PHYSICAL REVIEW A VOLUME 41, NUMBER 4 15 FEBRUARY 1990

Finite-size efFects on the characterization of fractai sets:f(a) construction via box counting on a finite two-scaieti Cantor set

Jan HkkanssonNordisk Institut for Teoretisk Atomfysik, Blegdamsuej 17, DK 210-0 Copenhagen 6i, Denmark

and Institute of Theoretical Physics, Chalmers University of Technology, S 412-96 Goteborg, Su&eden

Gunnar RussbergNordisk Institut for Teoretisk Atomfysik, Blegdamsuej 17, DK 2100-Copenhagen Q, Denmark

(Received 11 September 1989)

We study box counting on finite fractal sets and investigate how to obtain the generalized dimen-

sions and the spectrum of scaling indices with highest possible accuracy. As a model we use a sim-

ple one-dimensional Cantor set for which the f (a) spectrum may be found analytically —the exactresult is compared with the box-counting solution on the finite levels. There is a connection be-tween the q value and the size of the boxes giving the most accurate result for the f(a) spectrum on

any finite level.

I. INTRODUCTION

During the last few years it has become clear that mostfractals in nature are so-called multifractals. This meansthat the characteristic scaling properties of an object mayvary from point to point. For this reason the Hausdor+di'mension, which has extensively been used as a quantita-tive measure of the scaling in simple fractal objects, is notsufficient to characterize a multifractal. The Hausdorffdimension is only one in a continuum of dimensions, theRenyi dimensions D, ' introduced by Grassberger andHentschel and Procaccia in order to characterize frac-tals and strange attractors. In this formalism, Do is iden-tical to the Hausdorff dimension, while D& and D2 areknown as the information dimension and the correlationdimension, respectively. A similar characterization is ob-tained by the spectrum of scaling indices, the f (a) spec-trum, defined by Halsey et al. The generalized dimen-sions D and the f (a) spectrum are related to each othervia a Legendre transformation.

Real fractal objects, like aggregates, show fractal prop-erties only within a limited length scale interval. Suchfinite fractals may consist of a finite number of objectswith finite sizes. The structure becomes nonfractal insidethe single particles as well as outside some typical corre-lation length. The practical determination of the scalingproperties of an experimentally obtained fractal objectwill generally involve box counting on a two-dimensionalimage (e.g., a projection, like a transmission electron mi-crograph of aggregated Co particles ). When partitioningsuch an image, the effect of finite particle size will play animportant role for the determination of the f (a) spec-trum, especially for negative values of q (see below); thecalculation of f(a) for negative q has proven to be adifficult problem and a straightforward box-counting cal-culation will yield completely irrelevant spectra. Thisis due to the finite resolution and to the fact that theboxes will not be necessarily centered on particles of the

fractal; some of them will contain vanishingly small mea-sure giving unreasonably large contributions as q is takento minus infinity (the measure is being raised to the powerof q, see below).

In general, it is not possible to identify single particlesin an observed image. Instead one may define a smallest"particle" size, or resolution, being an estimate of the sizeof the smallest observable single particle, and then createa new image built up of these "particles. " This proceduregives a smallest relevant grid size for the box counting;on the particle scale, each box will completely cover oneparticle and thus have a uniform measure. If the image isdigitized there is a natural finest resolution given by thedigitization grid.

A number of digitizations with different grid sizes maybe constructed from an original image in order to createimages with varying degrees of resolution. Alternatively,new digitizations may be obtained from the original oneby some appropriate construction rule. This will be dis-cussed for one- and two-dimensional images in a laterwork. The basic idea is that by computing a set of (op-timal) f (a) approximations for different degrees of reso-lution, an extrapolation beyond the approximation ob-tained for the finest possible resolution will yield a goodestimate of the f (a) spectrum corresponding to infinitelyfine resolution.

In this work we will study the effect of finite particlesize on box-counting construction of the f (a) spectrum.We use a simple two-scaled Cantor set to model a finiteone-dimensional fractal with varying degrees of resolu-tion, and we show how to obtain on each level of resolu-tion the optimal f (a) approximation for both positiveand negative q. The approximations are compared withthe exact f (a) spectrum for the (infinite) Cantor set (cal-culated below). The complication of noncentered boxes isautomatically eliminated by the choice of set construc-tion; how to best avoid partially covered boxes for a gen-eral experimental fractal image will be a subject of thecontinued work.

