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4 Comparison to Other Measures

It may seem that we can also see the resolution length distribution in the Fourier spatial frequency spectrum of an image. But even a seemingly simple image like our first example, the checkers image, which has only a few components, the checkerboard, the checker pieces, and a few lines on the floor, has a complicated spectrum, as shown in Fig. 9 (left). Automatically finding the relevant scales is not easy. Although our eye may (given we know what to look for) catch the basic frequency and the higher harmonics of the tiles by looking at the full spectrum, once we sum up all the contributions of a particular frequency (Fig. 9 right) we lose themgif. So even in this extremely simple image, using the spectrum, a human can just barely find the characteristic frequencies. We claim that an algorithm to find relevant frequencies operating on either the 2D or the radially summed up spectrum would have to be significantly more complex than the one we use to find the information measure presented in this paper. We tried many other ways of using the Fourier spectrum in an information decomposition, but we found none nearly as good as the Kullback contrast.

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Figure 9: Spatial frequency spectrum plot of the checkerboard image (left) and its radial sum (right).

The computational costs of the power spectrum method and the algorithm for our information measure are comparable, both being tex2html_wrap_inline649 algorithms. In a non-optimized computer implementation on a one-processor Sparc station, both take around 20 seconds to process a size tex2html_wrap_inline651 image. Implementation on image processing hardware should allow real-time performance.