Klug, A.; Crowther, R. A.
Three-dimensional image reconstruction from the viewpoint of information theory
Gilbert, P.
Ph.D. Thesis
Crowther, R. A.; DeRosier, D. J.; Klug, A.
The reconstruction of a three-dimensional structure from projections and its application to electron microscopy
Diamond, R.
Filtering in the Method of Least-Squares
Holmes, K. C.; Angert, I.; Kull, F. J.; Jahn, W.; Schröder, R. R.
Electron cryo-microscopy shows how strong binding of myosin to actin releases nucleotide

Axial Tomography by filtered least squares

A transformation and back transformation yields the answer

The significance of equations 6, 7 and is as follows: equation 6 is a transformation of the back projection using a set of orthonormal functions that are the eigenfunctions of the outer product of the projector matrix with itself. The result of this tranformation is a list of numbers (with associated errors). To proceed we divide each of these numbers by the appropriate eigenfunction arranged from large to small in order to determine how much of each eigenfunction contribute to the final answer (7). At some point the eigenfunction becomes as small as the errors and we have to stop, otherwise we amplify up noise – there is no point in going on – a truncation. This yields a list of numbers telling us how much of each eigenfunction to put into the answer. The result is calculated by a back transformation (8). The effect of truncation is to limit the resolution in the answer. It transpires that the eigenfunctions with the longest spatial frequencies have the largest eigenvalues so that the resolution goes up with more eigenvalues. Since the base we have used is orthonormal and final we would not get another answer if we considered a larger subspace – just more noise.

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