Generally, yes: if the source image is truly in the public domain, you can usually use it without permission and create a derivative work from it. In that case, you would typically own copyright in the original parts you add to the new work, but not in the public-domain material itself.

A key point is that copyright law varies by country, so the safest answer is to check the law where you live or consult a copyright lawyer if it matters commercially.

Also be careful to verify that the source really is public domain. People often confuse public domain with free to view, free to download, or openly licensed. Those are not always the same thing.

So the practical answer is: if the background image is genuinely public domain, your added creative contribution can be protected, but you do not gain exclusive rights over the public-domain portion on its own.