I have a photograph that was taken on stage in a theater. The photo was taken with a Nikon D3100 camera with flash in auto mode and stored as both raw (.NEF) and jpeg file. Because spotlights were on when the picture was taken, a the center of the image is overexposed compared to the sides. In the jpeg, the sides of the picture appear properly exposed with acceptable color balance. Is there any way to change the exposure and color balance of the center of the image while keeping the sides unchanged?
Yes, it is possible, especially if you also have a NEF file besides the JPEG - Adobe Camera RAW (inside Photoshop or Lightroom) allows you to bring down highlights without affecting the darker areas of the frame - and if the said highlights are not blown out (i.e. clipped to a point where information is lost), you've got a fair chance of ending up with a usable picture - this depends mainly on the ISO setting that was used - the higher the setting, the narrower dynamic range you have to work with. Do not judge the result by the JPEG file - consider it just a snapshot with zero adjustments.
As for why this happened in the first place - if you've got the camera set to Matrix metering, it will try to expose the whole image properly, i.e. it won't know that you want to keep the shadows, well, shadowy. Next time put the flash down (yes, you will have enough light when the spotlights are on) and try to set metering to Spot - this way the camera will only consider a small portion of the image around the selected focus point (if it's a single one selected) or around the centre of the frame (if you have Auto area selected).
I'd say you would be looking for some kind of HDR plugin - one that can work from a single image rather than a pre-determined set of bracketed exposures.
The Nik Collection by Google has something that can at least have a go at that type of processing. I've used it, but not on a rescue mission, only from photo series designed to be processed that way.
The plugin suite recently became freeware - maybe as a parting gift with no further support, but I'm not sure on that.
Overexposure may be correctable in a RAW processor as noted. There are also numerous ways to change tones in an image editor such as Photoshop.
Normally, you would use a masking or selection process to only correct the brighter areas. Levels, Curves, some layer blending modes, burn tool, Exposure, Highlight/Shadow, and more can be used to change the actual brightness of that area.
I'd recommend looking at some video tutorials because there are a number of possible ways to approach this problem.
You can use plug ins, but this is very doable even w/o them. Below is the procedure if using Photoshop. It assumes that the center is NOT so overburned that you can't work with it:
1) In PS, create a duplicate layer of your image
2) adjust the duplicate to the correct exposure for the center. Don't worry right now if the sides get too dark.
3) go to the "Layer" menu item, select "Layer Mask" and "Hide All". This will place a mask in front of your duplicate layer and hide the previous edits.
4) in the PS toolbar (on the left), set the foreground color to pure white and the background color to pure black.
5) in the toolbar, select the Gradient tool
6) select the Radial Gradient type in the top toolbar just below the menu.
7) now fill the duplicate image with the from center to outside. This will make the areas with the white 'show through', showing the edited image work. The dark/black part of the gradient will not show, so PS will display the original sides/corners, which were properly exposed.
You can now save as a JPG. If you want you also select the two layers and press E on Windows to create another layer that combines the selected layers.
This technique works with everything btw.. selective sharpening, color tricks, .. you name it.