Danbooru-based image board with a specialization in high-quality images.
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kanzaki_kaori to_aru_majutsu_no_index transparent_png vector_trace

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Well, I definitely like the picture, but the edges of the hair could use some work. They probably didn't on a white background and so the white blended in with the background better. In any case, it looks good otherwise and is far better than I could ever do, having no artistic talent and all that.
It looks fine to me: try viewing it full size.
It does look find at full size. Unfortunately, It's not like I'm ever going to view it that way. My monitor is 1920x1200, so I have a pretty good size monitor, but the pic - like many around here - is so large that there's no way that much of anyone - if anyone - is going to be able to view the whole thing at once at full resolution. And the edges of this pic don't seem to fair well when the pic is reduced in size. Maybe it's my software, but they definitely look bad when shrunk. Still, it's a good pic, and I don't have a clue what the artist could have done to make it not have that problem when not at full size.
Are you viewing it in your browser? I don't understand what the problem you're having is. Here's a scaled down 1920x1200 (cropped to fix aspect ratio):

transparent: http://sadpanda.us/images/69198-CXQFP63.png
black bg: http://sadpanda.us/images/69199-GCVBROM.png
white bg: http://sadpanda.us/images/69200-WRRRU70.png

Save the image and apply it as your background wallpaper. Unless you're viewing at like 1000% or something, the edges look fine.

Edit: I just realized that you're probably seeing the sample image. The site's image scaling isn't the best and as such will do that to images. Make sure to view the full size version first, then save it to your computer and resize from there.