Using an adaptation of generative topographic mapping (GTM) (Bishop et al., 1998 ; Svensén, 1998 ), Fung et al. (2009 ) published the first successful recovery of the structure of a molecule from simulated diffraction snapshots of unknown orientation at signal levels expected from a 500 kD molecule by utilizing the information content of the entire ensemble of diffraction snapshots. Subsequently, Loh & Elser (2009 ) demonstrated structure recovery from simulated diffraction snapshots by an apparently different approach, using a so-called expansion–maximization–compression (EMC) algorithm (Loh & Elser, 2009 ). Both approaches have been validated with experimental data. Loh et al. (2010 ) have oriented snapshots from iron oxide nanoparticles obtained by single-shot diffraction. Using GTM, Fung et al. (2010 ) and Schwander et al. (2010 ) have determined the orientation of diffraction snapshots from gold nanofoam with ~8 × 10⁻² scattered photons per Shannon pixel with an orientational accuracy of about one Shannon angle. Using a variety of manifold embedding approaches, Giannakis et al. (2010 ) have demonstrated orientation recovery from diffraction snapshots of superoxide dismutase crystals with 1° accuracy compared with the goniometer step size of 0.5° and the crystal mosaicity of 0.8°. Using recently discovered symmetries of image formation, Giannakis et al. (2010 ) have used manifold approaches for orientation recovery and three-dimensional reconstruction of single chaperonin molecules with experimental cryo-electron microscopy snapshots as well as experimental snapshots processed to represent doses 10× lower than is possible with existing techniques.

Here we show the two Bayesian approaches of Loh & Elser (2009 ) and Fung et al. (2009 ) are fundamentally the same, and discuss their capabilities and limitations. Issues to do with the way each approach is implemented and performs under different conditions are beyond the scope of the present paper, if only because these aspects are under active development. In order to facilitate the discussion, the structure-recovery process is divided into two steps: (a) orienting the diffraction snapshots and assembling the three-dimensional diffraction volume; and (b) recovering the structure by a phasing algorithm. Since we are concerned with orientation recovery, the discussion will be confined to the first step.

The differences in presentation and notation notwithstanding, the Fung et al. (2009 ) and the Loh & Elser (2009 ) approaches are the same in all essential features. Specifically, they both:

(a) exploit the information content of the entire data set;

(b) recognize that a nonlinear mapping function relates the space of object orientations to the space of scattered intensities;

(c) determine the mapping function by Bayesian inference;

(d) use the well established expectation–maximization (EM) iterative algorithm (Dempster et al., 1977 ) to maximize likelihood;

(e) apply a constraint to guide likelihood maximization; and

(f) implement noise-robust algorithms with essentially the same computational scaling behaviors.

The representation of object orientations bears careful consideration. Despite their widespread use, Euler angles are not a good representation of orientational similarity, because an object can be rotated through large Euler angles () and end at an orientation very close to its starting point. As the Euclidean distance in quaternion space is a good measure of (dis)similarity between orientations, both Fung et al. (2009 ) and Loh & Elser (2009 ) use unit quaternions (Kuipers, 2002 ) to represent orientations. Diffraction to a point in reciprocal space, therefore, can be thought of as a functional , with representing a unit quaternion.

A diffraction snapshot consists of p intensity values. The mapping thus takes an orientation to generate a model snapshot . These are to be compared with experimental snapshots , but will, in general, not be identical to any single snapshot owing to (experimental) noise.²

Because a given object has only three orientational degrees of freedom, the points representing the diffraction snapshots in the so-called manifest intensity space trace out a three-dimensional manifold, which is a nonlinear map of the SO(3) manifold of orientations. At a conceptual level, given the ‘input’ and ‘output’ manifolds, it is possible to discover the nonlinear map between them. This links (‘maps’) a diffraction snapshot to a given orientation, and thus assigns an orientation to each diffraction snapshot (Fung et al., 2009 ; Giannakis et al., 2010 ). Once this has been accomplished, snapshots of similar orientation can be averaged to boost the signal, and structure recovery can proceed by standard techniques. In fact, appropriately wielded, manifold embedding can improve the signal far more efficiently than simple averaging of similar snapshots (Schwander et al., 2010 ; Giannakis et al., 2010 ), but this is beyond the scope of the present paper.

