Fractals, Vol. 19, No. 2 (2011) 195–201 c World Scientific Publishing Company DOI: 10.1142/S0218348X11005336 FRACTAL DIMENSION OF COALESCENCE HIDDEN-VARIABLE FRACTAL INTERPOLATION SURFACE SRIJANANI ANURAG PRASAD ∗ and G. P. KAPOOR † Department of Mathematics and Statistics Indian Institute of Technology Kanpur Kanpur 208016, India ∗ [email protected]† [email protected]Received March 23, 2010 Accepted February 10, 2011 Abstract In the present paper, the bounds on fractal dimension of Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) in R 3 on a equispaced mesh are found. These bounds determine the conditions on the free parameters for fractal dimension of the constructed CHFIS to become close to 3. The results derived here are tested on a tsunami wave surface by computing the lower and upper bounds of the fractal dimension of its CHFIS simulation. Keywords : Interpolation; IFS; Attractor; Fractal Surface; Fractal Dimension; Tsunami Wave; H¨ older Exponent. 1. INTRODUCTION Among the major recent developments in under- standing the structures of objects found in nature, the notion of fractals occupies an important place. Since the introduction of the term Fractal by Man- delbrot, 1 an increasing number of research papers have demonstrated the fractal nature of many systems with different physical properties. Fractal dimension is widely used to quantify the roughness of natural objects and structures. It was demon- strated by Mandelbrot 2 himself that the notion of fractal dimension is quite useful in quantifying the roughness of irregular patterns such as that of tortuous lines, crumpled surfaces, intricate shapes. 195 Fractals 2011.19:195-201. Downloaded from www.worldscientific.com by INDIAN INSTITUTE OF TECHNOLOGY KANPUR P K KELAR LIBRARY on 12/23/12. For personal use only.
AbstractIn the present paper, the bounds on fractal dimension of Coalescence Hidden-variable FractalInterpolation Surface (CHFIS) in R
3 on a equispaced mesh are found. These bounds determinethe conditions on the free parameters for fractal dimension of the constructed CHFIS to becomeclose to 3. The results derived here are tested on a tsunami wave surface by computing the lowerand upper bounds of the fractal dimension of its CHFIS simulation.
Among the major recent developments in under-standing the structures of objects found in nature,the notion of fractals occupies an important place.Since the introduction of the term Fractal by Man-delbrot,1 an increasing number of research papershave demonstrated the fractal nature of many
systems with different physical properties. Fractaldimension is widely used to quantify the roughnessof natural objects and structures. It was demon-strated by Mandelbrot2 himself that the notionof fractal dimension is quite useful in quantifyingthe roughness of irregular patterns such as that oftortuous lines, crumpled surfaces, intricate shapes.
Breslin and Belward3 used fractal dimension tomodel rainfall time series and discussed suitabilityof fractal analysis for these type of data. Arakawaand Krotkov4 discussed natural terrain modelingusing fractal geometry and gave methods of estima-tion of fractal dimension and fractal surface recon-struction. Lung and Zhang5 discussed the origin ofthe negative correlation between fractal dimensionand toughness of fractured surfaces of materials.Zahouani et al.6 modeled a random surface topogra-phy and showed that fractal dimension can be usedas an indicator of the real values of different scale-dependent parameters such as length, surfaces andvolume of roughness.
Barnsley and Harrington7 used shifted com-position to express affine Fractal InterpolationFunctions (FIFs) and computed their fractal dimen-sions. Bedford8 extended Barnsley’s definition ofself-affine fractal function to use non-linear scal-ings and showed that for a class of such functions,the Holder exponents are related to the box dimen-sion of the function. However, most of the naturalobjects like surfaces of rocks, sea surfaces, cloudsand many naturally occurring structures are madeup of both self-affine and non-self-affine parts. Inmost of the cases, the computation of fractal dimen-sion of these surfaces by existing methods is notpractically feasible. The present work is aimed atovercoming this inadequacy by finding the boundson Fractal Dimension of a Coalescence Hidden Vari-able Fractal Interpolation Surface (CHFIS), whichis generated from a non-diagonal IFS on a equis-paced mesh that generates both self-affine and non-self affine FIS simultaneously, depending on freevariables and constrained variable.
