I use a custom css class on slides.  I would like to use the same class on pages/posts that use Page Header Images, but not on pages/posts that use a just a Page Header Background Color.  

Is it possible to target just the headers that use background images?  So far I've only been able to target both images and backgrounds.  I'm using .caption-outline {

text-shadow: 2px 2px 8px #000000, -2px -2px 8px #000000;

}

on slides and tried 

.caption-outline, #page-header-bg h1, .single #single-below-header, #single-meta {

text-shadow: 2px 2px 8px #000000, -2px -2px 8px #000000;

} 

to get the headers on pages and posts but it targets both background color and background images as well.

Edit: Is this something that can be targeted with css or would it require an update to the theme to add the custom css to the headers on demand, like the sliders currently do?

Thank you, Tahir!  the .has-bg was the key.