Method for computing the three-dimensional capacity dimension from two-dimensional projections
of fractal aggregates
F. Maggi1,*and J. C. Winterwerp1,2,†
1Environmental Fluid Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, P.O. Box 5048, 2600 GA, Delft, The Netherlands
2WL兩Delft Hydraulics, P.O. Box 177, 2600 MH, Delft, The Netherlands
共Received 20 August 2003; published 30 January 2004兲
The current theory of projections of fractals is considered in this paper with application to fractal aggregates. In particular, this theory does not accurately enable the computation of the capacity dimension of three-dimensional aggregates from the capacity dimension of their two-three-dimensional projections. Herein we propose to compute the three-dimensional capacity dimension from the perimeter-based fractal dimension, using a semiempirical equation, an approach not applied earlier.
DOI: 10.1103/PhysRevE.69.011405 PACS number共s兲: 61.43.Hv
I. INTRODUCTION
Fractal geometry is widely recognized to be fundamental in scaling a variety of properties of aggregates of various nature. The fractal approach has been fruitfully employed, for instance, in sedimentology 共mud flocs, bed structure 关1–3兴兲, chemistry 共polymers, colloidal aggregates 关4,5兴兲, medicine共cancer growth, cell structure 关6兴兲, cosmology 共gal-axy distributions and patterns at large scales关7,8兴兲, and many other disciplines dealing with fractal aggregates.
In general, analysis of fractal aggregates inR3is based on optical projections in R2. However, the transformation of projectionP:R3→R2 distorts the three-dimensional共3D兲 in-formation of an aggregate, especially concerning its geomet-ric organization. One of the most convenient tools to de-scribe the geometric structure of a fractal set is the generalized dimensionality dq, proposed by Hentschel and
Procaccia in关9兴. It is defined as follows: if we consider an⑀ covering 傺Rn by means of boxes of size ⑀ of a fractal of length scale L, then N⫽(L/⑀)n⫽ᐉn is the total number of
boxes in the domain. If Niis the number of measuring points
in the ith box and Nfthe total number of points of the fractal, then pi⫽Ni/Nf determines the probability of a measuring
point lying in the ith box. Consequently, the generalized di-mensionality of the qth order is written as follows:
dq⫽ 1 1⫺q⑀→0lim ln
兺
i⫽1 N 共pi兲q ln⑀ , 共1兲where q is a moment that gives strength to the probability pi. The capacity dimension dC 关10兴, the information
dimen-sions dI 关11–13兴 and the correlation dimension dK 关14兴 are
special cases of dq: d0⫽dC, d1⫽dI, and d2⫽dK, with
d2⭐d1⭐d0⭐n. A basic property of the generalized dimen-sionality is that dq⫽const ᭙q for fully self-similar and ho-mogeneous fractals 共monofractal sets兲, while an infinite
number of dimensions 共all represented by dq) is required to
describe statistical self-similar, nonhomogeneous fractals 共multifractal sets兲.
The dimensionality dq has been further elaborated for
ap-plication purposes into the corresponding multifractal spec-trum f (␣) and singularity strength ␣(q) 关9,15–17兴. These quantities describe any arbitrary mass-density distribution of a nonhomogeneous fractal 关17,18兴 and its growth rate 关14,19–22兴. The relevance of evaluating the fractal proper-ties of aggregates stems from those fundamental works. In substance, the possibility of determining the fractal dimen-sions of an aggregate from its projection would make the characterization of the aggregate more complete. In particu-lar, the capacity dimension d0should be accessed as it plays a role in the nonlinear relationship between the mass M of an aggregate and its length scale L: M⬃Ld0, and nevertheless in a number of other quantities such as the effective density, porosity, etc.关3,5,23兴.
However, direct computation of dq from projections of
real aggregates is limited by geometric constraints. This was shown by early investigations which were addressed to un-derstand mathematically how projectionsP affect dq 关8,24兴. In particular, Hunt and Kaloshin关24兴 have elaborated that P preserves the 3D information only for a limited range of moments q (1⬍q⭐2), thus leaving unsolved questions re-lated to d0, which have direct implication to applied sciences and measurement techniques.
