Contrast Resonance

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Here is a phenomenon you may have noticed, if your sight is not sharp. In that case, (which is mine) when you look at a striped shirt, you sometimes have the surprise to see the strips, whereas you would rather expect not to see them.

Black and white strips Here are evenly spaced vertical strips
Fuzzy strips If your sight is not sharp, you will rather see that.
Fuzzier strips Even more fuzzy strips Very fuzzy strips These pictures get more and more fuzzy.
Gray picutre By increasing the fuzzyness a bit, you will have an all gray picture.
Inversion of the contrast The surprise comes when you go further in fuzzyness. The strips reappear (resonance)! Not only they do, but they are inverted !
There are other contrast inversions beyond this one, but they are weaker and weaker. At the limit, you get an all gray picture.

The following picture summarizes (click on it for a bigger version, 105 KO) the phenomenons.

105 KO !

The mathematical modelisation of the phenomenon is quite simple: fix a radius, which I call the fuzzyness radius, or r. Then the brightness of one given point is the average of the brightness of the points in the disk of radius r centered on the given point. (But don't take a gaussian weight, the resonance would not occur, or if it does it is much less visible !). For more simplicity, and getting most regular images, I only made the fuzz in the horizontal direction (i.e., I took the average of the points on horizontal lines, intead of disks). In that case the model is even simpler (as brightness in fuction of the horizontal coordinate, you get periodic piecewise affine functions). If you take a sine instead of a square for the starting brightness distribution, then the contrast (difference between max and min brightness) as a function of r is a cardinal sine function.

Two more pictures, the one on the right is a fuzzy version of the one on the left Pattern Fuzzy pattern

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