As I learned it, Photograph 51 was so good that anybody with any crystallography experience would have been able to tell the structure at a glance. Exactly, like you said she wanted to sit on it and get more data because, allegedly, she had observed Hoogsteen base pairing, or some other non-canonical base pair that escapes me.
But as https://www.tandfonline.com/doi/pdf/10.1080/1464518031000160... points out, the analysis technique that makes it possible to deduce the structure from the image was first developed 2 years earlier in a paper by Crick, Cochran and Vand. Note the lead author. In 1953, Francis Crick was one of a handful of people on the planet who would have made the connection. In fact he was able to make it from James Watson's description of the photograph! Rosalind Franklin can be pardoned for having failed to make the connection.
But, but... we were always told that Franklin was an expert crystallographer and that Watson and Crick were bumbling amateurs who knew nothing, NOTHING, about crystallography!
Franklin was. Which is why she was the one who actually produced that image. And if the problem could have been solved by deducing the crystalline structure from variant A, she would have been a much better choice.
The fact that Crick was the world expert on the one obscure thing about crystallography that mattered here doesn't mean that Franklin wasn't the real expert. Every expert has gaps in their knowledge.
Using Fourier transforms doesn't seem obscure to me. It seems to me -- today -- that it is an obvious thing to do. It probably was back then, too, if you were a physicist who was good at math.
The photograph tells you the gross structure ("it's a helix") and also that it's a double helix. It doesn't have any real information of the specific structure/location of the bases. That only came later when full x-ray crystallography of 3D crystals (not 2D pulled fibers) was done.