The main contribution of this paper is to show that it is possible to extract the 3D capacity dimension of fractal ag-gregates from their projections by following an alternative path. The paper is organized as follows. Section II summa-rizes a literature-based survey on the theoretical limits to which dq is subject in the case of projections of
nonhomo-geneous, extensive fractals. Furthermore, we give numerical evidence that the application of the theory to nonhomoge-neous finite fractals yields distorted results, especially for the capacity dimension d0. We infer that the finiteness of the sets causes these distortions共real fractal aggregates in contrast to extensive fractals兲. Section III is dedicated to the analysis of those results. In addition, we propose an analytical formula-tion capable of circumventing the limits exposed in Sec. II. This formulation is founded upon geometry arguments and fitting numerical results.
*Email address: f.maggi@ct.tudelft.nl †Email address: han.winterwerp@wldelft.nl
II. PROJECTIONS AND GENERALIZED DIMENSIONALITY
A. Problem definition
Hunt and Kaloshin关24兴 have observed that the projection
P:Rm→Rn of a fractal S
m傺Rm of generalized
dimensional-ity dq(Sm) ontoRn 共with n⬍m), yields to
dq„Sn⫽P共Sm兲…⫽min兵n,dq共Sm兲其. 共2兲
If dq(Sm)⬎n then the projection Sn has dimensionality
dq(Sn)⫽n, otherwise dq(Sn)⫽dq(Sm). When the projection
has the same dimensionality as the original (dq(Sm)⭐n),
thenP is called a dimension-preserving transformation. This relation has been proven analytically in 关24兴 only for values of the moment q:1⬍q⭐2. P is not dimension-preserving for q⭐1 and q⬎2. In other words, only the correlation di-mension d2 and the infinite number of dimensions between d2and the information dimension d1are preserved. All other dimensions are not preserved, the capacity dimension d0 in-cluded. This implies that for projections of real objects (m⫽3 and n⫽2), the capacity dimension d0(S3) (q⫽0) cannot be found from Eq. 共2兲:
d0共S2兲⫽min兵2, d0共S3兲其. 共3兲 It follows that we cannot use the 2D capacity dimension d0(S2) of a projection to characterize the 3D capacity dimen-sion d0(S3) of the original object in a direct way, at least theoretically, even when d0(S3)⬍2.
Moreover, Eq. 共2兲 is deduced from the literature to be valid for indefinitely extensive fractals. However, no precise distinction has been made in literature for finite fractals, such as aggregates. For this reason, we must first consider whether Eq. 共2兲 is applicable to 2D projections of fractal aggregates, because these are nonhomogeneous, finite, and closed 共com-pact兲 sets. Indeed, images of fractal aggregates consist of the complete closed sets, that is the sets themselves and their boundaries. Furthermore, such sets are self-similar only over limited ranges of length scales. This is in contrast to self-similar 共mono兲 fractal sets. Monofractals are, at least theo-retically, open and homogeneous sets because any observa-tion window replicates any other window, at any scale, according to the concept of self-similarity. The consequence of dealing with non-homogeneous finite and closed objects is that the application of Eq.共2兲 becomes less clear and liable to divergencies and misunderstandings关22兴.
We therefore compare in the next section theoretical val-ues of d0(S2) 关via Eq. 共2兲兴 and numerical values 共via com-puter simulations兲 of nonhomogeneous random fractals. From this test we will learn that Eq.共2兲 is not able to return accurate results, thus preventing the computation of d0(S3) from d0(S2). Next, we propose an alternative semi-empirical equation to compute d0(S3) accurately.
B. Application of Eq.„2… to artificial fractal aggregates Fractal aggregates are nonhomogeneous, finite, and closed random sets of connected seeds distributed within the do-main. Furthermore, they are statistically self-similar with
multifractal properties. Herein, we test the applicability of Eq. 共2兲 on artificial aggregates.
