4. Generating artificial molecular cloud images
The concept of fractional Brownian motion structures allows easy generation of synthetic images. Various methods are discussed in Peitgen & Saupe (1988). We generated synthesized images in the following way: specify the power law index of the power spectrum ; for an image with pixels calculate the Fourier amplitudes at the spatial frequency points in the Fourier domain (according to the power spectrum power law), and select completely random phases, i.e. uniformly distributed over the interval with a random number generator. As the image should be real valued, the Fourier amplitudes and phases have to match the appropriate symmetry conditions of hermiticity, . With the circular symmetry of the power spectrum , and hence the Fourier amplitudes, this results in the condition for the phases to be odd: . This is most easily realized in a numerical simulation by first generating an unconditioned set of phases, , from which the odd phases are generated as . Fourier transform (or fast Fourier transform, FFT, in case the number of points in each dimension matches ) then generates the fBm -image.
A slight complication arises when we deal with intensity images, like those of molecular line maps. In this case the image values have to be positive definite at all points. As there is no prescription on how to select Fourier amplitudes and phases in order to get a positive definite image, this condition is not easy to match. One possibility is to simply add an offset to the resulting image, so that its minimum value is equal to 0. This approach is, however, rather arbitrary. A second approach is based on the fact that the square of a real valued image is always positive definite. Squaring the image corresponds to self-convolution in the Fourier-domain. For an fBm image, squaring preserves the power law shape of the power spectrum and its spectral index, though only on average: considering the fact that the phases are completely random, it is plausible that the convolution process, i.e. the coherent adding up of the image amplitudes for all possible spatial lags, leads to close cancellation for all spatial lags except for the 0-lag point (and the points with appropriate symmetry). The amplitudes are reproduced, but with a random fluctuation due to the near cancellation of the random phases. We verified this by analyzing synthesized images generated this way. A proper proof would be highly desirable, but has to await future work.
The result crucially depends on the randomness of the phases in the original image. Due to the self-convolution in the Fourier-domain, this procedure is sensitive to possible aliasing effects. This complication results from the unavoidable periodic repetition of the Fourier transform due to the discrete sampling of a numerical realization of an image. The effect is the weaker, the steeper the power spectrum , i.e. the higher the value of . The magnitude of the power spectrum in the overlap region is then correspondingly lower. Large pixel numbers also help, as they increase the dynamic range over which the power law power spectrum gets sampled. In our numerical simulations, pixel numbers of at least 64 in each dimension turned out to be sufficient for fBm -images not too close to the ones with the most shallow power spectrum possible, i.e. . For pixel numbers of 256 in each dimension, still easy to handle even for 3-dimensional images, these aliasing effects turned out to be clearly confined to only the lowest spatial frequencies. The -variances of the images generated still show a wide range in spatial lags which follow the theoretical power law behavior derived in Appendix C.
Other methods for generating artificial fractal structures such as creating log-normal random distributions of density fluctuations, e.g. applied when synthesizing multifractals or often used in simulating primordial cosmological density fluctuations, might be used, but would have to be carefully checked with regard to the randomness of the phases of the resulting images, crucial in order for them to represent true fractional Brownian motion structures.
Fig. 1 (right side) shows an example of an fBm -structure generated this way, where we choose , close to the value derived from the observed image of the Polaris Flare. In addition, we varied the seed value of the random number generator used for generating the random phases until we obtained an image whose large scale structure accidentally resembles the Polaris Flare section displayed in Fig. 1 (left). The comparison shows that such a structure actually gives a good representation of the structure observed for molecular cloud images.
Fig. 5 shows a series of synthesized fBm images covering the full range of , i.e. . Considering the pixel size of these synthesized images as the resolution of the maps, simple eye inspection of this series shows them to be most similar to real molecular cloud images for in the range , in good agreement with the value derived above from the -variance analysis. This test, though admittedly not a quantitative analysis, has been performed successfully with several experienced molecular cloud observers.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998