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*Astron. Astrophys. 336, 697-720 (1998)*
## 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 , 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 is the
signal, i.e. image intensity. The drift behavior (mean variation over
distance) is correspondingly . Note that the
range for with *H* between 0 and 1 spans
exactly the regime from to
for which the *E*-dimensional *two point
correlation function* follows the behavior
(see Appendix B). Outside the *fBm* -range, it turns, for higher
values of , over into a behavior
, independent of the value of
. For smaller values of ,
the *two point correlation function* does not have a simple power
law behavior for small .
The drift behavior of an *fBm* structure,
, implies a fractal dimension of the
*E*-dimensional "mountain surface" of
, e.g. measured as a box coverage dimension (see
Voss 1988): at a linear scale a single
-dimensional box of volume
is needed to enclose the "mountain surface". At
a times smaller scale
each enclosing box has a volume of , and
such boxes are needed to enclose the "mountain
surface" over the original extent. These boxes thus cover a total
volume of . In units of scaled down original
boxes, each having a volume of , one thus
requires of such boxes to enclose the "mountain
range". According to the definition of the box-coverage fractal
dimension , , so that
comparison with the above results shows that
.
Inserting the above relation between and
*H*, namely , we get
as the relation between the fractal dimension of the *fractional
Brownian motion* structure and the power law index of its *power
spectrum* . As expected, thus varies between
and *E* for varying
between *E* and . Outside this range,
settles at these extreme values according to
the settling of the drift behavior at and
for too small and too large power law indices
.
As was discussed in the previous section, the
-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
-dimensional image, as is obvious from the fact
that the phases in the -plane of the
*E*-dimensional Fourier image are randomly distributed. The fact
that 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 stays constant on projection
implies that the drift index changes: from
we conclude that . The
fractal dimension as defined above then changes from
to , 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 (, ), on each
projection. Even if it starts out in *E* dimensions close to
being very broken up, with *H* close to 0, i.e.
, the first projection, that is the
-dimensional structure has already
; and after the second projection the resulting
-dimensional structure formally has already
. Its power law index thus falls outside the
range where ; the drift behavior rather settles
at , i.e. . 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 . For these
1-dimensional structures the fractal dimension *d* varies between
in the Euclidean limit ()
up to the maximum value of , 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 , 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 and correspondingly a box
coverage fractal dimension , its
-dimensional iso-density surfaces, being
effectively "zero-cuts" of the *E*-dimensional structure, have an
area-perimeter (resp. volume-surface) fractal dimension
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
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 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 . Its box coverage dimensions
as defined above is then , the (volume-surface)
dimension of its 2-dimensional iso-density surfaces is correspondingly
by 1 lower, i.e. . Due to the constancy of
on projection, the corresponding 2-dimensional
column density image has, as derived above, ;
its box coverage dimension is , 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, , 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
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998
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