We first generate nr fractal aggregates S3傺R3 by means of a simple algorithm which produces self-correlated random structures with known capacity dimension d0. This technique is a static aggregation algorithm. Seed-by-seed diffusion or cluster-cluster reactions are not accounted for. Starting with a single共cubical兲 seed i⫽1, a second seed i⫽2 is placed ran-domly in one of the 3-by-3-by-3 free, neighbor locations. Then, one of the existing seeds is chosen from an exponen-tial distribution, and a new seed is attached to it. In this algorithm, recent seeds 共large indexes i) have higher prob-abilities to receive a new seed. This procedure is repeated for 1000 seeds. The capacity dimension d0(S3) of the aggregate under construction is tuned by means of the exponent of the exponential distribution. The resulting sets S3are aggregates with few open branches, more similar to CCA aggregates than DLA aggregates关10,21兴. The nr sets S3 are afterwards projected along the three Cartesian directions, thus obtaining the projections
兵S2,x( j)其,兵S( j)2,y其,兵S2,z( j)其, 共4兲 with j⫽兵1, . . . ,nr其 the repetition index. Three examples of
S3( j) and their projections are given in Fig. 1. For each of the 3nr projections, we compute the capacity dimension d0 ac-cording to Vicsek关10兴:
d0共S2,兵x,y ,z}
( j)
,X兲⫽log关N兴
log关X兴, 共5兲
where N is the number of seeds in the projection and X ⫽兵L2,L3,D2,D3其are the length scales taken into account. L is the size of the minimum hypercube enveloping S and D is the hydraulic diameter, in R2 and R3, respectively. The length scales兵L3,D3其傺R3 are known from the construction of the aggregates while the length scale兵L2,D2其傺R2 result from the transformation P. Next, we compute the average capacity dimensions d0(S2
( j)) of the projections for each jth
aggregate as follows: d0共S2 ( j) ,X兲⫽1 3关d0共S2,x ( j) ,X兲⫹d0共S2,y ( j) ,X兲⫹d0共S2,z ( j) ,X兲兴. 共6兲 depending on the used length scales. The reason to consider different length scales comes from a misuse of them in the application to real cases.
Now, we consider the capacity dimension d0(S3,兵L3,D3其), computed as a function of L3 and D3 solely.
The relationship between d0(S2 ( j)
,X) and d0(S3 ( j)
,L3) is given in Fig. 2, while the relationship between d0(S2
( j) ,X) and d0(S3
( j)
,D3) is given in Fig. 3. Both these experimental sets deviate largely from Eq.共2兲 for all the length scales here considered. In particular, Eq. 共2兲 tends to overestimate the real values both for low- and high-dimensional aggregates.
Thus we have shown that, for nonhomogeneous, finite, and closed fractal aggregates, Eq.共2兲 does not enable a direct
extraction of d0(S3) from d0(S2), even for d0(S3)⬍2. This was already stated in Eq.共2兲 and derived analytically in 关24兴 for extensive fractals.
III. DIRECT COMPUTATION OF d0„S3… FROM THE
PROJECTION S2
A. Perimeter of fractal sets
From the previous results, we have found confirmation that information concerning the capacity 共the capacity di-mension, that is the space-filling ability兲 is polluted by the projection itself, even for d0(S3)⬍2. Hence, we analyze
an-other set belonging to the projection, which is independent or nearly independent of the transformation: the contour of the projected set S2. The contour is a subset of the surface of S3. The measure of the contour, that is the perimeter, does not represent a capacity of S3. The perimeter, or better the pe-rimeter segmentation reflects the roughness of the object in R3.
Herein, we investigate to which extent the information of the structure inR3 can be found in the projected, perimeter-based fractal dimension dP, which is defined for instance in 关21兴. dP does not belong 共or give evidence of belonging兲 to
the set of dimensions in dq. As a consequence, the theory of FIG. 1. 共a兲 Example of a high-fractal-dimension aggregate, S3
(1) , d0(S3
(1)
)⫽2.49. The projections show a massive and round-shaped organization of the primary particles.共b兲 Example of a mid-fractal-dimension aggregate, S3(14), d0(S3
(14)
)⫽2.09. The projections show a less massive and irregular-shaped organisation of the primary particles. 共c兲 Example of a low-fractal-dimension aggregate, S3
(30) , d0(S3
(30) )
⫽1.81. The projections show a weak and irregular-shaped organization of the primary particles.
FIG. 2. 2D capacity dimension d0(S2 ( j)
,X) of the projections
S2
( j)共dots兲 as a function of the 3D capacity dimension d 0(S3
( j) ,L3). They have been compared to the theoretical relation, computed through Eq.共2兲 共solid line兲.
