[Big Snip]
Does this make sense to you guys on the other side of the pond?
Nope.
And I believe the technology is the same here as over there (except that over here 2400 dpi is the most common resolution whilst 2540 dpi is most common over in Europe). Also, it's not the imagesetter that builds the screens - it's the RIP.
The cell size in which the halftone dot is formed is variable - it is not fixed at a 4x4 pixel matrix as you suggest. On a fixed resolution output device it can range, depending on the requested lpi, from about a 3x3 pixel matrix (capable of 10 possible levels of gray) to a 23 x 23 pixel matrix (capable of 530 possible levels of gray).
Let's take a simple example to make the math easier. We'll make our halftone dot cell a 2x2 pixel matrix. By turning on pixels within that cell we are only capable of simulating 5 levels of gray:
To increase the number of gray levels we could divide the halftone cell into a grid of smaller pixels:
What we've done is increased the dpi of the device - e.g. gone from 2400 dpi to 4800 dpi.
However if we can't, or won't, increase the device resolution to achieve more gray levels, there is another solution.
The way that virtually all modern halftone screening gets around the gray levels limitation imposed by AM screening and device resolution is by using a process called "dithering" - which is what I think you are trying to describe. "Dithering" leverages the fact that we see tone areas rather than individual dots.
Rather than using one halftone dot to generate a tone, dithering uses a "supercell" made up of a number of individual halftone dot cells. Then by alternating (dithering) between the different halftone dots that can be achieved, we create a tone area representing the average tones in the supercell and therefore a tone value we could not otherwise represent.
So, taking our original 2x2 halftone cell, we might define a Supercell made up of, for example, alternating 25% and 50% dots. The result will be a tone area that simulates a 37.5% value. A tone value we could not otherwise represent with our 2x2 pixel halftone cell.
There are other techniques - but the Supercell method is by far the dominant one.
It has nothing to do with "generations of imagesetters" Supercell screening was introduced around 1990 or so. It was adopted, and very well marketed, by Agfa (in their Agfa Balanced Screening) as a viable alternative to, for example, the likes of Heidelberg's IS hardware based screening.
Regarding: "AMPLITUDE MODULATION & FREQUENCY MODULATION Marketing terms to describe (obfuscate) how the dots are placed in the 4x4 grid."
IMHO, these are very good terms that describe the basics of these two approaches to screening. In one, the amplitude is indeed varied and not the frequency. In the other, the frequency is indeed varied and not the amplitude. Both AM and FM have hybrid variants - where the technologies are mixed, however using the AM and FM models is still useful for understanding a specific screen's underlying structure.
Regarding: "So Photoshop tells you to make your image twice the value of your LPI simply because each line in the Lines Per Inch is made up of a grid of 4x4 (=16) dots."
IMHO, nope. I think the PShop programmers simply implemented the industry convention that ppi should be 2x lpi. It has nothing whatsoever to do with gray levels. It has to do with the spacial resolution of the frequency of halftone dots as illustrated in my previous post with the sailboat image. Both halftones have the same resolution, both have the same number of gray levels. One resolves the pixels of the original image because the screen has a higher resolution than the image. The other does not resolve the pixels of the image because the screen has a lower resolution than the image.
hope this clarifies, gordon p
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