Talk:Difference-map algorithm

Fair use rationale for Image:Wiki reconstruct.png
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BetacommandBot (talk) 05:39, 30 November 2007 (UTC)

3 to the fourth power is not 729. 3 to the fourth is 81, and 3 to the sixth is 729. Which one was intended?

Fixed points and solution set
I've commented out the following statement:
 * The set of fixed points in a particular application will normally have a large dimension, even when the solution set is a single point.

The iterated map is x ↦ x + β S, where S is the difference between the two projections PA(fB(x)), which is an element of A, and PB(fA(x)), an element of B. We see that x is a fixed point of the map iff S = 0, which means that PA(fB(x)) = PB(fA(x)), so this is a common point of A and B. Then, furthermore, PA(x) = x, so fA(x) = PA(x) − (PA(x)−x)/β = x. Likewise, PB(x) = x. So the common point in the intersection of A and B is x itself. This shows that x is a fixed point iff it is in the solution set of x ∈ A ∩ B. The set of fixed points and the solution set coincide, so one cannot have a larger dimension than the other. --Lambiam 22:10, 10 December 2012 (UTC)

Phase Retrieval Section
I restored this section because the algorithm's origins can be traced to this particular application. Also, the reconstruction figure now has the necessary explanatory text. --Veit Elser 10:42, 23 January 2014 (UTC)