I dont feel that is the case. Xbox version is darker, so it gives this illusion of a crispiness while it is not the case when you look closely. It has some missing textures as well compared to ps4/pc version, someone posted a gif in the previous page. Look at that those blue railing cover in the gif below some very bad aliasing in both, but xbox version is noticebly worse. 

Edit: noticable differences in larger images here http://i.minus.com/ibp8tqOwsjlAou.gif  http://i.minus.com/igaJKLNteE5Mv.gif Now imagine on a much Larger TV screen. 

They need to tone down the gamma in ps4 version, that will make a fair comparison.

Another video comparision... no weird dark too.

http://www.videogamer.com/features/article/battlefield_4_which_is_better_xbox_one_or_ps4.html

"Battlefield 4 on PlayStation 4 looks stunning, with DICE clearly utilising the console’s additional firepower to yield a superior image quality on Sony’s machine. Images are pin-sharp, appearing to run at a native 1080p and fluid 60 frames per second. But while the Xbox One version effortlessly maintains the same frame-rate, the image quality is perceptibly worse, sporting a softer, jaggier look that carries all the hallmarks of a game upscaling from a sub full-HD resolution. My suspicion is that the Xbox One version is rendering natively at 720p before upscaling to 1920x1080, although DICE opted not to confirm numbers when confronted about the resolutions of the two next-gen versions directly."

Just a fix.

Sorry but those pics only back up my belief that X1 looks superior. PS4 looks blurry and washed out. Surprising considering how superior PS4 is supose to be.

Again though, this is my opinion going by what is shown. Just look at the water in the botton shot. Some of the effects are missing in the PS4 version.

Xbox one has one great graphical advantage though. You can see all the soldiers a lot better, because their camo is missing

http://i.minus.com/ibmS2Qk8v0t44g.gif

Like I said, click the links to see bigger pictures (or just go to gaf)

I feel a litle cautious about linking neogaf as you xbox fans have problem with it but here if you want more pictures, then make your judgement.

http://www.neogaf.com/forum/showthread.php?t=705961

Then there's that frame rate difference. 

LOL typical damamge control. Call me a MS fan because that way I must be bias and not because its an honest and true opinion.I have never owned nor will I ever own a MS console.

You will have to find some other way of disproving me other than trying to claim I hold allegiance to MS.

Wow... opinion is differant than flat out denial. Now give up the sham and give credit where its do.