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Scientists have shown that the visual system is built to most efficiently
represent natural stimulus. Natural images are noted to have power spectra
that follow an
f^-2 trend. Since visualizations attempt to provide insight
by sending data through the visual system, we tested visualization screenshots
to see if they have properties of natural images. By running the results of the IEEE
InfoVis contest through some image analysis, we found that the winners of the
competition tend to be more natural.
Spatial Frequencies
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Spatial frequencies are similar to sound frequencies. Sound frequencies are a measurement
of compression varied over time, whereas spatial frequencies are a measurement of
intensity varied over distance. Since spatial frequencies can only measure a single
intensity value, brightness is commonly used. One way to measure the spatial frequencies of a function is by using Fourier transforms. Essentially, sine and cosine waves of different amplitude and frequency are added together to form the intended function. These sine and cosine functions make up a Fourier series. For a two dimensional image, the Fourier transforms are performed over each line in the horizontal axis then over each line in the vertical axis or vice versa. In turn, to find the two-dimensional Fourier transform of an n-by-n image, one must find 2n one-dimensional Fourier transforms. |
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Natural and Unnatural Images
A natural image is any picture of nature. Pictures of a forest scene, a mountain, or a dog would be considered natural images. This class of images constitutes an infinitely small fraction of all possible images, yet our visual system is precisely tuned to perceive them rather than some larger range of image types. In fact, recent studies have shown that the eye actually regulates its growth to maintain a natural image on the retina. To measure the spatial frequency distribution of these images, one begins by computing the Fourier transform. The rotational average of the two dimensional result yields a more manageable, one dimensional series also known as a power spectrum. When the amplitude of this spectrum is plotted on a log-log scale as a function of frequency, the spatial frequency distribution can be visualized.
Plots of the power spectra from the natural images are shown above. These plots have nearly straight lines with slopes of approximately -2, which corresponds to an f^-2 trend. The consistency between the plots is not trivial, as these images appear quite dissimilar. Unnatural images have very different power spectra. The images below show three unnatural images and their corresponding spatial frequency plots. The distinctness of natural images becomes more evident in these plots, as the unnatural images do not show the f^-2 trend. These plots have been observed for tens of thousands of images, and the f^-2 trend consistently differentiates natural images.
Testing Visualizations' Naturalness
To test if effective visualizations have natural properties, we looked at the InfoVis contest results. In the contests, a single dataset is used by all participants to create a variety of visualizations. Judges then rank these visualizations. We compared this rank with the visualizations' naturalness. To measure the level of naturalness, we measured the distance of the spatial frequency distribution from f^-2. The results can be seen below.
Coming Soon: A Downloadable Spatial Frequency Analyzer