Bijective transformations of images

One calls Bijective Transformation of Image a transformation of an image finished of nxm pixels on itself: each pixel is thus moved and no pixel is lost, what is called in mathematics a permutation of the whole of the pixels and in language running a mixture.

The transformation of the image $(nxm)$ which moves the pixel $(I, J)$ in $((i+1) %n, m)$ , corresponds to a shift of a pixel towards the line of the image. In a commonplace way, N applications of this transformation gives again the initial image.

This property is general: for any Bijective transformation of Image, there exists a smaller entirety $k$ as applied $k$ time, the transformation gives again the initial image.

This result is an immediate consequence owing to the fact that the whole of the bijective transformations of an image $(nxm)$ is a group.

The bijective transformations of the most known images are

the transformation of PhotoMaton
the transformation of the Baker
the transformation of Hilbert

The number of stages before seeing reappearing the image is sometimes very large and depends on the one hand on the transformation and on the other hand size of the image. For example a square image whose east coast a power of 2 will return very quickly, whereas with two unspecified numbers, the return perhaps extremely long. During these stages, one passes sometimes by reconstitutions very close to the initial image.

Example: PhotoMaton

This transformation was introduced by JP Delahaye and P Mathieu of LIFL/USTL into PourLaScience 242, DEC 1997.

See the detailed article: Transformation of the photomaton.

References

JP Delahaye and P Mathieu. Scrambled images, found Images. review For Science 242, DEC 1997, PPP 102-106
JP Delahaye and P Mathieu. Data-processing Scytale. review For Science 359, seven 2007, p 90-95
JP Delahaye and P Mathieu. Scrambled images, found Images. Mathematical and Mathematical plays of Plays, Belin/For Science, 1998

External bonds

A software is available in the form of Applet Java to test various transformations on the images of its choice was carried out with the LIFL/CNRS of Lille: Applet with the LIFL

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