My latest View from the Pennines (a regular piece for Mathematics Today, the magazine of the Institute of Mathematics and Its Applications) is about invisibility and optics/scanning; if you want to see the details go to the View from the Pennines Homepage. As always my starting point was an observation on life on the moors around us, or our pond -- in this case the amazing silver and black carapace of the water boatman.
Conventional camouflage is about matching colour and shade. The surprising thing about the water boatman in the picture is that the metallic sheen seems to match material/refractive properties of the water surface seen from below. This suggests that a good theory of camouflage would include some measure of the 'feel' of the background. It would be interesting to see a mathematical description of camouflage. An obvious comment is that stripes should not be too regular (I have a great picture of a dragonfly I'll dig out) -- does this mean we should be using Fourier space? Another aspect any good theory should encompass is behaviour. The caddis fly larva actually uses its environment to provide it with camouflage -- it glues bits of leaf and mud to its back -- but seen from the side of the pond it is relatively easy to spot because it does not move with the local currents. I suppose its prey doesn't have the advantage of position (or brain processor power) to see this flaw.
Both the water boatman and the caddis fly larva show that camouflage needn't be perfect from all vantage points, only those of the object being stalked or avoided. Thus any serious mathematical description would have to specify what the camouflage needs to achieve as well as how to achieve it. I wonder whether there is any serious work in this area?