Analytical procedure to determine the self-referred lacunarity function for simple shapes. Supplementary material

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by  Erbe P. Rodrigues

Analyti
al pro
edure to determine the self-referred la
unarity
fun
tion for simple shapes. Supplementary material.
Erbe P. Rodrigues, Mar
oni. S. Barbosa and Lu
iano da F. Costa
Instituto de Físi
a de São Carlos
Universidade de São Paulo Caixa Postal 369 CEP 13560.970São Carlos, SP, Brazil
5
0 e-mail: mar
oniif.s
.usp.br
0
2

n
a
J Abstra
t

0
2 Theanalyti
al
al
ulationoftheself-referredla
unarityisusedasavalidationstandardofthe

omputationalalgorithm. Inthissupplementarymaterialtoourarti
le(see
ond-mat/0407079)
]
h we present a detailed
al
ulation for two simple shapes, namely a square box and a
ross.
c
e
m
- 1 Introdu
tion: self-referred la
unarity
t
a
t
s The la
unarity of an obje
t is de(cid:28)ned as
.
t Z(2) σ2(r)
a
Λ(r) = = +1,
m [Z(1)]2 [µ(r)]2
-
d
Z(1) Z(2) σ2(r) µ(r)
n where and are the (cid:28)rst and se
ond moments of a mass distribution and and
o µ(r) σ2(r) A(x,y,r)
c are its varian
e and mean. We
an
al
ulate and by de(cid:28)ning a fun
tion that
[ r (x,y)
gives the area of the obje
t inside a window of radius , whose
enter is in the point , in the

1
v following way
1 A(x,y,r)dxdy
0 µ(r) = x y ,
R R
5 dxdy (1)
x y
1 R R
[µ(r)−A(x,y,r)]2dxdy
0 σ2(r) = x y .
5 R R
dxdy (2)
0 x y
R R
/
t x y
a
For the self-referred approa
h of la
unarity the integral is
al
ulated only for values of and that
m
- fall over the obje
t, see Figure 1.
d
n
o
c 2 Analyti
al
al
ulation for simple shapes
:
v
i
X 2.1 Obje
t: a square box
r L
a We (cid:28)rst show how to determine the self-referred la
unarity for a solid square of side shown in
Figure 1(a). In order to
al
ulate the analyti
al la
unarity of a square it is ne
essary to determine
A(x,y,r)
the area fun
ition . This fun
tion is determined by dividing the square in regions labeled
I II III I II
by , and as in Figure 1(a) and 1(b). There is one region of type and four of type and
III r = [0,L/2] r = [L/2,L]
. It is also ne
essary to divide this
al
ulation into two intervals: and .
1
r
y
y
III
III
r
I
I
II
II
0 r L - r L x 0 L - r r L x
(a)
(b)
r∈[0,L/2]
Figure 1: Squareregionsthat
orrespondtotheself-referred la
unarity
al
ulationfortheintervals(a)
r∈[L/2,L] I
and (b) . The sliding windows in the region are
ompletely (cid:28)lled in (a) and
apture all the
II III
squarein (b). Intheregions and they are partially (cid:28)lledin (a) as well as in (b).
r ∈ [0,L/2] A1(x,y,r)
i
For the (cid:28)rst interval, , the fun
tions and respe
tive integration ranges for
x y
and are
A1(x,y,r) = 4r2, x∈ [r,(L−r)], ∈[r,(L−r)]
I
A1 (x,y,r) = 2r2+2ry, x ∈ [r,(L−r)], ∈ [0,r]
II
A1 (x,y,r) = r2+[r+(L−y)]x+r(L−y),
III
x∈ [0,r], y ∈ [(L−r),r]
A1
i
The mean value is determined by the integration of the over their respe
tive regions divided by
the total area of the square, a

ording the Equation 1
A1(x,y,r)dxdy
x y i i
µ (r) =
1 R R P
dxdy
x y
R R
1
= A1(x,y,r)+4A1 (x,y,r)+4A1 (x,y,r)dxdy
L2 Z Z I II III
r2(r−2L)2
=
L2
II III
The four regions and are equivalent and it is ne
essary only to integrate over one of region
II III
and and multiply the result by 4. The varian
e is
al
ulated in analogous way a