41 1855 1990 The American Physical Society

1856 JAN HAKANSSON AND GUNNAR RUSSBERG

II. GENERAL FORMALISM

r(q)=(q —1)D, . (2)

Now, assume the following scaling relation for the proba-bility of the ith piece in the limit I~0:

a,i (3)

This relation defines the scaling index a;. The sane scal-ing relation may be found in many points, and all pointson the multifractal having the same scaling index are saidto be a subfractal with a pointwise dimension a;. Thissubfractal has the dimension f (a; ). In other words, thefunction f(a) can be interpreted as the Hausdorff dimen-sion of the set of points with the same pointwise dimen-sion a. For a simple fractal, e.g., a self-similar object likea one-scaled Cantor set (not multifractal) f (a)=DO fora=DO, and zero elsewhere, whereas for a multifractal aassumes values over an interval and f (a) is a continuousfunction on this interval. If we now divide the systeminto pieces of size 1 and express the partition sum (1) asan integral over a, we get

I"(r,q)=l I da p(a )i~'-~"', (4)

where da'p(a'}l ~( ' is the number of times a' assumesa value in the interval [a', a'+da']. In the limit 1—+0,the dominant contribution to the integral is receivedwhen the exponent qa' —f (a') is close to its minimumvalue, so we perform a saddle-point approximation

d, [qa' f (a')]. .(q)

—0. ——da (5)

This 1eads to the following I.egendre transformation,which is used to determine f (a):

dv =a,

r(q) =aq f, —

The starting point in the analysis of fractal objects,such as aggregates, strange attractors, and other complexsets, is the construction of a partition function I . Dividethe set into N pieces with a size l,- and a probabilityweight p, of the ith piece. The partition sum is thengiven by4

N pqPr, q)= g

1 1,'As /:—max(l;)~0, three things may happen: (i) Ifr & r(q), the partition sum diverges; (ii) if r ( r(q), thepartition sum becomes zero; only when (iii) r=r(q), thepartition sum may approach a hte, nonzero value.Thus, by requiring I (r, q)=const, we define the relationbetween r and q. The new dimensions r(q) are simply re-lated to the generalized dimensions Dq through

III. EXACT RESULTS FOR T%0-SCALED RECURSIVESETS

If a measure is constructed from an exact recursiverule, one can easily determine its r(q}, Dq, and f (a) spec-trum. Suppose that the measure is generated by the fol-lowing process. Start with the original region with mea-sure one and size one. Divide the region into pieces oftwo sizes and probabilities, where N of them are nonemp-ty. I.et n, denote the number of pieces of length /, andnz the number of length lz. Further, let the respectiveprobabilities be p, and p2. At this first level the partitionsum is given by

P1 PZI 1=n1 - ' +n2—I' I'1 2

(10)

where n1+n2 =N. To get the next level of the set, eachpiece is further divided into N nonempty pieces. At thissecond level the partition sum will be

n2 1 +2n n1 2 +n2 2

2 1 lz 1 2 lz lz 2 lr1 1 2 2

We now see that the first partition function will generateall the others and I „=I"",. For this reason I, is called agenerator for the set. If the probabilities are normalized(n ip i +n zpz

= 1), then r(q) is defined throughI „(q,r)=1. This gives us the following equation (whichcan be solved numerically) for determining r(q):

~q ~qn +n =1.i ir 2 ir

1 2

If we for simplicity choose the lengths I, and lz such thatl, =L and l2=l =I and the probabilities to be propor-tional to the corresponding box size, i.e., p, =L "/(n, Ld+nzL ) and pz=L /(n, L +nzL "), where d isthe geometrical (Euclidean) dimension of the set, we canrewrite Eq. (12) as a second-order equation in L q

The solution to this equation gives

(n, +4nzy~)' n, —r(q) =dq — ln

lnL 2n2(13)

where

From Eqs. (8) and (9) we note that the f (a) spectrum is aconvex function with a slope q in each dense point. Asq ~ ao the largest p, (i.e., the most concentrated part ofthe multifractal) dominates the partition sum. This cor-responds to a point where the f (a) curve vanishes withinfinite slope, and where a has its minimum value. Asq~ —Do the smallest p; dominates and at the corre-sponding maximum a value the f(a) curve vanishes withnegative infinite slope. We also note that the maximumvalue of f (a) is equal to the Hausdorff dimension sincef (a) has its maximum for q =0 and Do = r(0—)