To render the formalism tractable, the SO(3) space of orientations is represented by a discrete set of K orientations (‘nodes’) , distributed nearly uniformly on the three-sphere (Lovisolo & da Silva, 2001 ; Coxeter, 1973 ). The inter-node spacing is chosen to satisfy the Shannon–Nyquist sampling criterion, determined as follows. Consider recov­ering the structure of an object with largest diameter D (radius R) to resolution r (Fig. 1 ). The orientational accuracy needed is thenwith the number of independent orientations in three dimensions given byThe pre-factor of accounts for the fact that the three-sphere is a double-cover of SO(3). The Shannon element in terms of quaternions isleading towhere S is the number of symmetry elements of the molecule being reconstructed.

Both LE and Fung maximize the log-likelihood iteratively by the well known EM algorithm (Dempster et al., 1977 ). Each iteration modifies the model snapshots, effectively moving the manifold defined by them closer to the experimental data. There is no guarantee that the final solution is a global maximum.

Once the mapping corresponding to maximum likelihood has been determined, the orientation of each measured diffraction pattern is taken to correspond to that x_k which maximizes the probability of ‘belonging’ to . Thus we choose the orientation x _k which maximizes the probability . The conditional probability is determined using equation (5).

Having assigned the N diffraction snapshots to the K orientational bins, the diffraction volume can be reconstructed. In standard ‘classification and averaging’, diffraction patterns assigned to the same orientation x_k are averaged so that there is one representative diffraction pattern for each x_k. So-called generative models such as that used by Fung allow one to construct (‘generate’) model snapshots for each orientation directly from the manifold. As the manifold represents the information content of the entire data set, the generative approach offers significantly greater noise reduction than classification and averaging, which relies on the information in the neighborhood of a given orientation only (Schwander et al., 2010 ; Giannakis et al., 2010 ).

In order to eliminate this problem, both the GTM and EMC algorithms place a ‘contiguity constraint’ on the map y. This constraint demands that two nodes which are close to each other in the space of orientations be mapped to points close to each other in data space. Fung and LE impose this contiguity constraint differently. In the GTM approach used by Fung, the map is expanded in terms of a set of basis functions:where is one of M basis functions ( the number of independent orientations K) and represent the expansion coefficients (weights).

Likelihood maximization proceeds by adjusting the M sets of p coefficients. The basis functions are chosen so as to vary slowly with x. In the current implementation of GTM, they are Gaussians (Bishop et al., 1998 ). The map in equation (7) varies slowly, provided the weights are small. This is achieved by imposing a zero-centered Gaussian distribution on the sum of the squares of the weights. This strategy helps ensure that, topologically, the neighborhood assignments in manifest (intensity) space reflect the neighborhood assignments in latent (orientation) space, i.e. is close to when x_k is close to x_k′.

The EMC algorithm of LE, in contrast, uses the model diffraction patterns themselves (rather than the weights ) as fitting parameters. After each expectation–maximization step, a so-called ‘compression’ step inserts the model diffraction patterns into reciprocal space according to their orientations, and the resulting irregularly spaced points are interpolated onto a uniform grid to determine a new diffraction volume by local averaging. Next, an ‘expansion’ step uses the new diffraction volume as the source for a fresh set of model diffraction snapshots by interpolating back onto the irregularly spaced points corresponding to the pixels of each of the model diffraction patterns. In this approach, both the compression and expansion steps act as low-pass filters; replacing two diffraction patterns by their average and then deducing two diffraction patterns from the average removes sharp variations between diffraction patterns mapped to similar points in reciprocal space. In essence, the so-called compression–expansion cycle is an alternative implementation of the contiguity constraint, whereby neighboring orientations in latent space give rise to neighboring points in manifest intensity space.