When working on photos with transparent background in Photoshop, which file type would you recommend is best to use to import to InDesign, TIFF or PSD?
Why and when choose one over the other in InDesign?
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In my opinion. If the next step is done by a person that has all adobe suite, including Photoshop, you can leave the PSD. If the other person does not have it, send a TIF.
If the next person needs to print directly from Indesign to a RIP, ask them. Probably they prefer a TIFF.
If you alone are exporting to a PDF, choose either one.
I would not use Layered PSD for this, but a flattened copy.
For a transparent background... .psd may be slightly better. Although there isn't a great deal of difference.
A .tiff with a transparent background is slightly non-standard. Adobe introduced that a few years ago, so it's fine with Adobe software, but there's no telling if a RIP somewhere may have a problem with it. I'm not 100% certain of the output logic Adobe used for a tiff with transparency. So, I can't really state if there's a problem with it or not. It's merely one of those things that might cause issues for someone else later. If you do use tif, it's sometimes best to not use the LZW compression option with tiffs.
For my work, I've exclusively used .psd files for many years. Never worry about anything other than proper color mode. Then just export to PDF/X-1a with InDesign.
There's nothing wrong with the tif format. So there's no hard and fast reason to avoid it. Just as, if working with Adobe software, there's nothing wrong with the .psd format.
In the end each workflow may be different and one format may work better than the other for your work. For my files, I much prefer using .psd.
Maybe these differences can help you make the choice. There might be more technical differences, but there should be no real difference in the results produced by these two formats.
I personally prefer using the TIFF format for two reasons: The possibility of LZW compression (to save just a little disc space) and the fact that I have had problems using PSD files with our RIP software (even though it uses software from Adobe). But that was in a very specific case involving 1-bit images tinted with spot colors. I wouldn't advice against using PSD in general.
(Do not hesitate to correct the list if I am mistaking or add points if you know any other differences.)