The organization of the paper is as follows: Abrief introduction on construction of CHFIS andits smoothness is given in Sec. 2. Our main resultson bounds of Fractal Dimension of CHFIS on equi-spaced mesh are derived in Sec. 3. Using thesebounds, certain conditions on the free parame-ters are determined that lead the fractal dimensionof the constructed CHFIS to become close to 3.Finally, in Sec. 4, to substantiate our results, thebounds on fractal dimension of CHFIS of a Tsunamiwave surface are computed.
2. CONSTRUCTION OF CHFIS
Let (x0, y0, z0,0), (x1, y0, z1,0), . . . , (x0, y1, z0,1),. . . , (x0, yN , z0,1), . . . , (xN , yN , zN,N ), N ∈ N, bean interpolation data in R
3 such that
x0 < x1 < · · · < xN , y0 < y1 < · · · <yN and the independent variables on X and Yaxis are equally spaced on a square mesh [0, 1
2 ] ×[0, 1
2 ]. For the construction of Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS),a set of real parameters ti,j, called hidden-variables, are introduced and the generalized inter-polation data (xi, yj , zi,j , ti,j) : i, j = 0, 1, . . . ,N is considered. Define the Iterated FunctionSystem (IFS)
R4, ωn,m = (φn(x), ψm(y), Gn,m(x, y, z, t)) : n,m = 1, 2, . . . N, (2.1)
where the functions φn : [x0, xN ] → [xn−1, xn],ψm : [y0, yN ] → [ym−1, ym] are
φn(x) = xn−1 +xn − xn−1
xN − x0(x− x0),
ψm(y) = ym−1 +ym − ym−1
yN − y0(y − y0)
and the function Gn,m : [x0, xN ] × [y0, yN ] × R2 →
R2 is
Gn,m(x, y, z, t)
=
Fn+1,m(x0, y, z, t), x = xN , n = 1, . . . , N − 1,m = 1, . . . , N
Fn,m+1(x, y0, z, t), y = yN , n = 1, . . . , N,m = 1, . . . , N − 1
Fn,m(x, y, z, t), otherwise,
with the function Fn,m(x, y, z, t) = (F 1n,m(x, y),
F 2n,m(x, y)) given by
F 1n,m(x, y, z, t) = αn,m z + βn,m t+ en,m x
+ fn,m y + gn,m xy + kn,m,
F 2n,m(x, y, t) = γn,m t+ en,m x+ fn,m y
+ gn,m xy + kn,m.
(2.2)
In Eq. (2.2), αn,m and γn,m are free variables chosensuch that |αn,m| < 1 and |γn,m| < 1, |βn,m|is a constrained variable chosen such that |βn,m| +|γn,m| < 1 and the real coefficients en,m, fn,m, gn,m,
kn,m, en,m, fn,m, gn,m and kn,m are obtained by thejoin-up conditions:
Fractal Dimension of Coalescence Hidden-Variable Fractal Interpolation Surface 197
It is known9 that there exists a metric τ onR
4, equivalent to the Euclidean metric, such thatthe IFS, given by Eq. (2.1), is hyperbolic withrespect to the metric τ and there exists a uniquenon-empty compact set G ⊆ R
4 with respectto the metric τ such that G = ∪N
n=1 ∪Nm=1
ωn,m(G). The set G, called the attractor of theIFS for the given interpolation data, is the graphof a continuous function F : [x0, xN ] × [y0, yN ] →R
2 such that F (xi, yj) = (zi,j , ti,j) for i, j =0, 1, . . . , N, i.e, G = (x, y, F (x, y)) : (x, y) ∈[x0, xN ] × [y0, yN ] and F (x, y) = (z(x, y), t(x, y)).Now, expressing the function F (x, y) component-wise as F (x, y) = (F1(x, y), F2(x, y)), the Coales-cence Hidden-variable Fractal Interpolation Surface(CHFIS) for the given interpolation data is definedas follows:
Definition 2.1. The Coalescence Hidden-variable Fractal Interpolation Surface(CHFIS) for the given interpolation data (xi,yj, zi,j) : i, j = 0, 1, . . . , N is defined as the func-tion F1(x, y) whose graph is the projection of thegraph of the function F (x, y) on R
3.
A set S of points x = (x1, x2, . . . , xn) in a Euclideanspace of dimension n is called self-affine if S isunion of N distinct subsets, each identical withaS = (a1x1, a2x2, . . . , anxn) : a = (a1, a2, . . . , an),ai > 0 and x ∈ S up to translation and rotation. IfS is not self-affine, then it is called non-self-affine.The function F1(x, y) occurring in Definition 2.1 iscalled a CHFIS as it exhibits both self-affine andnon-self-affine nature. We observe that the functionF2(x, y) for the same interpolation data is always aself-affine function.
