If you take enough images with your digital camera, they can all be compared together and a unique signature can be determined. This means that even when you think that you are posting a photo anonymously to the internet, you are actually providing clues for the government to better tell who you are. The larger the sample size of images they have, the easier it is them to track down images coming from the same camera. Once they know all the images are coming from the same camera, all they then have to do is find that camera and take a picture to confirm it beyond a reasonable doubt.
It is important to remove this noise signature so that you cannot be tracked down. I cannot guarantee any of these methods will work beyond the shadow of a doubt because the woman doing research for the government on how to find the signature is very good. Check out her papers if you're really good at dense math, and pass along what you learn. I can only promise that this will make their work more difficult.
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Step 1: Remove the Camera Info
When you take an image with a digital camera, the camera itself leaves a data file which automatically identifies itself. This file is called an EXIF and usually has information like the make of the camera, the ISO setting, the date and time, the pixel setting, etc. This needs to be removed. The easiest way to remove this is to open photoshop and save it for web.
When you save, I recommend using a JPG compression with a quality setting of no better than 60%. This will add a lot of ugly noise make it slightly harder to create a noise signature.
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Step 2: Clean the Lens
Clean your camera's lens constantly with a non-scratch cloth that you can get for cleaning camera lenses or eye glasses. This will remove specks of dirt that will show up from image to image and never change. If there are pixels never changing on your camera then it makes it really easy to identify you. This is why it is important to crop the image.
Step 3: Crop and Resize the Image
If you have dirt on year lens, dead pixels or "hot" pixels, then it is really easy to find images from the same camera since they will all have static pixels. This means, from image to image there will always be a handful of pixels that don't change.
The way to get around this is by cropping and resizing the image. This will shift the position static pixels and when images are compared together make them just seem like noise.
This alone will not always remove the noise signature itself, but it is one way to make it harder to identify it.
Step 4: About the Noise Signature
So far I've only told you how to make it hard to tell a picture has come from your camera, but not how to remove the noise signature. The marks made by dirt on the lens and dead pixels is actually different than the noise signature that they use to keep track of what pictures you are taking.
The noise signature itself is actually due to thermal noise (environmental heat) and imperfections on the CCD itself. Professional photographers and astrophysicists have tried to compensate for this by creating cameras that evenly lights a CCD a moment before the image is taken to create a noise reference against the image being taken or by super-cooling the camera itself so that heat can't distort the CCD.
In your store-bought camera, it won't have either of these functions and so you will end up getting a lot of pseudo-random noise mainly on the last 3 bits of every pixel. This noise isn't really noticeable to the naked eye, but with a large enough image sample, they can create a noise profile and match almost with certain accuracy pictures from the same camera. Since the noise is dispersed throughout the image itself and tends to remain constant through certain standard image manipulations, it is hard to remove without destroying the image.
Step 5: Removing the Noise Signature
Ideally, you would remove the noise signature by making the image cleaner. It is noise after all. Although, determining which noise to remove isn't always clear.
It is probably more effective to remove the noise signature by creating more noise. One thing to try is to open Photoshop and save an image as a jpg using bicubic compression, resize it, crop it and then save it for web, again as a jpg (removing the exif) and finally save it for web again as a jpg using a quality of about 50. This should destroy the noise signature beyond recognition, but the image might not look too good.
It wouldn't be good to do this ever time either since you will end up creating a new "signature." The best thing to do is to save each image using a different elaborate method every time. For instance, you might convert something to a flash video, take a screenshot and then save for web as a jpg compressed with a quality of about 50.
It may not work, but you could also try adding a small amount of random noise to the red and green channels. This might actually clean up the image and remove the signature, but I'm not sure.
Step 6: Common Sense
If you're bothering to make your image anonymous, post it anonymously. Use a TOR server and if you have to register for an account to post it, create a new one for every couple of images. Make the images seem like they are coming from as many places as possible.
Step 7: Use a Film Camera
If you are really that concerned, use a film camera and scan your images using multiple scanners at multiple different corporate copy shops. The only downfall to this is that you actually have to go somewhere to scan the images and store employees or video cameras could maybe identify you. If you wait a few days before posting things online, it will make it harder for them to connect it with you.