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Astron. Astrophys. 336, 697-720 (1998)

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3. Connection with the theory of fractal images

It is well established within the work that has been done in analyzing and synthesizing fractal images (e.g. in order to generate artificial landscapes for computer animations etc.), that such structures can easily be generated as so called fractional Brownian motion (fBm )-images (see e.g. Peitgen & Saupe 1988). These are generated by specifying a power law power spectrum with a given index [FORMULA], where E is the Euclidean dimension of the image and H, the drift exponent (see below; sometimes also called Hurst exponent ) ranges between 0 and 1, and completely random phases of the image 1. Such structures have a two point correlation function


as derived in Appendix B, where [FORMULA] is the signal, i.e. image intensity. The drift behavior (mean variation over distance) is correspondingly [FORMULA]. Note that the range for [FORMULA] with H between 0 and 1 spans exactly the regime from [FORMULA] to [FORMULA] for which the E-dimensional two point correlation function follows the [FORMULA] behavior (see Appendix B). Outside the fBm -range, it turns, for higher values of [FORMULA], over into a behavior [FORMULA], independent of the value of [FORMULA]. For smaller values of [FORMULA], the two point correlation function does not have a simple power law behavior for small [FORMULA].

The drift behavior of an fBm structure, [FORMULA], implies a fractal dimension of the E-dimensional "mountain surface" [FORMULA] of [FORMULA], e.g. measured as a box coverage dimension (see Voss 1988): at a linear scale [FORMULA] a single [FORMULA]-dimensional box of volume [FORMULA] is needed to enclose the "mountain surface". At a [FORMULA] times smaller scale [FORMULA] each enclosing box has a volume of [FORMULA], and [FORMULA] such boxes are needed to enclose the "mountain surface" over the original extent. These boxes thus cover a total volume of [FORMULA]. In units of scaled down original boxes, each having a volume of [FORMULA], one thus requires [FORMULA] of such boxes to enclose the "mountain range". According to the definition of the box-coverage fractal dimension [FORMULA], [FORMULA], so that comparison with the above results shows that [FORMULA].

Inserting the above relation between [FORMULA] and H, namely [FORMULA], we get [FORMULA] as the relation between the fractal dimension of the fractional Brownian motion structure and the power law index of its power spectrum . As expected, [FORMULA] thus varies between [FORMULA] and E for [FORMULA] varying between E and [FORMULA]. Outside this range, [FORMULA] settles at these extreme values according to the settling of the drift behavior at [FORMULA] and [FORMULA] for too small and too large power law indices [FORMULA].

As was discussed in the previous section, the [FORMULA]-dimensional projection of an E-dimensional structure with a power law power spectrum has again a power law power spectrum with the same spectral index (note that in the case of infinitely extended images, this is even independent of whether the phases are randomly distributed or not). The randomness of the phases for the E-dimensional image implies randomness of the phases of the projected [FORMULA]-dimensional image, as is obvious from the fact that the phases in the [FORMULA]-plane of the E-dimensional Fourier image are randomly distributed. The fact that [FORMULA] stays constant on projection is in contrast to the commonly formulated, though never proven, hypothesis, that the fractal dimension of the projection of a fractal structure is lower by 1 than that of the original structure. With the relations between the various parameters of an fBm -structure as discussed above, we explicitly show that this does not hold for a fractional Brownian motion fractal structure. The fact that the power spectrum spectral index [FORMULA] stays constant on projection implies that the drift index [FORMULA] changes: from [FORMULA] we conclude that [FORMULA]. The fractal dimension as defined above then changes from [FORMULA] to [FORMULA], that is by 1.5, and not by 1.

This implies that a fractional Brownian motion structure becomes smoother, i.e. gets closer to the limit of a regular Euclidean structure ([FORMULA], [FORMULA]), on each projection. Even if it starts out in E dimensions close to being very broken up, with H close to 0, i.e. [FORMULA], the first projection, that is the [FORMULA]-dimensional structure has already [FORMULA]; and after the second projection the resulting [FORMULA]-dimensional structure formally has already [FORMULA]. Its power law index thus falls outside the range where [FORMULA]; the drift behavior rather settles at [FORMULA], i.e. [FORMULA]. A fractional Brownian motion structure can thus be projected at maximum once before becoming smooth enough to be indistinguishable from a Euclidean structure: on projection, the small scale structure overlaps to form larger, smoother structures.

Another measure of the fractal dimension of e.g. a 2-dimensional image is given by the area-perimeter relation of its iso-intensity contours, which scale as [FORMULA]. For these 1-dimensional structures the fractal dimension d varies between [FORMULA] in the Euclidean limit ([FORMULA]) up to the maximum value of [FORMULA], where the perimeter increases linearly with the area enclosed. Similarly for a 3-dimensional structure: consider a 3-dimensional fractal density distribution; the area of its iso-density 2-dimensional surfaces scales with their enclosed volume as [FORMULA], with d ranging between 2 and 3.

Voss (1988), p. 65, states that in general, given an E-dimensional fractional Brownian motion density structure with [FORMULA] and correspondingly a box coverage fractal dimension [FORMULA], its [FORMULA]-dimensional iso-density surfaces, being effectively "zero-cuts" of the E-dimensional structure, have an area-perimeter (resp. volume-surface) fractal dimension [FORMULA] by exactly 1 lower than the box coverage fractal dimension of the structure they result from. Vogelaar & Wakker (1994) checked this numerically over a limited range of [FORMULA] by analyzing fBm -structures generated to calibrate their area-perimeter relation code. We developed a similar code (Zielinsky et al., in prep. ) and also compared various numerical methods to determine the area and perimeter of contour lines. Within the systematic errors associated with each method, the area-perimeter fractal dimension of the Polaris Flare image shown in Fig. 1 is [FORMULA] or slightly higher, consistent with the fractal parameters determined by the other methods discussed above. Note that this value for the fractal dimension is higher than that determined for several other clouds in various tracers, which typically gives values in the range 1.3 to 1.5. At this place a cautioning remark seems appropriate, encouraging careful distinction between the different fractal dimensions in use. To illustrate this, consider a 3-dimensional fBm density structure with a certain value of [FORMULA]. Its box coverage dimensions as defined above is then [FORMULA], the (volume-surface) dimension of its 2-dimensional iso-density surfaces is correspondingly by 1 lower, i.e. [FORMULA]. Due to the constancy of [FORMULA] on projection, the corresponding 2-dimensional column density image has, as derived above, [FORMULA]; its box coverage dimension is [FORMULA], and is 3/2 less than that of the 3-dimensional structure. The (area-perimeter) dimension of the (1-dimensional) iso-column-density contours is again by 1 lower than the box coverage dimension of the 2-dimensional column density structure, [FORMULA], and thus also 3/2 less than the (volume-surface) dimension of the (2-dimensional) iso-density surfaces.

Table 1 summarizes the connection between the different parameters. It also includes the range of parameters derived from observations of the Polaris Flare in the present paper. The values for the Polaris Flare are in very good agreement with the relations derived for an fBm structure and a value of H near 0.4.


Table 1. Fractal parameters characterizing Molecular Clouds

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998