LPI = Lines per Inch
It is a measure of the fineness or coarseness of the screening in photographs and tints in a printed piece. The term comes from back when we halftoned images by photographing them through screens, which are pieces of film with a pattern of dots on them--you measured off an inch and counted the number of dots in a straight line. Or, actually, someone at the screen factory did it; they were marked, so if you wanted a 133-line screen you just took it out of the box--which you ALWAYS put it back in after you were done using it, because they scratch easily and they're hundreds of dollars each. You can still get these, by the way--people still shoot halftones on cameras.
Four things go into determining your linescreen: the reproduction process, substrate, prepress imaging device and source image all play a role.
As to the reproduction process, some ways to print will "hold" a higher linescreen than others. Offset's higher than flexo which is higher than screenprinting.
The substrate is what the product is being printed on. If you're running on Kromekote, which is a paper with a mirrorlike surface that really doesn't absorb ink, you can get a far higher LPI than if you're running on a cotton t-shirt.
The prepress imaging device is obviously important, and here's how you calculate the maximum screen frequency you can get out of it: divide the resolution you're using by 15. A dot in a digital halftone screen is made of a group of device dots (aka a "supercell") and you need 100 device dots per supercell so you can get all the dot percentages from 1% to 100%. You need to have more dots per cell available because not all dots are the same shape--some run up-and-down, some side-to-side--so we've found a supercell that's 15 dots square, or some multiple thereof, works well. So...if you have a 2400dpi or 2540dpi imager, you can get a smooth 150-line screen out of it.
And for the source image...the rule of thumb here is, "maximum LPI is one-half the DPI." In other Words, a 300dpi image gives a good 150-line screen.
The resolution of certain output (and input) devices is sometimes defined in LPI
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