I am not sure this topic was not discussed so if I am double-posting - sorry!
Anyway I was informed someone expects problem with camera after update to 4.1.2.
The reason is the firmware contains updated driver which is not pre-installed in the phone because the firmware is actually a dump.
I would like to explain the way I performed to solve the problem (now I have GT-i9300 with 4.1.2 and working stock Camera (both video/photo) and other camera applications also).
First of all: make a Nandroid backup! I have tested all the steps but I cannot guaranty everything will work properly in your case.
Second - I can help with GT-i9300 only. If you have other device - sorry. Maybe this receipt works - maybe not, I don't know.
1. Please dial in stock dialer the following: *#34971539#
2. In menu please choose Phone/CAM FW Ver Check
We need to know your two parameters: Cam FW Ver and Phone FW Ver.
Cam FW Ver is a version of driver installed in your phone. Phone FW Ver is a version which comes with a firmware. Usually firmware's one is newer. In my case it was
Rear Camera ZDFE02
Phone FW ZDFI02
Be careful! There are other camera vendors exists - and it is easy to detect by first two letters of driver version (in our case it is ZD). If you have other vendor - please download appropriate driver.
After you have download the archive you have to unpack the file named "SlimISP.bin" on your sdcard (internal one).
Then dial *#34971539# once again and choose Phone to CAM FW Write (Rear) (if you have unpacked the file properly - this button will become active). Flashing will take about 20-30 seconds - please be patient.
After flashing finishes exit the Camera Firmware menu and enter it once again (*#34971539#). Choose Phone/CAM FW Ver Check
Check again above-mentioned two parameters: Cam FW Ver and Phone FW Ver. If you are lucky enough they will be equal and the process completed successfully.
The source is 4pda forum (and all thanks should be addressed to them and Pako777 individually).