Capacity Dimensions and Correlation Dimensions

Fractal dimensions for all the 89 spiral galaxies and 14 elliptical galaxies were computed and data similar to Figure 8 were obtained. For all the computations, R filter (centered around 650 nanometer) images were used. The R filter was chosen for the following reasons. Different filter images contain different magnitude distributions, depending on the material contained within the galaxies. Therefore, the contours created around the range of intensity values have slightly different shapes in one filter image than the other for the same galaxy. This gives different fractal dimensions for the same galaxy for a different filter image. Since our goal is to compare fractal dimensions of various classes of galaxies, it would be advisable to use galaxy images from only one frequency band. R was a common band in all the images taken at the Palomar Observatory and the Lowell Observatory.

Fractal dimensions, found for the R filter images, were averaged over the intensity range in each galaxy for comparison. Histograms in Figure 9 show average fractal dimensions (capacity dimension and correlation dimension) for all the spiral and elliptical galaxies.

 

Figure 9: Number of spirals and ellipticals versus avg. cap. and corr. dimensions

As mentioned earlier, one would expect the average fractal dimension for the elliptical galaxies to be lower than the average fractal dimension for the spiral galaxies because of their less complex structure. Histograms of the number of galaxies versus the capacity dimensions, in Figure 9, show a tendency opposite to that expectation, showing higher average capacity dimensions for ellipticals than spirals. Histograms of number of galaxies versus the correlation dimensions, on the other hand, show an overlap in peaks of the histograms for both classes of galaxies. The Kolmogorov-Smirnov test on these histograms confirms that the differences between the distributions for spirals and ellipticals are not statistically significant. Although in the example of Figure 8, correlation dimension seems to be working as a separator between the two classes, the histograms show that, in general, we cannot rely on either of the two average fractal dimensions computed for the intensity range selected here for classification.

 

Figure 10: Number of galaxies versus fractal dim. for a selected intensity range

The number of galaxies in the histograms in Figure 9, is plotted versus the average of the fractal dimensions for the entire intensity range starting from the sky value plus 4 times the standard deviation around the sky value, to the point where the contours run out of a minimum number of points required for the fractal dimension program. When we reexamine Figure 8, we notice that both capacity and correlation dimensions are higher for the spiral galaxy than the elliptical galaxy around the center of the intensity range. We therefore expect that if we have a more selective range, a fraction of the entire range around the center, different results for the averages of the fractal dimensions would be obtained. Figure 10 shows histograms for 20% of the intensity range around the center of the entire range. The Kolmogorov-Smirnov test on these histograms confirms that the difference between the distributions for spirals and ellipticals are statistically significant for average correlation dimension. We conclude that the average correlation dimension, for a selected intensity range around the center of the entire intensity range, could have possible use for galaxy classification.