Calculation of Fractal Dimensions of the Contours

Running the program Fd3 (described in chapter 3) on the contour data files calculated capacity dimensions and correlation dimensions of the contours. 50 contours were created for a range of intensities for each galaxy. The range was determined by the sky value and the number of points required by the fractal dimension program. As the background of a galaxy image is the sky, at the intensity value of the sky most of the image would be filled with the contours around this value. This would make the fractal dimensions very high (close to 2) for all the images and would be useless for classification being common in both elliptical and spiral galaxies. The intensity values around the mean sky value, therefore, could be considered noise. As one moves to higher intensity values from the mean sky value, this noise starts dropping and one begins to get contours from the galaxy. Therefore, the lower side of the range for contour generation was set to the mean intensity level of the sky plus four times the standard deviation around the mean sky value. The minimum number of data points required to obtain reliable fractal dimensions is 10^d, where d is the true fractal dimensions of the object (Liebovitch and Toth 1989). Since we are dealing with the geometrical objects with unknown fractal dimensions, deciding the minimum number of data points is difficult. Therefore, knowing that the fractal dimension of a two dimensional object cannot be greater than 2, a minimum of 200(significantly greater than 10² = 100) points were initially used. Running the program on several galaxies led to the conclusion that their fractal dimension, on average, is significantly less than 1.7. This gives us the minimum of $10^{1.7} \approx 50$. To be on the conservative side, a minimum of 80 was chosen. At high intensity levels, closer to the center of a galaxy, the contours start to become smaller, containing fewer data points. At one point, the contours run out of the minimum required data points (80) setting the higher side of the intensity range for generating contours. This range was divided into 50 intensity levels and fractal dimensions were computed for contours around each level.

Figure 8 shows example of fractal dimensions for two of the galaxies computed using the Fd3 program. The examples show fractal dimensions of 50 contours generated for different intensity values for an elliptical galaxy (NGC 4374) and a spiral galaxy (NGC4303). The average correlation dimension is significantly lower for the elliptical galaxy than the spiral relating to its less complex structure. The average capacity dimension on the other hand is higher for the elliptical than the spiral galaxy, which contradicts our expectation.

 

Figure 8: Fractal Dimensions of NGC 4374 (Elliptical) and NGC 4303 (Spiral)