Theory of Projection onto Convex Sets

What follows is a brief introduction to the philosophy of the POCS method which can be seen as a generalization of a method introduced by Gerchberg and Saxton (1971) and Gerchberg (1974). A similar method of iterative single isomorphous replacement (Wang, 1985) in X-ray crystallography is also known under the name of solvent flattening. Still another, related method of constrained thickness in reconstructions of 1D membrane profiles was proposed earlier on by Stroud and Agard (1979). An excellent primer for the POCS method was given by Sezan (1992).

Similarly as in multivariate data analysis of images (chapter 4), which takes place in the space of all functions with finite 2D support, we now consider the space of all functions with finite 3D support. In the new space (Hilbert space), every conceivable bounded 3D structure is represented by a (vector) end point. Constraints can be represented by sets. For instance, one conceivable set might be the set of all structures that have zero density outside a given radius R. The idea behind restoration by POCS is that the enforcement of known constraints that were not used in the reconstruction method itself will yield an improved version of the structure. This version will lie in the intersection of all constraint sets and thus closer to the true solution than any version outside of it. In Fourier space, the angular gap will tend to be filled. The only problem to solve is how to find a pathway from the approximate solution, reconstructed, for instance, by back-projection or any other conventional technique, to one of the solutions lying in the intersection of the constraint sets.

Among all sets representing constraints, those that are both closed and convex have proved of particular interest. Youla and Webb (1982) showed that for such sets the intersection can be reached by an iterative method of consecutive *projections. A *projection from a function f(r) onto a set C in Hilbert space is defined as an operation that determines a function g(r) in C with the following property: "of all functions in C, g(r) is the one closest to f(r).'' Here, "closeness" is defined by the size of a distance; for instance, by the generalized Euclidean distance in Hilbert space:

(Note that, by implication, repeated applications of *projection onto the same set lead to the same result.) In symbolic notation, if Pi (i = 1,..., n) denotes the operation of *projection onto set Ci so that f0 = Pf is the function obtained by *projecting f onto Ci, the iterative restoration proceeds as follows: we start from a "seed" function f, and obtain successively

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