(8) y=nL+nL" (14)d2

, (0.da

(9) From the relation (2) the whole spectrum of generalizeddimensions is known as well. In particular, the values of

41 FINITE-SIZE EI'I ACTS ON THE CHARACTERIZATION OF. . . 1857

D „—:u,„and D „=a;„aregiven by

lny lnp2

2 lnI. 2 lnL

and

lny lnp r

lnL lnL

Equations (6) and (7) give us the f(a) spectrum

a(q)=d- ~ln 2' 2

n2+4n yv n (nz+4n yI)f/z

(n f +4nzy'r)' n,—f(a(q)) = ln 2' 2

1 2qnzy lny&rz

'inL n f+4nzy~ n, (—n f+4n~y~)'

For the Cantor set in Fig. 1 we have

[1+4(-,' P]'"—1

D = q+ lnq

—1 ln2 2

and

(15)

(16)

(17)

(19)

0 8

0. 7

0. 6

0. 6

0. 4

0. P.

0. 0

-50. 0

0. 6

0. 0

0. 7

50. 0

0. 8

ln —,'

D „=,=0.7925. . . ,ln —' (20)

FIG. 2. D~ and the f (a) spectrum for the two-scaled Cantorset in Fig. 1. The zeros off are a,„=0.5850 and a,„=Q.7925,see text.

ln —',D„=,=0.5850. . . .ln —'

2

(21)

In Fig. 2 we show Dq and the f (a) spectrum for thisCantor set. The Hausdorff dimension is

ln[( &5+ 1)/2]0 ln2

(22)

The value (&5+1)/2 is the golden mean.Another soluble example with I, =1& is the two-

dimensional multifractal in Fig. 3. This set is constructed

by the following rule. Start with a square and divide itinto 16 pieces. Remove all pieces except the four in themiddle, which now form a large square, and the foursquares in the corners. Then continue the procedure anddivide each of the five squares into 6ve new ones, and soon. For this object y =

—,', d =2, n, =1, and nz =4. D is

L2 P2

L, P

Ll, Pp

1, p

1L, pP 1,p

n=1

n=2

L P L1Pn=3

M 0 MO QI n=4

FIG. 1. A Cantor set construction with two rescalingsI, =L =

zand 1~=1=

4 and respective probability rescalings

p& =P= —, and p~=p =—,'. The division of the set continues

self-similarly.FIG. 3. Example of a two-dimensional fractal with two re-

scalings. The fractal object is shown on level 4.

1858 JAN HAKANSSON AND GUNNAR RUSSBERG 41

then given by

1 1 (1+16X2 ~)'~~ —1

q—1 ln2

2q + ln8

and the f (a) spectrum by

8X2 ~a(q) =2-

1+16X2~—(1+16X2 ~)'

(~(q)) = Sq X21+16X 2-~ —(1+16X 2-~)'"

r

(23)

(24)

1.3

1.2

1.0-40. 0 -20. 0 0. 0 20. 0 40. 0

1ln

ln2

1+16X2 —(1+16X2 ~)'

8

(25) 1.0

The function r(q) is shown in Fig. 4, while D and the

f (a) spectrum are shown in Fig. 5. We can use Eq. (10)in order to understand the shape of r(q). Suppose p~ (pz(l, (lz) and let q~ —~. Then the first term will dom-inate, i.e.,

0. 5

and

l)~n)p~ (26} 0. 01.0 1.2

inn, +q lnp,r(q) =qD „—y „lnl )

(27)F&G. 5. Dq and f(a) for the two-dimensional fractal in

Fig. 3.

by definition. Here f „=f(a(—~)}. For q~oo thesecond term in Eq. (10) dominates and

lnnp+q lnppr(q)~ =qD„f„. —

lnl ~

(28)

We now see that the function r(q) has two asymptoteswith slopes D„=a;„and D =a,„,which cross the ~axis at f„and f „, respectively. We also know thatr( 1 }=0 and r(0) = —Do = f,„. Explicitly, —we get forthe two limits of a

and

ln —,'

D„=,=1,ln —,

' (30)

(31)

and the Hausdorff dimension for this fractal is given by

in[( &17—1 ) /8]1n2

ln —,'

Dln —' 2

20. 0

10. 0

Furthermore, one gets the following limits of f:29

ln4

ln —,'

and

ln 1

,=0.

1n—,'

(32)

(33)

0. 0

-10~ 0

-20. 0

-30. 0

-20. 0 -10.0 0. 0q

10.0 20. 0

FIG. 4. ~(q) for tke two-dimensional fractal in Fig. 3.

