Yes, this capability exists to some extent, but not through "signing" the image in the normal sense. It's based on the sensor noise patterns. Jan Lukáš, Jessica Fridrich, and Miroslav Golja (and a few others) at SUNY Binghamton have done work relating to two fields - identification of digital cameras using sensor noise patterns and identification of digital image forgeries using sensor noise patterns.
Something like this paper probably discusses what you are looking for. By detecting interruptions and inconsistencies in the noise pattern produced by the sensors that capture an image, it is possible to detect what parts of a digital image may have been manipulated. It's not a trivial process by any means, but research has been done on using the characteristics of the hardware to perform this type of task.
The last time I did work in this field was about 5 years ago, so I'm a bit out of touch with the latest and greatest, but I do know that law enforcement and the press are both interested in this capability (or at least were 5 years ago). You might have to do some digging to see if/how this has advanced, but it seems to be the best bet at proving authenticity. I just don't see it as something an individual would have on their own.