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

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 367-368   | 1 | 2 |

Section 2.5.6.2. 3D reconstruction in the general case

B. K. Vainshteinc and P. A. Penczekg

2.5.6.2. 3D reconstruction in the general case

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In the general case of the 3D reconstruction of [\varphi _3 ( {\bf r})] from projections [\varphi _2 ( {\bf x}_\boldtau)], the projection vector [\boldtau] occupies arbitrary positions on the projection sphere (Fig. 2.5.6.2[link]). First, let us consider the case of a 2D function [\rho _2 ( {\bf x})] and its ray transform [\varphi _1 ( x,\psi )]. We introduce an operation of backprojection b, which is stretching along [\boldtau _\psi ] each 1D projection [\varphi _1 ( x_\psi )] (Fig. 2.5.6.3[link]). When the result is integrated over the full angular range of projections [\varphi _1 ( x,\psi )], we obtain the projection synthesis defined as[b(x,y ) = \textstyle \int \limits_0^\pi {\varphi _1 ( {x\cos \psi + y\sin \psi ,\psi } )} \,{\rm d}\psi. \eqno(2.5.6.9)]However, the backprojection operator is not the inverse of a 2D ray transform, as the resulting image b is blurred by the point-spread function [( x^2 + y^2 )^{-1/2}] (Vainshtein, 1971[link]):[b(x,y) = \rho ( x,y ) * ( {x^2 + y^2 } )^{-1/2}.\eqno(2.5.6.10)]By noting that the Fourier transform of [( x^2 + y^2 )^{-1/2}] is [( u^2\ +] [ v^2 )^{ - 1/2}] and by using the convolution theorem [{\scr F}[ f * g] =] [ {\scr F}[ f ]{\scr F}[ g ]], we obtain the `backprojection-filtering' inversion formula:[\eqalignno{\rho ( x,y ) &= b( x,y ) * ( x^2 + y^2 )^{1/2} = {\scr F}^{ - 1}\big[ | {\bf u} |{\scr F}[ b ] \big] &\cr& ={\rm Filtration}_{| {\bf u} |} [ {\rm Backprojection}( \varphi _1 ) ].&(2.5.6.11)}]The more commonly used `filtered-backprojection' inversion is based on the 2D version of the central section theorem (2.5.6.8)[link]:[{\scr F}[ \varphi _1 ( x_\psi ) ] = \Upsilon _2 ( {\bf u}_\psi ) = \Upsilon _2 ( R,\Psi ),\eqno(2.5.6.12)]where [{\scr F}[ \rho _2 ] = \Upsilon _2 ]. With this in mind, [\rho _2 ( {\bf x} )] can be related to its ray transform by evaluating the Fourier transform of [\rho _2 ] in polar coordinates:[\eqalignno{ \rho _2 ( {\bf x} ) &= \textstyle \int \Upsilon _2 ( {\bf u} )\exp ( - 2\pi i{\bf ux} )\, {\rm d}{\bf u} &\cr & = \textstyle \int \limits_0^\pi \textstyle \int \limits_{ - \infty }^\infty \Upsilon _2 ( R,\Psi )\exp ( - 2\pi i{\bf x}\boldtau )| R |\,{\rm d}R\,{\rm d}\Psi&\cr & = \textstyle \int \limits_0^\pi \textstyle \int \limits_{ - \infty }^\infty {\scr F}[ \varphi _1 ( x_\psi ) ]\exp ( - 2\pi i{\bf x}\boldtau )| R |\,{\rm d}R\,{\rm d}\Psi &\cr & =\textstyle \int \limits_0^\pi {\scr F}^{ - 1}\big [ | R |{\scr F}[ \varphi _1 ( x_\psi ) ] \big]\,{\rm d}\Psi &\cr & = {\rm Backprojection}[ {\rm Filtration}_{| R |} ( \varphi _1 )]. &(2.5.6.13)} ]In three dimensions, the backprojection stretches each 2D projection [\varphi _{2_i} [ {\bf x},{\boldtau }( \theta ,\psi )_i ]] along the projection direction [\boldtau( \theta ,\psi )_i ]. A 3D synthesis is the integral over the hemisphere (Fig. 2.5.6.2[link])[b( {\bf r} ) = \textstyle \int \limits_\omega \varphi _2 ( {\bf x},\omega _\boldtau )\,{\rm d}\omega _\boldtau = \varphi _3 ( {\bf r} ) * ( x^2 + y^2 + z^2 )^{ - 1}. \eqno(2.5.6.14)]Thus, in three dimensions the image b obtained using the backprojection operator is blurred by the point-spread function [1/( x^2 + y^2 + z^2 )] (Vainshtein, 1971[link]). It is possible to derive inversion formulae analogous to (2.5.6.11)[link] and (2.5.6.13)[link].

[Figure 2.5.6.3]

Figure 2.5.6.3 | top | pdf |

(a) Formation of a backprojection function; (b) projection synthesis (2.5.6.9)[link] is a superposition of these functions.

The inversion formulae demonstrate that it is possible to invert the ray transform for continuous functions and for a uniform distribution of projections. In electron microscopy, the projections are never distributed uniformly in three dimensions. Indeed, a uniform distribution is not even desirable, as only certain subsets of projection directions are required for the successful inversion of a 3D ray transform, as follows from the central section theorem (2.5.6.8)[link]. In practice, we always deal with sampled data and with discrete, random and nonuniform distributions of projection directions. Therefore, the inversion formulae can be considered only as a starting point for the development of the numerical (and practical!) reconstruction algorithms. According to (2.5.6.10)[link] and (2.5.6.14)[link], a simple backprojection step results in reconstruction that corresponds to a convolution of the original function with a point-spread function that depends only on the distribution of projections, but not on the structure itself. Taking into account the linearity of the backprojection operation, one has to conclude that for any practically encountered distribution of projections it should be possible to derive the corresponding point-spread function and then, using either deconvolution or Fourier filtration (with a `weighting function'), arrive at a good approximation of the structure. This observation forms the basis of the weighted backprojection algorithm (Section 2.5.6.5[link]). Similarly, the central section theorem gives rise to direct Fourier inversion algorithms (Section 2.5.6.6[link]). Nevertheless, since the data are discrete, the most straightforward methodology is to discretize and approach the reconstruction problem as an algebraic problem (Section 2.5.6.4[link]).

References

Vainshtein, B. K. (1971). Finding the structure of objects from projections. Sov. Phys. Crystallogr. 15, 781–787.








































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