How to remove paper colour from a scan? - theory
 

I'm trying to reconstruct the original image from a scan of an image printed on a coloured paper.

Originally I thought that I simply remove the background paper colour by "dividing" by it - I assumed the CMYK subtractive colour model. However my experiments contradict to this model and show strange results. I'd be glad if someone could explain the results to me, or could show me the correct model that I could use for my purposes.

I printed a sample RGB image with an inkjet printer on papers of various colour (white,green,blue,orange,...), and then I scanned them back. I expected (in accordance with the subtractive colour model) that any printed colour can make the paper only darker:

But in fact, when printing e.g. purple on green paper I got r/R = ~1.1, and for purple on blue paper there was even r/R =~2.2. The other colour components ratios (g/G and b/B) were < 1.0 as expected. The same happened also for some other colour combinations, and I noticed also some other strange observations.

Can you explain these results to me?

Some of this depends on the image itself.
Can we see a preview of it? Attach it to a reply post and I'll take a look.

I've had mixed results with this over the past 20 years. Some work, some are a lost cause.

My primary aim is to scan old offset printed magazines, not inkjet printed images. But as I've had unsatisfying results with that, I began to experiment to see how exactly the printed colours change when printed on papers of various colour. As I have no access to an offset device, I performed my experiments with an inkjet printer. And I was surprised that my experimental results contradict to my expectations.

I started with the assumption that the used CMYK colors are pure inks (that is transparent colour filters), and even verified this assumption to an extent with a microscope :-). But my experiments suggest that the printing colours contain inegligible pigmental (that is nontransparent, reflective) part. Especially the red (magenta?) colour.
Since my previous post I've learned that there exists a bunch of colour models that describe CMYK printings - the precise modelling is much more complicated than I expected. The model I was using is probably called the Neugebauer model, but currently I know very little about these topics. But in fact, I don't need these models in their completeness, as currently I would be satisfied with removing the paper colour only. Though sometime I may need to make the complete decomposition of the image into CMYK colours.

Here is the sample image I used for my experiments; it has lots of facets of almost the same colour. I printed it on several (ordinary, coloured) papers and scanned back, but the result is a .png of size ~5MB, so not suitable for publishing here.
Then I measured the average colour of areas of size ~150px at identical places on all the printed copies. Here are some of the results (in RGB):

• original:
  * violet: 151,40,111
  * yellow: 235,240,96
  * blue: 23,102,213
  * green: 31,132,53
  * magenta: 210,79,171
  * orange: 215,66,34
• white paper: 236,227,233; 223,212,220
  * violet: 186,109,143
  * yellow: 224,214,120
  * blue: 91,135,93
  * green: 108,165,118
  * magenta: 217,121,165
  * orange: 221,127,116
• blue paper: 56,184,212; 70,183,209
  * violet: 103,96,137
  * yellow: 82,176,127
  * blue: 50,123,180
  * green: 53,146,116
  * magenta: 104,107,154
  * orange: 107,116,112
• green paper: 121,191,104; 123,189,104
  * violet: 129,104,84
  * yellow: 127,183,78
  * blue: 79,129,100
  * green: 85,148,77
  * magenta: 136,117,88
  * orange: 138,117,75
• orange paper: 233,144,92; 222,138,87
  * violet: 187,96,78
  * yellow: 222,140,73
  * blue: 114,108,74
  * green: 123,118,62
  * magenta: 214,99,78
  * orange: 216,106,72

In bold are the values that I don't understand.
Note: The original image is in RGB (but this is not important; actually it's YCbCr, but printed from an RGB editor), it is printed on a CMYK printer, and scanned back on an RGB scanner.