This is pretty easy to do. Each control on the form has a Size and a Location property, which you can use to instantiate a new Rectangle, like so:
Rectangle rect = new Rectangle(button1.Location, button1.Size);
To get a Bitmap that contains the portion of the background image located behind the control, you first create a Bitmap of the proper dimensions:
Bitmap bmp = new Bitmap(rect.Width, rect.Height);
You then create a Graphics object for the new Bitmap, and use that object's DrawImage method to copy a portion of the background image:
using (Graphics g = Graphics.FromImage(bmp))
{
g.DrawImage(...); // sorry, I don't recall which of the 30 overloads
// you need here, but it will be one that uses form1.Image as
// the source, and rect for the coordinates of the source
}
This will leave you with the new Bitmap (bmp) containing the portion of the background image underneath that control.
Sorry I can't be more specific in the code - I'm at a public terminal. But the intellisense info will tell you what you need to pass in for the DrawImage method.