Fractal Dimension of a CHFIS F1(x, y) is the boxcounting dimension10 of its graph defined as
D(Graph(F1(x, y)))
= limn→∞
log(Nn(Graph(F1(x, y)))log 2n
;
provided the limit exist, Nn(Graph(F1(x, y))) beingthe smallest number of closed boxes in R
3 of side 12n
that intersect the graph of CHFIS F1(x, y).A function F : R2 → R is said to be a Lip-
schitz function of order δ (written as Lip δ) if|F (X) − F (X)| ≤ K (dM (X, X))δ where, K is aconstant, δ ∈ (0, 1] and dM (X, X) = |x− x|+ |y− y|for X = (x, y), X = (x, y). The function F issaid to be a Lip∗
k δ function, if |F (X) − F (X)| ≤K([dM (X, X)]δ[1 + log(dM (X, X))k]), where K is a
constant, δ ∈ (0, 1] and k ∈ N. Denote,
pn,m(x, y) = en,m x+ fn,m y + gn,m xy + kn,m
qn,m(x, y) = en,m x+ fn,m y + gn,m xy + kn,m.
(2.4)
It is observed that the functions pn,m(x, y) andqn,m(x, y), given by Eq. (2.4), belong to the classesLip λn,m and Lip µn,m (0 < λn,m, µn,m ≤ 1)respectively.
Definition 2.2. The Bivariate CHFIS F1(x, y) iscalled a Critical CHFIS if any one of the condi-tions Ω = 1, Γ = 1 and Θ = 1 holds.
3. FRACTAL DIMENSION OFCHFIS
In this section, the bounds on the fractal dimen-sion of CHFIS F1(x, y) for different critical casesare obtained in Theorems 3.1 and 3.2. Using thesebounds, certain conditions on the free parametersare determined that lead the fractal dimension ofthe constructed CHFIS to become close to 3. Also,these bounds give us a range of the free parame-ters that ensure the fractal dimension of the con-structed CHFIS to be strictly greater than 2. LetIr1,...,rn ≡ φrn(0)+ |Irn | Ir1,...,rn−1 = φrn · · ·φr1(I)and Js1,...,sn ≡ ψsn(0)+ |Jsn | Js1,...,sn−1 = ψsn · · · ψs1(J), ri, sj ∈ 1, 2, . . . , N, where |Iri | and |Jsj |denote the length of the intervals Iri and Jsj respec-tively. Hence, the area of square Sr1,...,rn,s1,...,sn =Ir1,...,rn × Js1,...,sn is |Sr1,...,rn,s1,...,sn | = |Ir1,...,rn | ×|Js1,...,sn| = |Ir1 | × |Js1 | × · · · × |Irn | × |Jsn | =|Sr1,s1| × · · · × |Srn,sn| and the diameter diamM
(Sr1,...,rn,s1,...,sn) of the square Sr1,...,rn,s1,...,sn is|Ir1,...,rn | + |Js1,...,sn |.
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198 S. A. Prasad & G. P. Kapoor
The following theorem give the bounds on thefractal dimension of CHFIS F1(x, y) when Θ = 1and Ω = 1 or Γ = 1.
Theorem 3.1. Let F1(x, y) be a CHFIS withΘ = 1. Then, for the critical condition Ω = 1,
ζ(αi,j) ≤ D(Graph(F1(x, y))) ≤ 3 − δ(Γ) (3.1)
where ζ(αi,j) = max1+log(
PNi=1
PNj=1 |αi,j |)
log N , 2 andδ(Γ) ∈ (0, 1]. Further, for the critical conditionΓ = 1,
η(γi,j) ≤ D(Graph(F1(x, y))) ≤ 3 − δ(Ω), (3.2)
where η(γi,j) = max1 +log(
PNi=1
PNj=1 |γi,j |)
log N , 2 andδ(Ω) ∈ (0, 1].