FIG. 3. 2D capacity dimension d0(S2 ( j)
,X) of the projections
S2
( j)共dots兲 as a function of the 3D capacity dimension d 0(S3
( j) ,D3). They have been compared to the theoretical relation, computed through Eq.共2兲 共solid line兲.
projection does not apply to dP, and therefore it is not
sub-ject to the rule of Eq.共2兲. However, it still gives information on the fractal structures of aggregates. For this reason, and because of a lack of theoretical work dealing with this prob-lem, we perform a simple numerical experiment on the cor-relation between d0(S3
( j),L
3) and dP(S2
( j)), thus neglecting the length scales L2, D2, and D3.
B. Perimeter-based fractal dimension of the projections The perimeter-based fractal dimension dP is defined
ac-cording to 关21兴:
dP⫽2
log关P兴
log关A兴, 共7兲
where P and A represent the perimeter and the area of a projection. Within our context, A is given by the number of seeds within the projected area and P is given by the number of seeds on the contour.
By means of simple geometry arguments, we compute the values of dP for the two extreme cases of thin line and
mas-sive box projections. Let us therefore consider an⑀covering of the set S2of length scale L2by means of boxes of size⑀, corresponding to a resolution ᐉ⫽L2/⑀. The values of dP then depend on the resolution ᐉ, as elaborated in the two following cases.
Thin line. Let us consider the case of a projection which becomes a thin line for increasing resolution ᐉ⑀关1,⬁), Fig. 4共a兲. In that case
P⬅A⫽ᐉ, ᐉ⑀关1,⬁兲,
and, using Eq.共7兲, the resulting dP becomes
dP⫽2
log关P兴 log关A兴⫽2
log关ᐉ兴
log关ᐉ兴⫽2, ᐉ⑀关1,⬁兲. 共8兲
Massive box. Let us now consider a projection consisting of a massive box for resolutions ᐉ⑀关1,⬁), Fig. 4共b兲. The generalized forms expressing the perimeter P and the area A are
P⬅A⫽ᐉ, ᐉ⫽1 P⫽4ᐉ⫺4, A⫽ᐉ2, ᐉ⑀关2,⬁兲, from which we write Eq. 共7兲 as a function of ᐉ
dP共ᐉ⫽1兲⫽2 log关ᐉ兴 log关ᐉ兴⫽2, ᐉ⫽1 dP共ᐉ⫽2兲⫽2log关4ᐉ⫺4兴 log关ᐉ2兴 ⫽2, ᐉ⫽2 共9兲 dP共ᐉ⭓3兲⫽2 log关4ᐉ⫺4兴 log关ᐉ2兴 ⬍2, ᐉ⑀关3,⬁兲,
where the cases ᐉ⫽1 (⑀⫽L) and ᐉ⫽2 (⑀⫽L/2) represent two trivial solutions for dP that can be referred to as a
patho-logical effect caused by the low resolution. It is possible to see from Fig. 4共b兲 that P⫽A for resolutions ᐉ⭓3 (⑀ ⭐L/3). Hence, dP decreases for increasing resolution. For
ᐉ→⬁ (⑀→0) we obtain the lower limit
lim ᐉ→⬁ dP⫽ lim ᐉ→⬁ log关4ᐉ⫺4兴 log关ᐉ兴 ⫽ limᐉ→⬁
冉
log关4兴 log关ᐉ兴⫹ log关ᐉ⫺1兴 log关ᐉ兴冊
⫽1, 共10兲 which represents an asymptotic case for infinitely high reso-lutions of fully massive aggregates.The limiting values of dP are then represented by dP⫽2
for linelike projections and dP⫽1 for massive projections
and infinitely high resolution.
C. Correlation analysis of d0„S3… and dP„S2…
In order to investigate how dP(S2) relates to d0(S3), we first normalize the projections of Eq. 共4兲 with a reference resolution ᐉr. This is performed by using a magnification factor fmdefined as
fm⫽ᐉr
ᐉ , 共11兲
in such a way that
L2m⫽ fmL2⫽ᐉr⑀᭙S2 ( j)
. 共12兲
We compute the average perimeter-based fractal dimen-sion dP(S2( j)) for the jth set S2( j) as follows:
dP共S2 ( j)兲⫽1 3关dP共S2,x ( j)兲⫹d P共S2,y ( j)兲⫹d P共S2,z ( j)兲兴, 共13兲 FIG. 4. Geometric representation of the limiting cases of 共a兲
linelike projection and共b兲 fully massive projection as functions of the resolutionᐉ. Dark gray boxes represent regions of perimeter/ area overlapping, while light gray boxes belong solely to the area.
where dP is defined in Eq.共7兲. In this computation we
con-sider the external perimeter only, therefore neglecting inner empties.
