant and that inside the object [\varphi] = constant = 1, and outside it [\varphi = 0], then Ω is the volume of an object. The volume of an object, say, of molecules, viruses and so on, is usually known from data on the density or molecular mass.


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