IV. BOX COUNTING

Box counting is maybe the simplest and the most corn-mon method to calculate D and the f(a) spectrum for ageneral multifractal. %hen using box counting we dividethe d-dimensional objects into boxes, all with the samesize l, i.e., l; = l for all i.

Let N be the total number of nonempty boxes, M thetotal number of particles (the total mass) and N; the totalnumber of measures (particles) in the ith box. Thenp;=N;/M. Further suppose that the probabilities are

41 FINITE-SIZE EFFECTS ON THE CHARACTERIZATION OF. . . 1859

normalized to give 1(r,q)=1. We then obtain the fol-

lowing partition sum:

I'(r, q) = Ii=1

(34}

~ ~

(35)

and from Eqs. (6) and (35) we get

By taking the logarithm of the partition sum we find therelationship between v. and q,

r(q)= ln ginl, .) M

Il=3

FIG. 6. A fractal with finite resolution given by level 3 of theCantor set in Fig. 1, together with the relevant box partitions.

a(q) = d7

dq

N;ln (36)

Having ~ and a as functions of q, we may now calculateD and f (a) from

and

f(a(q) )=qa(q) —r(q)

[qa(q) —f(a(q))] .r(q) 1

q—1 q

—1

(37)

(38}

If the N; are allowed to take fractional values, we see thatthere will be problems for large negative q; the sum abovemay be totally dominated by any small N; and will get avalue which strongly depends on the position of the grid.

V. EXACT BOX COUNTING ON A FINITE SET

We will now discuss box counting on different levels ofthe Cantor set in Fig. 1; the levels n E[0,n„]CN of theCantor set will be used to model images with resolutions4 " for a fractal object whose j7nest resolution (e.g. , parti-

)icle size in an aggregate) is 4 '. On a certain level ofresolution, the resolution is given by the size of the small-est observable (coverable) object and defines the size ofthe particles out of which the remaining parts of the ob-ject are built. Only fully covered particles should becounted; partially covered particles must be excluded inorder to avoid terms that blow up for large negative q.

For the simple Cantor set being considered, it is easy tofind explicit functions r(q), D, and an f (a) spectrum onany finite resolution level n of the s-t and with any boxlength 1=2 '"+ ', where m E[—n, n] labels the size ofthe boxes on each resolution level. In Fig. 6, a fractal im-

age with a resolution 4 is modeled by level 3 of theCantor set. The relevant set of (nonempty) boxes neededto cover this object is also shown in the figure. For agiven resolution level n, there are 2n different box parti-tions with box lengths 2 '"+ ', m = —n, —n + 1, . . . , n.On each box level m one gets a spectrum of n —~m ~+ 1

different probability measures. Up to box level m =0,there are no finite size effects. If 0&m n, some boxescover parts of the large sized objects consisting of manyparticles; these boxes thus have a uniform and equal mea-sure. If m =n, all boxes contain the same measure and afurther subdivision of the boxes will yield no further in-

f'Lk PLk ) +pL—k

f'Lk Lk-f'Lo =2P 'pLo

k=1, . . . , n . (39)

Here P =—', and p =—,' build up the probability measures.

We may now generate the probability distribution on anybox level m E- [ n, n] t—hrough

gn +mLn (40)

0 8 I I W

0. 6

f (a) 0. 4

0. 2

0 P I a ~

0. 6 0. 7 0. 8

FIG. 7. Exact f (a) spectrum (thick line) compared with thesolutions from the box counting with different box sizes on reso-lution level 16 (thin lines).

formation.The set of boxes can be constructed by a simple pro-

cedure. We have two types of boxes, L, covering parts ofthe image with all of its length, and L', covering a holewith its left half and parts of the image with its right half.On the second level (m = n+—1) the set of (nonempty)boxes can thus be represented by (L,L'). When we con-struct the next level we note that the box L is divided intotwo new pieces L and L' (m & 0), and L' turns into L; therepresentation is (L,L', L). When m &0 we have tomodify the rule since some of the boxes L are divided intotwo equal parts L.

