Back-projection is an operation that is the inverse of the projection operation: while the latter produces a 2D image of the 3D object, back-projection "smears out'' a 2D image into a 3D volume ["back-projection body,'' see Hoppe et al. (1986), where this term was coined] by translation into the direction normal to the plane of the image (figure 5.14). The topic of modified back-projection, as it is applied to the reconstruction of single particles, has been systematically

Figure 5.14 Illustration of the back-projection method of 3D reconstruction. The density distribution across a projection is ''smeared out'' in the original direction of projection, forming a ''back-projection body.'' Summation of these back-projection bodies generated for all projections yields an approximation to the object. For reasons that become clear from an analysis of the problem in Fourier space, the resulting reconstruction is dominated by low-spatial frequency terms. This problem is solved by Fourier weighting of the projections prior to the back-projection step, or by weighting the 3D Fourier transform appropriately. From Frank et al. (1985), reproduced with permission of van Nostrand-Reinhold.

Figure 5.14 Illustration of the back-projection method of 3D reconstruction. The density distribution across a projection is ''smeared out'' in the original direction of projection, forming a ''back-projection body.'' Summation of these back-projection bodies generated for all projections yields an approximation to the object. For reasons that become clear from an analysis of the problem in Fourier space, the resulting reconstruction is dominated by low-spatial frequency terms. This problem is solved by Fourier weighting of the projections prior to the back-projection step, or by weighting the 3D Fourier transform appropriately. From Frank et al. (1985), reproduced with permission of van Nostrand-Reinhold.

presented by Radermacher (1988, 1991, 1992), and some of this work will be paraphrased here.

Let us consider a set of N projections into arbitrary angles. As a notational convention, we keep track of the different 2D coordinate systems of the projections by a superscript; thus, p,(r(i)) is the ith projection, r(i) = {x(i), y(i)} are the coordinates in the ith projection plane, and z(i) is the coordinate perpendicular to that.

With this convention, the back-projection body belonging to the ith projection is b i( r(i), z(i))=pi(r'(i))t(z(i)) (5.17)

0 elsewhere

Thus, bi is the result of translating the projection by D (a distance that should be chosen larger than the anticipated object diameter). As more and more such back-projection bodies for different angles are added together, a crude reconstruction of the object will emerge:

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