Figure 5 shows the relationship between dP(S2) and d0(S3) for various resolutions, ᐉr⫽兵16,256,1024其 pixels. Therein, we have evaluated the boundary points Z at d0(S3)⫽3 共massive box兲,
Z16⫽„3,z共ᐉr⫽16兲…,
Z256⫽„3,z共ᐉr⫽256兲…, 共14兲
Z1024⫽„3,z共ᐉr⫽1024兲…,
known by the analytical solution of Eq. 共9兲, where we have applied the notation
z共ᐉ兲⫽dP共S2,ᐉ兲⫽
log关4ᐉ⫺4兴
log关ᐉ兴 . 共15兲
There are three major features that we can observe from the results given in Fig. 5. The first is that low dimensional structures, with a high level of branching at the left-hand side of the plot, possess projections with high values of dP. In contrast, high dimensional structures, with massive and round-shaped masses at the right-hand side of the plot, have low values of dP. The second is that dP(S2) does not reach a constant value for d0(S3)⬎2, in contrast to the rule of Eq. 共2兲. Rather, a hyperboliclike correlation does appear in the full range 1⭐d0(S3)⭐3. The third is that low resolutions 共16 pixels, for instance兲 move the points towards the upper limit dP⫽2. An increase in resolution lowers the points
as-ymptotically towards the limit dP⫽1, as shown in Eq. 共10兲. These are valuable results that can be used to derive a semiempirical equation to relate d0(S3) and dP(S2) as a function of the resolution and with a hyperboliclike structure.
D. Semiempirical relation for dP„S2… and d0„S3…
By considering the fully known points Z of Eq. 共14兲 and assuming a function of the form
dP共S2兲⫽ a 关d0共S3兲兴2
⫹b, 共16兲
we correlate the results in Fig. 5 by solving the following system in correspondence of the two points Z and K:
z共ᐉ兲⫽ a
32⫹b at Z⫽„3,z共ᐉ兲…, 2⫽ a
关k共ᐉ兲兴2⫹b at K⫽„k共ᐉ兲,2…, 共17兲 with z(ᐉ) defined in Eq. 共15兲. The coordinates k(ᐉ) of the boundary points K at dP⫽2 for a given resolution ᐉ have been expressed as a function of z(ᐉ) by fitting the data points in Fig. 5 at the upper limit dP⫽2:
k共ᐉ兲⫽k„z共ᐉ兲…⫽z共ᐉ兲关z共ᐉ兲⫺1兴⫹1, 共18兲 which results in
K16⫽„k共ᐉr⫽16兲,2…,
K256⫽„k共ᐉr⫽256兲,2…, 共19兲
K1024⫽„k共ᐉr⫽1024兲,2….
Hence, the coefficients a and b are
a共ᐉ兲⫽9
冉
z共ᐉ兲⫺2关k共ᐉ兲兴 2⫺9z共ᐉ兲 关k共ᐉ兲兴2⫺9冊
, b共ᐉ兲⫽2关k共ᐉ兲兴 2⫺9z共ᐉ兲 关k共ᐉ兲兴2⫺9 . 共20兲Finally, Eq.共16兲 reads as a function of d0(S3) andᐉ, dP共S2兲⫽
再
a共ᐉ兲 关d0共S3兲兴2 ⫹b共ᐉ兲 for d0共S3兲⬎k„z共ᐉ兲…, 2 for d0共S3兲⭐k„z共ᐉ兲…. 共21兲 Figure 6 shows the numerical results 共dots兲 and the em-pirical fit 共solid curves兲 obtained from Eq. 共21兲 for resolu-tions ᐉr⫽兵16,256,1024其 pixels. The fit for ᐉr⫽16 pixels isacceptable though not perfect (R2⫽0.970). A better fit is obtained for resolutionsᐉr⫽256 pixels (R2⫽0.975) and for
ᐉ⫽1024 pixels (R2⫽0.973), see Fig. 7. The correlation co-efficients R2 appear to have a maximum for a given resolu-tion (ᐉr⫽256 in this case兲. Consequently, the reader can
argue that the optimal determination of d0(S3) occurs for a resolution ᐉ⬍⬁. However, we note that the fluctuation of the correlation coefficient is in the order of 10⫺3; therefore, FIG. 5. Variation of dP(S2 ( j) ) as a function of d0(S3 ( j) ,L3) at different resolutionsᐉr.