In order to construct the full box set and calculate theprobability measure in the boxes we use the operator f'and the generators Lk and Lk having the following prop-erties:

1860 JAN HAKANSSON AND GUNNAR RUSSBERG 41

by identifying the coefficient and probability measureconnected to each Lk and L„' (L„represents the initial in-terval of length one). As an illustration we write the dis-tribution given by F L3 for the box level m =1 in Fig. 6

0. 8

0. 6

I L3=5P pLo+2Pp L', +p L, . (41) f(~) o. 4

If we put the probability distribution generated by Eq.(40) into the partition function Eq. (I), we get 0. 2

n+mI (m) 2(n + m)T ~ (m)( (m) )q

n

i =)m~+m

where

(42) 0. 00. 6 0. 7 0. 8

(m) pn+m —i [(i+1)/2]I (43)

n(m) n(m —1) +& (n™—I)+2$ n(m —1))i i —

1 I I i —2m i —2

n' "'—:1, n '=0, i 6[0,n+m] .

The symbols e; and 5; in Eq. (44) are defined throughT

1, for i even

0, for i odd

1, fori =00, otherwise .

(44)

(45)

(46)

With the partition function known on any box level onany level of resolution of the Cantor set, we can now con-struct the scaling functions

r(m)(q)—

(D(m))n q

1lnX'

(n+m) ln2

(m)(q)

1

(47)

(4&)

y(m)a'„'(q}=-

(n +m) ln2 g'm' (49)

The square brackets here denote the integer part. ThecoeScients n ' are given for m = —n, . . . , n;i =0, . . . , n +m by the recursion relation

FIG. 8. Exact f (a) spectrum compared with successive, op-timal approximations on different levels of resolution. The thincurves are fs, f,6, and f32.

fore depends on the value of q. For positive q values, f ', 6'

is the most accurate approximation for all q, and thecurves fI6', m (0 will all be on the right-hand side off(0)

The main conclusion of the above results is that if wewant to approximate f (a) for the Cantor set with boxcounting at a finite level n, we should (i) let the envelopeof the curves f„' I(a), m =0, 1, . . . , n —1 approximate

f (a) for negative q, i.e., we select the left-most curve fornegative q values by calculating the crossing points of thecurves, and (ii) let f„' '(a) approximate f (a) for all posi-tive q. The optimal f (a) spectrum thus obtained, f„'(o.),is shown in Fig. 8 for n =8, 16, and 32 compared withthe exact curve. We have that lim„„f„"=f.

To illustrate the convergence of the Hausdorff dimen-sion we have in Fig. 9 plotted Do as a function of the boxsize on level 32. Note the fast convergence for smallboxes. The minimum of the curve is the best approxima-tion of the Hausdorff dimension. Also note that the exactvalue of D „can be found by calculating Do with a boxsize I of the same size as the smallest interval in the set.

f„' '(a'„'(q))=q~'„'(q) —q'„'(q), (50)

where

y(m)a

y(m)A

n+m(m)(p(m))q

i =~m~+m

n+mI )( ( )) 1 ( I ))

i =/m/+m

(51)

0. 804

0. 75 - ~

In Fig. 7 we show the exact f (a) curve and succes-sive approximations f„' '(a'„'), for n = 16 and

m =0, 1,2, . . . , 15 [f', 6' just gives a single point at

(D,D „)]. Note that the curves from the box-counting calculation cross each other. %'e observe thatfor large negative q, fI6

' is the best approximation.When decreasing the magnitude of ~q~, the f'&'6 ' curvecrosses f', 6

' and becomes the best approximation. If wecontinue to decrease ~q ~, f",6 ' crosses the f",6 ' curve andso on until f 36' crosses fI6'. The optimal box size there-

0. 70

1~ ~~4

0. 0 0. 1 0. 2 0. 3

FIG. 9. Successive approximations of the Hausdorff dimen-sion Do plotted vs the box size I on level 32. The dashed linerepresents the exact value.

41 FINITE-SIZE EFFECTS ON THE CHARACTERIZATION OF. . . 1861

VI. CONCLUSIONS

We have used a Cantor set at finite levels to model afinite fractal. The simple choice of length scales of theCantor set made it possible to find the generalized dimen-sions and the spectrum of scaling indices analytically.For this particular choice we have also been able to solvethe complete box-counting problem. By comparing thebox-counting solution with the exact one we find a rela-

tionship between the value of q and the size of the boxesgiving the most accurate result. For positive q, the bestresult is obtained for a grid size equal to the size of thelargest connected object, i.e., the length scale wherefinite-size effects begin to play a role. For negative q, theinfluence of finite sizes is crucial for the determination ofthe f (a) spectrum, such that all the way down to theparticle level of resolution, each partition level (box size)gives a contribution in a corresponding q interval.

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