Hi @rbenton,
I do see how shooting a white background would help with processing, and agree that a black background would allow a similar process, but doing the processing at all should be avoided IMO. Part of the reason I do "disappearing" backgrounds is to avoid such processing. You could also avoid it with the white background, simply doing a manual crop around the coin. Allowing Photoshop to mess with it creates an unnatural edge treatment. I've never done a post-processed black background on a coin that was shot on black, so am not sure it is any better, but I would expect so, since the transition from the darker coin edges to the background is a more natural one versus the dark edges to white. Publishing in black may mitigate this, even if you shoot backlit.
As examples, your 1915 edges with white background look very processed, but the 1955 edges do not, see below. However, your 1955 image is titled "55 ddo phone", and is lower quality vs the 1915 image. Was it shot and processed the same way? Can you process the 1915 (backlit) image with a final black background for comparison?
And not to be nitpicky, but your 1915 image seems a bit oversharpened. Sharpening level is a personal choice of course, and the images I am showing below are at 300% so suffer from pixelation, but at these zoom levels it's usually more apparent if oversharpening has occurred. You didn't mention how much sharpening you're using, or how it's being done. I see this type of oversharpening happen with Sony cameras (using Sony software), but I don't have any experience with the D810 and PhotoShop. Using LightRoom, I can create such adverse effects most easily using the Local Contrast adjustments, or with the Contrast by Detail Level adjustments.
If you want to try shooting with a black background, I use ProtoStar. It is the least reflective surface I have found which can stand up to background use, ie having a coin sit directly on it without damaging it. There are other materials and paints and such that are much blacker, but they are too delicate for such use. Even the ProtoStar material must have some maintenance to remove any dust before shooting, but it holds up well.
One more thing about the metadata...I find it odd that you're not doing MWB. How can you trust the color of your output?
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Edited to add: I see a couple other items in the metadata to discuss:
ISO 320. Because you're shooting hot, this works OK, but if you start shooting to avoid over-exposure, you will want to go lower in ISO (100 or lower) to improve shadow noise. This is especially important as you map shadows from raw to jpg. Note that if you use a black background like the ProtoStar, you'll lose some shadow detail unless you change the processing a bit since even the blackest backgrounds don't go to 0,0,0. I believe what you're doing with the backlighting and over-exposure is called "shooting to the right", while what you end up doing with black backgrounds is "shooting to the left".
Aperture 16. I assume this is the readout from your AFS lens, which will report effective aperture rather than nominal. However, EA16 is still quite small, and while it improves DOF, it also degrades resolution and sharpness. Your D810 pixel pitch is 4.88um, so your DLA is ~f/7. You say D810, not D810E, so you have an active AA filter that will degrade the DLA by perhaps 50% to ~f10, so you are losing 60% of your available resolution and sharpness by shooting at EA16.
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