Check a few things
If you will check a few things you can probably find the answer to your rough idle. First off, when was the last time the valves were adjusted? My engine kind of ran rough to until I gave it a valve adjustment, seemed to make it happy after that. Before adjusting the valves there are a few things you can visually look at to see if perhaps you aren't sucking air into the fuel system as the engine is either trying to start or is running. The next time you go to start the car, lift the hood and prime the injector pump and lines with the manual pump. Do you see any air being pushed through the clear fuel return line when you do this? Trying starting the car right after priming and see if that makes the difference in how it runs at start-up. If it runs fine, then you are sucking air into the fuel system after it shuts down, and it's sputtering in the morning while it tries to rid itself of the non burning air gaps. We can go further into this as far as other troubleshooting ideas, but try this first. It could simply be a slight crack in the brittle like clear fuel lines, a leaking primer pump seal, loose injector lines etc. Look for leaking at any fuel line, including the bleed lines from injector to injector.