There are a lot of ways to do this. The very best, the most controlled and cleanest is to use the pen tool. It will give you a crisp edge free of any color you don't want. Another method you can use is the quick select and be sure to zoom way in to check your selection edges. You won't get them all but that's the next step.
Whichever selection tool you use, once you have your selection, add a layer mask. With the pen tool selection, you probably won't need to do much except clean up here and there. With other methods, depending on the care you take, it will probably be helpful to use the refine mask option.
Select the layer mask, it should look framed, right click on it and choose refine mask. Set your viewing choice as white Bg, check smart radius, and play with the other settings, mainly shift edge. Any of the others might make it worse, but you'll have to experiment. Now you should only have some small cleanup to do.
Here is a rough selection I made with the quick select tool and refine layer mask options No, I didn't try to get all the BG out of the middle and no, my selection was not careful and no, I didn't do the manual cleanup. :mrgreen:
