Projection onto Convex Sets in Practice

The numerical computation in the various steps of POCS has to alternate between real space and Fourier space for each cycle. Both support (mask) and value constraints are implemented as operations in real space, while the enforcement of the "replace" constraint takes place in Fourier space. For a typical size of a 3D array representing a macromolecule (between 64 x 64 x 64 and 128 x 128 x 128), the 3D Fourier transformations in both directions constitute the largest fraction of the computational effort.

The support-associated *projector is of the following form:

where M is the 3D mask defining the ''pass'' regions of the mask. In practice, a mask is represented by a binary-valued array with ''1'' representing ''pass'' and ''0'' representing ''stop.'' The mask array is simply interrogated, as the discrete argument range of the function f is being scanned in the computer, and only those values of f(r) are retained for which the mask M indicates ''pass.'' The support constraint is quite powerful if the mask is close to the actual boundary, and an important question is how to find a good estimate for the mask in the absence of information on the true boundary (which represents the normal situation). We will come back to this question later after the other constraints have been introduced.

The value constraint is effected by the *projector (Carazo, 1992):

0 elsewhere

0 elsewhere

a /(r,) < a Pv/r,-)=- /r,-) a < /(r ,-) < b - b /(r ¡) > b

2Gerchberg's (1974) method, not that of Gerchberg and Saxton (1971), is a true precursor of POCS since it provides for replacement of both modulus and phases.

The measurement constraint is supposed to enforce the consistency of the solution with the known projections. This is rather difficult to achieve in practice because the projection data in Fourier space are distributed on a polar grid, while the numerical Fourier transform is sampled on a Cartesian grid. Each POCS *projection would entail a complicated Fourier sinc interpolation. Instead, the measurement constraint is normally used in a weaker form, as a "global replace'' operation: within the range of the measurements (i.e., in the case of the random-conical data collection, within the cone complement that is covered with projections; see figure 5.13), all Fourier coefficients are replaced by the coefficients of the solution found by weighted back-projection.

This kind of implementation is somewhat problematic, however, because it reinforces a solution that is ultimately not consistent with the true solution as it incorporates a weighting that is designed to make up for the lack of data in the missing region. Recall (section 4.2) that the weighting function is tailored to the distribution of projections in a given experiment, so it will reflect the existence and exact shape of the missing region.

A much better "replace" operation is implicit in the iterative schemes in which agreement is enforced only between projection data and reprojections. In Fourier space, these enforcements are tantamount to a "replace" that is restricted to the Fourier components for which data are actually supplied. The use of the global replace operation also fails to realize an intriguing potential of POCS: the possibility of achieving anisotropic super-resolution, beyond the limit given by Crowther et al. (1970). Intuitively, the enforcement of "local replace'' (i.e., only along central sections covered with projection data) along with the other constraints will fill the very small "missing wedges'' between successive central sections on which measurements are available much more rapidly, and out to a much higher resolution, than the large missing wedge or cone associated with the data collection geometry. The gain in resolution might be quite significant.

What could be the use of anisotropic super-resolution? An example is the tomographic study of the mitochondrion (Mannella et al., 1994), so far hampered by the extremely large ratio between size (several micrometers) and the size of the smallest detail we wish to study (50 A). The mitochondrion is a large structure that encompasses, and is partially formed by, a convoluted membrane. We wish to obtain the spatial resolution in any direction that allows us to describe the spatial arrangements of the different portions of the membrane, and answer the following questions: Do they touch? Are compartments formed? What is the geometry of the diffusion-limiting channels? The fact that the important regions where membranes touch or form channels occur in different angular directions makes it highly likely in this application that the relevant information can be picked up in certain regions of the object. The subject of tomography is outside the scope of this book, but similar problems where even anisotropic resolution improvement may be a bonus could well be envisioned in the case of macromolecules.

Examples for the application of POCS to experimental data are found in the work of Akey and Radermacher (1993) and Radermacher and coworkers (1992b, 1994b). In those cases, only the measurement and the finite support constraints were used. In the first case, the nuclear pore complex was initially reconstructed from data obtained with merely 34° and 42° tilt and thus had an unusually large missing-cone volume. Radermacher et al. (1994b) observed that POCS applied to a reconstruction from a negatively stained specimen led to a substantial contraction in the z-direction, while the ice reconstruction was relatively unaffected.

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