Proof. Case (i): (Θ = 1, Ω = 1 and Γ = 1) In thiscase, there exist constants C1 and C2 such that11
C1(dM (X, X))δ(Γ) ≤ dM (F1(X), F1(X))
≤ C2(dM (X, X))δ(Γ)
×[1 + log(dM (X, X))], (3.3)
for some δ(Γ) ∈ (0, 1], X = (x, y), X = (x, y),0 ≤ x < x ≤ 1
2 and 0 ≤ y < y ≤ 12 . In
fact, δ(Γ) = min(λ, µ) or min(λ, τ1) if Γ ≤ 1 andδ(Γ) = min(λ, τ2) or δ = min(λ, τ3) if Γ > 1 where,τ1, τ2 and τ3 are non-negative real numbers in theinterval (0, 1] such that τ1 ≤ log α
log |Smin| , τ2 ≤ log γlog |Smin|
and τ3 ≤ log(αγ)log |Smin| − µ.
Let Gr1,...,rm,s1,...,sm = (X,F1(X), F2(X)) :X ∈Sr1,...,rm,s1,...,sm. Then, max(dM (X, X)) :X, X ∈Sr1,...,rm,s1,...,sm = diamM (Sr1,...,rm,s1,...,sm) and|Gr1,r2,...,rm,s1,s2,...,sm| ≡ maxdM (F1(X), F1(X)) :(X, X) ∈ Sr1,...,rm,s1,...,sm. Then, by Eq. (3.3),
Taking summation over r1, r2, . . . , rm and s1,s2, . . . , sm from 1 to N we have
C1Nm
∑r1,...,rm
∑s1,...sm
m∏i=1
|αri,si |
≤ Nm∑
r1,...,rm
∑s1,...,sm
|Gr1,...,rm,s1,...,sm|
≤C2
(1N
)m(δ(Γ)−1) [1 + log
(1N
)m]
×∑
r1,...,rm
∑s1,...,sm
1.
The above inequality implies
C1Nm
N∑
i=1
N∑j=1
(|αi,j |)
m
≤ Nn(Graph(F1(x, y)))
≤C2
(1N
)m(δ(Γ)−1) [1 + log
(1N
)m]·N2m.
The bound on fractal dimension of CHFIS F1(x, y)given by Eq. (3.1) follow from the above inequalities
and ζ(αi,j) = max1 +log(
PNi=1
PNj=1 |αi,j |)
log N , 2.
Case (ii): (Θ = 1, Γ = 1 and Ω = 1) The proof issimilar to case (i) up to Eq. (3.4). In the presentcase, since Γ = 1, γri,sj ≤ (|Sri,sj |)µ ≤ (|Sri,sj |)δ(Ω),where, δ(Ω) ∈ (0, 1] is given by δ(Ω) = min(λ, µ)or min(λ, τ1) for Ω ≤ 1 and δ(Ω) = τ1 for Ω > 1.11
Hence,
m∏i=1
|γri,si | ≤m∏
i=1
|Sri,si |δ(Ω)
= |Sr1,r2,...,rm,s1,s2,...,sm|δ(Ω) ≤(
1N
)mδ(Ω)
.
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Fractal Dimension of Coalescence Hidden-Variable Fractal Interpolation Surface 199
Therefore, it follows from (3.4),
C1
m∏i=1
|γri,si | ≤ |Gr1,r2,...,rm,s1,s2,...,sm|
≤ C2
(1N
)mδ(Ω) [1 + log
(1N
)m].
Taking summation over r1, r2, . . . , rm and s1,s2, . . . , sm from 1 to N , we have
C1Nm
∑r1,...,rm
∑s1,...sm
m∏i=1
|γri,si|
≤ Nm∑
r1,...,rm
∑s1,...,sm
|Gr1,...,rm,s1,...,sm|
≤C2
(1N
)m(δ(Ω)−1) [1 + log
(1N
)m]
×∑
r1,...,rm
∑s1,...,sm
1.
The above inequality implies
C1Nm
N∑
i=1
N∑j=1
(|γi,j |)
m
≤ Nn(Graph(F1(X)))
≤C2
(1N
)m(δ(Ω)−1) [1 + log
(1N
)m]·N2m.
The bound on fractal dimension of CHFIS F1(x, y)given by Eq. (3.2) follow from the above inequalities
and η(γi,j) = max1 +log(
PNi=1
PNj=1 |γi,j |)
log N , 2.
Theorem 3.2 give us the bounds on the fractaldimension of CHFIS F1(x, y) when Θ = 1.
