For this you'd need to make the inner lining of the umbrella out of something similar to projection screens - to adequately hold the image. Then, somewhere down near the handle there would be mounted a projector, processor, battery, etc.

The projector would be set up to project onto the curved surface (with intensity tapering off towards the edges for even lighting, etc), and I suspect the umbrella frame would have to be rather unobtrusive, or even external. Perhaps the inner lining would be somewhat elastic to allow for a smooth curved surface.

The baseline model would simply project a static scene of an idyllic blue sky, with perhaps a few intriguing clouds dotted about. Perhaps an integrated speaker could project some appropriate ambient sounds, like birdsong, or water running down a stream. More advanced models could project moving scenes, orient the scene dependant on the movement of the umbrella using tilt switches/gyroscopes, etc. Obviously you could have a library of scenes including a rainforest canopy, beautiful sunsets, clear starry nights (including a big bright romantic moon), or even something prosaic, like say the Sistine Chapel.

Next time you're stuck out in the rain, walking to the train station, with a dreary grey cityscape all around, you could turn on your SunBrella and be surrounded by a bright, sunny sky, birds chirping away over the murmur of a trickling brook. mmmm. Just the ticket.