I figured out how to do this about a year ago. A close friend's wife died of cancer in 2009, and he wanted to give his daughters a nice picture of their mother when she was young. Unfortunately, one of the few pictures he had dated from 1980 and was printed in color on that silk textured paper. It was badly faded as well. I scanned the print, and immediately ran into the texture issue. I thought about using the clone tool in photoshop, but that seemed pretty daunting and destructive to the image.
As I mentioned before, my work involves using signal processing techniques, and I knew that this regular pattern was signal, and the image itself was noise, at least in a statistical sense.
I sort of grocked this whole thing out because I knew what I wanted, and just happened to run across this application that did what I needed. I googled until I found the free ImageJ application mentioned above. It allowed me to go from this to this:
Basically, the workflow amounts to this:
1) Open the ImageJ (64bit) program and go to Edit->Options->Memory and set the memory to something pretty large - I went to 6GB.
2) Open your tiff file in imageJ and look at it at 100% (image->Zoom->View 100%)
You will see something like this:
3) Then use Process->FFT->FFT.
It will think a while. Maybe more than a while. The calculations are involved. Say 5-10 minutes for a 250Mb image file.
You will get separate FFTs on each channel.
The FFTs look like this:
This the same image's Red channel. The white lines are regular patterns with periodicity - the grain pattern you want to go away.
Use the brush tool to paint a mask over these areas so that they won't appear in the image. It will look like this:
3) Now do the inverse FFT (Process->FFT->InverseFFT) on each of the channels and save each channel separately as the ImageRED, ImageGREEN and ImageBLUE tiff file.
4) Load each of these channels into photoshop at the same time and merge them. This is done using the Merge Channels selection in the Channels pallet. You will have to look in the photoshop help menu about how to do this, but it is pretty straightforward. Merge the files into the appropriate channels and you have your image back without the noise.