statistically it is not relevant to infer any systematic trend or behavior. Besides this, the appreciable alignment of the data point supports the goodness of the technique proposed here. By inversion of Eq. 共21兲, we can write the following equation:
d0共S3兲⫽
冑
a共ᐉ兲 dP共S2兲⫺b共ᐉ兲
for dP共S2兲⬍2, 共22兲 which gives the 3D capacity dimension of the aggregates from the perimeter-based fractal dimension of their projec-tions and the adopted resolution.
E. The case of infinite resolution
Forᐉ→⬁, the coordinates z(ᐉ) and k(ᐉ) of the boundary points Z and K of Fig. 6 become
z⬁⫽ lim ᐉ→⬁ z共ᐉ兲⫽ lim ᐉ→⬁ log关4ᐉ⫺4兴 log关ᐉ兴 ⫽1, k⬁⫽ lim ᐉ→⬁ k„z共ᐉ兲…⫽ lim ᐉ→⬁ z共ᐉ兲关z共ᐉ兲⫺1兴⫹1⫽1. 共23兲
and the coefficients of Eq.共20兲 are consequently a⫽9
8, b⫽ 7
8. 共24兲
Equation共22兲 in the asymptotic limit ᐉ→⬁ then becomes d0共S3兲⫽
冑
9/8 dP共S2兲⫺7/8
for dP共S2兲⬍2. 共25兲 It matches the theoretical points Z⫽(3,z⬁) and K⫽(k⬁,2) as shown in Fig. 6.
F. Critical resolution
Equation 共22兲 allows us to detect a critical resolution ᐉc
below which the computation of d0(S3) is corrupted by low resolution. Let us consider a fractal aggregate with capacity dimension d0(S3)⫽d*. If we want to be able to detect d* by means of Eq.共22兲, then the condition
k共ᐉ兲⭐d* must be satisfied. By expanding we obtain
关z共ᐉ兲兴2⫺z共ᐉ兲⫹1⫺d*⭐0. Its corresponding solution is
z1共d*兲⭐z共ᐉ兲⭐z2共d*兲, with
FIG. 6. Comparison of the numerical and analytical results from Eq.共21兲 at different resolutions ᐉr.
FIG. 7. Comparison of the parametrized values of d0(S3) according to Eq.共22兲 versus the measured ones for the tested resolutions ᐉr
z1共d*兲⫽1⫺
冑
⫺3⫹4d*2 ,
z2共d*兲⫽
1⫹
冑
⫺3⫹4d*2 . 共26兲
If we consider that real aggregates possess capacity di-mensions in the range 1⭐d*⭐3, then the discriminant ⌬ ⫽⫺3⫹4d* is limited to the range 1⭐⌬⭐9. In fact, if S3傺R3 then d*⭐3 for obvious physical limits. At the same time, if d*⬍1 then the aggregate would consist of, at least, two disjointed, thin masses. This is not a unique aggregate anymore but two or more individual aggregates, with inde-pendent fates. Therefore, for the considered range of ⌬, we obtain the ranges of validity of z1(d*) and z2(d*):
⫺1⫽z1 inf共d*⫽3兲⭐z 1共d*兲⭐z1 sup共d*⫽1兲⫽0, 1⫽z2inf共d*⫽1兲⭐z2共d*兲⭐z2 sup 共d*⫽3兲⫽2.
