You can't _register_ copyright. You automatically receive it with the creation of a new original work.
You can't provide it to the public domain. You can, if you have copyright on the work, publish it under a license which permits others the freedom to re-use it. The work is still copyright protected though.
The existence of copyrights on these photographs was the subject of the trial: Can these photographs be considered new (derivative) works? If so the author of the photos ―the museum― owns the copyright. If not the photos are protected by the same copyright as the paintings, although the paintings fell into the public domain as the original author has died more than 70 years ago.
If you make a 3d-model of an object which is still protected under copyright it's for a judge to decide whether you've infringed on that copyright or created a derivative work.
That's all, of course, completely separate from trademarks.
https://helpx.adobe.com/photoshop/kb/uptodate.html?sdid=XT3P...