This problem comes up a lot more than you might realize.

Detecting and redrawing only the parts that were covered is the wrong approach. Do not even try this.

The best approach, usually, is really to redrawn the whole scene each frame. This would involve remembering which pictures were placed and where they were placed, usually with a class for remembering that sort of thing and an ArrayList of those type of objects. This is fairly complicated, but is the most flexible way if you need to make changes to your sketch later.

Capturing the background and saving it as an image, however, is also a viable option and is fairly straight forward.
Here's an example:

Thank you for your insight and code.  I will implement it that way.

My only concern is that processing may bog down when I have it redraw many images all over my 30" monitor, or saving and redrawing that entire 4MP image every frame (probably worse).  We'll see how it goes.

P.S.  How does windows handle dragging folders around or even moving the cursor around on the screen?  Is the entire screen redrawn all the time?

If you don't need flexibility and you worry about fps, then just use the method in tfguy44's code example. Apart from a black screen there is nothing more efficient than displaying a single image (no matter how many images were used for it's creation).

It's not much the size of the screen, but the number of objects needed to be redrawn!
For more than 95% of cases, clearing the whole screen to redraw everything is fast enough.

An algorithm for selective redrawings is complicated, and only needed for programs w/ lotsa objects!