I'm bad with colors... Are the corner uprights purple colored? If it's them, then the ones on the end (not front/back) are the more important ones, since both of the horizontals rest on them.
Personally, I think it would look kinda funny without both pieces in the corners, unless you end up skinning the whole frame or something similar.
Also, eliminating those four pieces will only add up to 1, maybe 2 2x4s, so that's only a couple bucks saved.
Maybe try using 2x3s, they are pretty cheap, usually kiln-dried (for certain values of "dry"...), and I think they look better in some places then 2x4s. I usually do my corner verticals with paired 2x3 and 2x4.
And, as Dustin87 mentioned, you can skip the pieces on the inside corners, but it will make placing fasteners a bit trickier. If you have a pockethole jig, that wouldn't be an issue, otherwise you are going to have to put some really long screws through the 4" width of a 2x4.