The quantity z(ᐉ) is z(ᐉ)⫽dP(S2) as defined in Eq.共15兲; it is proven to lie in the range关1,2兴 in Sec. III B. The result-ing solutions of z(ᐉ) are then limited to the positive range:
1⭐z共ᐉ兲⭐z2共d*兲, 共27兲 as represented in Fig. 8. In particular, since
z共ᐉ兲⭓1 ᭙ᐉ⑀关1,⬁兲, we must satisfy only the condition
z共ᐉ兲⭐z2共d*兲. 共28兲
Therefore, in order to computeᐉcwe substitute Eqs.共15兲 and共26兲 into Eq. 共28兲,
z共ᐉ兲⫽log关4ᐉ⫺4兴 log关ᐉ兴 ⭐ 1⫹
冑
⫺3⫹4d* 2 ⫽z2共d*兲, log关4ᐉ⫺4兴⭐1⫹冑
⫺3⫹4d* 2 log关ᐉ兴, 共4ᐉ⫺4兲⭐ᐉ共1⫹冑⫺3⫹4d*兲/2. 共29兲Eventually, the transcendental function in ᐉ of Eq. 共29兲 can be rewritten for simplicity in the following form:
f共ᐉ兲⭐g共ᐉ,d*兲, 共30兲
with f (ᐉ)⫽(4ᐉ⫺4) and g(ᐉ,d*)⫽ᐉ(1⫹冑⫺3⫹4d*)/2. In Fig. 9 we have represented the critical resolutions ᐉc computed
from the intersection of f (ᐉ) and g(ᐉ,d*) through Eq.共30兲, for aggregates of various capacity dimensions d*inR3. Fig-ure 9 provides a practical tool for computing the minimum resolution required to be able to extract d0(S3) from dP(S2), once the observer can estimate the minimum expected d0(S3).
IV. CONCLUSION
Current theory of the projection of fractals does not al-ways enable direct computation of the capacity dimension of fractal sets embedded in R3 from the capacity dimension of projections inR2. This occurs in particular when the fractals under investigation are aggregates, that is finite and closed objects, with nonhomogeneous mass density distributions. In general, theoretical research tends to refer mostly to exten-sive fractals. However, in practice finite fractals are more likely to occur. Fractal aggregates differ considerably from indefinitely extended fractals. We have given evidence of the impact of the finite extent of fractals by means of compari-sons of theoretical and numerical results in Figs. 2 and 3.
For these reasons, we have developed a method, circum-venting the rule of Eq.共2兲 to obtain the 3D capacity dimen-sion of aggregates from their projections. We neglect the information of ‘‘capacity’’ d0(S2) present in the projections S2, in favor of information concealed in the perimeter of the projection solely. To this end, a correlation analysis has been carried out to relate d0(S3) to dP(S2), using the perimeter-based fractal dimension dP in R2. The results show that
FIG. 8. Representation of the solution interval of the quantity
z(ᐉ). In particular, the range of validity of z(ᐉ) is shown to be
bounded in the range关1,z2(d*)兴.
FIG. 9. Representation of the functions f (ᐉ) and g(ᐉ,d*). The intersection points define the critical resolutionᐉcbelow which the
estimation of d0(S3) through Eq.共21兲 is distorted by the low reso-lution.
d0(S3) and dP(S2) are related to each other by means of a hyperboliclike resolution-dependent function, defined in Eq. 共21兲.
The expression here proposed to compute d0(S3) from dP(S2) allows us to derive analytically a critical resolution below which d0(S3) cannot be calculated accurately. This has resulted in the nomogram of Fig. 9, which can be directly employed to estimateᐉc.
The accurate extraction of the capacity dimension of frac-tal aggregates obtained with Eq. 共22兲 does not mean, how-ever, that it can be successfully applied to any type of aggre-gated structure. The concept of universality is here involved for two reasons. The first is that Eq.共22兲 considers the infor-mation of perimeter segmentation, so that the internal area of the projections can be of any type: Euclidian or non-Euclidian. For Euclidian aggregates, only the external sur-face can be considered fractal, and not the complete object,
thus there is no sense in computing a fractal dimension of a regular共Euclidian兲 structure. The second reason is that DLA and CCA processes produce different aggregate geometries 关10,21兴. At the moment we cannot state whether the perim-eter segmentation is effectively capable of incorporating in-formation on the geometrical structure in addition to the ca-pacity of the structure. Therefore, future investigation must be oriented to understand whether Eq. 共22兲 is valid for dif-ferent aggregation kinematics 共that is different structures兲, and not only for various capacity dimensions.
ACKNOWLEDGMENTS
The authors thank Professor Jurjen Battjes for his critical suggestions addressed during the development of the content exposed in this paper. This study was financed with Delft University Research funds through the BEO Program.
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