The 3D variance V(R) can be estimated by back-projecting the weighted-projection 2D noise variances (now leaving out the drawing index j ):

The only remaining task is to estimate the projection noise n(i)(r) for each projection direction. This can be done in two different ways. One way is by obtaining a reconstruction from the existing experimental projections by weighted back-projection, and then reprojecting this reconstruction in the original directions. The difference between the experimental projection and the reprojection can be taken as an estimate of the projection noise.

The other way (figure 6.2), closely examined by Liu and Frank (1995) and Liu et al. (1995), is by taking advantage of the fact that in normal data collections, the angular range is strongly oversampled, so that for each projection, close-neighbor projections that differ little in the signal are available. In other words, an ensemble of projections, of the kind assumed in the gedankenexperiment, actually exists in a close angular neighborhood of any given projection.

It is obviously much easier to go the first way, by using reprojections (see figure 6.2). The detailed analysis by Liu and coworkers, in the works cited above, needs to be consulted for an evaluation of systematic errors that might favor the second way in certain situations. It is also important to realize that there are certain systematic errors, such as errors due to interpolation, inherent to all reprojection methods, that cause discrepancies unrelated to the 3D variance (see Trussel et al., 1987). We have implicitly assumed in the foregoing that the weighted back-projection algorithm is being used for reconstruction.

It should be noted that the estimation of the 3D variance from the 2D projection variances works only for reconstruction schemes that, as the weighted back-projection, maintain linearity. Iterative reconstruction methods such as the algebraic reconstruction technique (ART) (Gordon et al., 1970) and the simultaneous iterative reconstruction technique (SIRT) (Gilbert, 1972) do maintain the linearity in the relationship between projections and reconstruction, unless

Figure 6.2 Flow diagram of 3D variance estimation algorithm. Provided the angular spacing is small, the difference between neighboring projections can be used as noise estimate n(i)(k, I). When such an estimate has been obtained for each projection, we can proceed to compute the 3D variance estimate by using the following steps: (i) convolution with the inverse of the back-projection weighting function (i.e., the real-space equivalent to the back-projection (BP) weighting in Fourier space); (ii) square the resulting weighted noise functions, which results in estimates of the noise variances; (iii) back-project (BP) the noise variance estimates which gives the desired 3D variance estimate. From Frank et al. (1992), reproduced with permission of Scanning Microscopy International.

Figure 6.2 Flow diagram of 3D variance estimation algorithm. Provided the angular spacing is small, the difference between neighboring projections can be used as noise estimate n(i)(k, I). When such an estimate has been obtained for each projection, we can proceed to compute the 3D variance estimate by using the following steps: (i) convolution with the inverse of the back-projection weighting function (i.e., the real-space equivalent to the back-projection (BP) weighting in Fourier space); (ii) square the resulting weighted noise functions, which results in estimates of the noise variances; (iii) back-project (BP) the noise variance estimates which gives the desired 3D variance estimate. From Frank et al. (1992), reproduced with permission of Scanning Microscopy International.

they are modified for the purpose of incorporating nonlinear constraints (e.g., Penczek et al., 1992). Weighted back-projection, as we have seen earlier (chapter 5, section 4), has the property of linearity for evenly spaced projections, for which |k|-weighting holds, while general weighting functions often incorporate (nonlinear) application of thresholds.

An application of the variance estimation method by adjacent projections is shown in figure 6.3 in section 2.3.2.

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