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artificial worlds this way on a conveyor belt, as it were. What comes next, however, exceeds this by far. |
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Let's look at an arbitrary image. To make this as uncomplicated as possible, the image you consider should consist of black and white points only. You'll be able to see structures inside the image, assuming, of course, that the image really does represent something. |
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The key here is the concept of self-similarity. If you look at a piece of the image under a magnifying glass, you'll see something that looks very much like the starting image. It might be rotated slightly, or even deformed, but basically it will be the same as the original. In practical terms, you need to detect so-called affine transformations Ti (i = 1, . . ., n) of the image B, which possess additional properties. Affine transformations are very easy to describe in mathematical terms. An affine transformation in a plane has the general form of |
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Here, (x,y) is the starting point, while (xnew, ynew) is the transformed point in the plane. The constants a, b, c, d, e, and f determine the exact character of the affine transformation. The pair (e,f) is responsible for the displacement of points, while the other parameters a, b, c, and d cause compressions, elongations, and rotations. |
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In the specific competing affine transformation T, you can interpret self-similarity in such a way that in the case of the image, T(B) coincides nicely with a part of B. If you take this a step further, it might be possible for you to detect a range Ti(i = 1, . . ., n) of competing affine transformations that recreate the total image in miniature. If you match all this in such a way that the relationship |
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is satisfied at least approximately, you get an output image that in some respects is completely coded by the transformations T1, . . ., Tn. This coding is extremely efficient, since each transformation is completely described by the six values a to f. Now, what about decoding? As it turns out, there is a brilliant solution: chance reconstructs the output image. |
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Before we look at an example, let's review the solution algorithm for the fractal description of structures: |
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A. Determine a range of competing affine transformations T1, . . ., Tn with the property described in Eq. 6.5. |
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