Theorem 3.2. Let F1(x, y) be a CHFIS withΘ = 1. Then,
The rest of the proof is similar to the proof ofTheorem 3.1 and hence is omitted.
Remark 3.1. Using Theorems 3.1 and 3.2, it fol-lows that the constructed CHFIS has fractal dimen-sion close to 3 if (i) |αi,j | is sufficiently close to 1 andΩ = 1 or Θ = 1 or, (ii) |γi,j| is sufficiently close to1 and Γ = 1. For, if Ω = 1 or Θ = 1 or Γ = 1, thenλ or µ is sufficiently close to 0 so that δ(Γ), δ(Ω)or δ(Ω,Γ) is also sufficiently close to 0 since thesevalues are bounded by λ or µ.
Remark 3.2. It follows from Theorems 3.1 and 3.2that
2 < D(Graph(F1(x, y))) < 3 (3.6)
if (i) ( 1N ) < |αi,j| ≤ ( 1
(2N))2p where p = λ for Ω = 1
and p = µ for Θ = 1 or (ii) ( 1N ) < |γi,j| ≤ ( 1
(2N))2µ
and Γ = 1.
4. EXAMPLE
Consider the sample data given by Table 1 takenfrom a tsunami wave surface (Fig. 1) and the gen-eralized interpolation data given by Table 2.
Table 3 gives choices of parameters αn,m, γn,m
and βn,m corresponding to critical conditions Θ = 1,
Table 1 Surface Values zn,m at (xn, ym) in Inter-polation Data.
(a) CHFIS simulation for Critical Case Θ = 1. (b) CHFIS simulation for Critical Case Ω = 1.
(c) CHFIS simulation for Critical Case Γ = 1.
Fig. 2 CHFIS simulation of tsunami wave surface.
Ω = 1 or Γ = 1 for the simulation of thetsunami wave surface as a CHFIS. The values ofthese parameters are determined by requiring thatthe functions pn,m and qn,m defined by Eq. (2.4)are in suitable classes Lip λn,m and Lip µn,m
respectively such that the value of Θ, Ω or Γ(c.f. Eq. (2.5)) equals 1. The last three columnsin Table 3 give corresponding δ-value of simu-lated CHFIS, lower bound and upper bound (c.f.Inequalities (3.1), (3.2) and (3.5)) of its fractaldimension. Figures 2(a)–2(c) give the simulations
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Fractal Dimension of Coalescence Hidden-Variable Fractal Interpolation Surface 201
of the tsunami wave surface (c.f. Fig. 1) as CHFISunder different critical conditions corresponding tothe choices of free parameters in Table 3.
It is observed in Table 3 that the critical con-dition Θ = 1 gives the largest lower bound onthe fractal dimension of the corresponding CHFIS(c.f. Fig. 2(a)) while the critical condition Ω = 1gives the least upper bound on the fractal dimen-sion of the corresponding CHFIS (c.f. Fig. 2(b)). Incase of the critical condition Γ = 1, the upper andlower bounds on the fractal dimension of the cor-responding CHFIS (c.f. Fig. 2(c)) are not the leastand largest bounds respectively. Thus, for the aboveTsunami wave surface (c.f. Fig. 1) and the gener-alized interpolation data (c.f. Table 2), the criticalcondition Θ = 1 or ω = 1 give closer bounds (loweror upper respectively) on the fractal dimension ofits simulated CHFIS.
The box counting dimension of the Tsunami wavesurface (c.f. Fig. 1) in our present example has alsobeen computed and found to be 2.1377, substanti-ating the inequalities (3.1), (3.2) and (3.5).
5. CONCLUSIONS
In this paper, the bounds on Fractal Dimensionof Coalescence Hidden-variable Fractal Interpola-tion Surface (CHFIS) are determined. Using thesebounds on Fractal Dimension of CHFIS, certainconditions on the free parameters are found thatlead the fractal dimension of the constructed CHFISto become close to 3. As a test case, a tsunami wavesurface is considered and the bounds, found in thepresent work, on fractal dimension of its simulatedCHFIS are computed to substantiate our results.Besides other applications in diverse fields of sci-ence and engineering (viz. Refs. 12–14), the boundson fractal dimensions of CHFIS found here are likelyto be helpful in the development of models for com-putation of tsunami intensity.
ACKNOWLEDGMENTS
The author Srijanani thanks CSIR for researchgrant (No:9/92(417)/2005-EMR-I) for the presentwork.
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