Ft Pf

f(1)= Pn Pn—1 ...Pf as the result of the first pass of the *projection scheme. Next, the procedure is repeated, this time with f1 as the starting function, yielding p2, etc. As the geometric analogy shows (figure 5.35), by virtue of the convex property of the

Figure 5.35 Principle of restoration using the method of *projection onto convex sets (POCS). Ci and C2 are convex sets in the space RN, representing constraints, and P1; P2 are associated *projected operators. Each element / is a 3D structure. We seek to find a pathway from a given blurred structure / (the seed in the iterative POCS) to the intersection set (shaded). Any structure in that set fulfills both constraints and is thus closer to the true solution than the initial structure /. From Sezan (1992), reproduced with permission of Elsevier.

Figure 5.35 Principle of restoration using the method of *projection onto convex sets (POCS). Ci and C2 are convex sets in the space RN, representing constraints, and P1; P2 are associated *projected operators. Each element / is a 3D structure. We seek to find a pathway from a given blurred structure / (the seed in the iterative POCS) to the intersection set (shaded). Any structure in that set fulfills both constraints and is thus closer to the true solution than the initial structure /. From Sezan (1992), reproduced with permission of Elsevier.

sets, each iteration brings the function (represented by a point in this diagram) closer to the intersection of all constraint sets. In this example, only n = 2 sets are used.

Carazo and Carrascosa (1987a,b) already discussed closed, convex constraint sets of potential interest in EM: (i) spatial boundedness (as defined by a binary mask); and (ii) agreement with the experimental measurements in the measured region of Fourier space, value boundedness, and energy boundedness. Thus far, in practical applications (see following), only the first two on this list have gained much importance, essentially comprising the two components of Gerchberg's (1974) method.2 Numerous other constraints of potential importance [see, for instance, Sezan (1992)] still await exploration.

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