ording to
Equation 2
1
σ2(r) = (A1(x,y,r)−µ (r))2
1 L2 Z Z I 1 (3)
+ 4(A1 (x,y,r)−µ (r))2
II 1
+ 4(A1 (x,y,r)−µ (r))2dxdy
III 1
r5(r−6L)(3r−4L)(3r−2L)
= −
9L4
2
µ (r) σ (r)
1 1
Knowing and we may determine the expression for self-referred la
unarity for the (cid:28)rst
interval σ2(r) 4L2(5r−6L)2
Λ (r) = 1 +1 =
1 [µ (r)]2 9(r−2L)4
1
r ∈ [L/2,L] A2
i
The
al
ulationforthese
ondinterval, , pro
eedsinthesameway,withthe fun
tions
and their respe
tives integration intervals given by
A2(x,y,r) = L2, x∈ [(L−r),r], y ∈ [(L−r),r]
I
A2 (x,y,r) = L(r+y), x ∈[(L−r),r], y ∈ [0,(L−r)]
II
A2 (x,y,r) = r2+[r+(L−y)]x+r(L−y),
III
x∈ [0,(L−r)], y ∈ [r,L]
and the mean, varian
e and La
unarity as in
r2(r−2L)2
µ (r) =
2 L2
(3L−r)(r−L)3(3r4−14r3L+12r2L2+6rL3−L4)
σ2(r) = −
2 9L4
L2(2r3−6rL2+L3)2
Λ (r) =
2 9r4(r−2L)4
The (cid:28)nal expression for the whole interval self-referred la
unarity is
4L2(5r−6L)2, r ∈ [0,L/2];
 9(r−2L)4
Λ(r) = 
L2(2r3−6rL2+L3)2, r ∈ [L/2,L].
 9r4(r−2L)4
L = 100
Figure 2 shows a plot of the expression above for a square of length , in absolute units of
length.
1.1
L=100
Λ (r)
1
1.075
s)
s
e
nl
nsio Λ2(r)
e
m
di 1.05
y (
arit
n
u
c
a
L
1.025
1
0 10 20 30 40 50 60 70 80 90 100
Radius (units of length)
L 100
Figure 2: Theanalyti
ally
al
ulatedself-referred la
unarity
urvefor asquarewith side of absoluteunitsof
length.
3
2.2 Obje
t: a
ross
k
The analyti
al pro
edure to
al
ulate the la
unarity of a
ross, whi
h o

upies a region , is similar
A(x,y,r)
to the pro
edure presented above for a square. Again we de(cid:28)ne a fun
tion . For this
ase,
there are seven intervals and ea
h one was divided into regions whi
h de(cid:28)ne an asso
iated fun
tion
A (x,y,r)
i
. Figure 3 shows the (cid:28)rst interval
onsidered in the self-referred la
unarity
al
ulation of
L1 L2
a
ross with dimensions and and the respe
tive regions. Figure 3 presents only non-repeated
i j
regions , however the
al
ulation regards all the possible regions . In the following we show the

al
ulation of the (cid:28)rst interval in more detail. For the remaining intervals we provide only the
A (x,y,r) µ(r) σ2(r) Λ(r)
i
fun
tions , , and . Figures 4 to 9 presents the divided regions for the
2 7 Λ(r) = (σ2(r)/µ2(r))+1
remaining intervals labeled from to . The expression for la
unarity,
µ σ2
where and is given by
A (x,y,r)dk
µ(r)= k j ,
R P dk
k
R
and [ A (x,y,r)−µ(r)]2dk
σ2(r)= k j ,
R P dk
k
R
dk k A = 4L1L2+L22
where k is the total area ( k ) of the
ross.
R
0 ≤ r ≤ L2
2.2.1 Interval 2
r L2−r
L1+r
IV III
L1
r
V II
2r
VI
r
VII
0
1
I L2
−(L2−2r)
L1
Figure 3: The
ross dividedto regions for the(cid:28)rst interval.
A (x,y,r) = 4r2 x ∈[r,L2−r], y ∈ [−(L2−2r),0]
I
A (x,y,r) = 4r2 x ∈[r,L2−r], y ∈ [0,L1]
II
A (x,y,r) = (2r+L1−y)2r x ∈ [r,L2−r], y ∈ [L1,L1+r]
III
4
A (x,y,r) = (r+x)(2r+L1−y) x ∈[0,r], y ∈ [L1,L1+r]
IV
A (x,y,r) = (r+x)(2r) x∈ [0,r], y ∈ [2r,L1]
V
A (x,y,r) = 4r2−y(r−x) x∈ [0,r], y ∈[r,2r]
VI
A (x,y,r) = 4r2−y(r−x) x∈ [0,r], y ∈[0,r]
VII
µ
The Cal
ulation of the mean for the (cid:28)rst region gives
(A +4A +4A +8A +8A +8A +4A )dk
µ (r) = k I II III IV V VI VII
I R A
k
r2(4L22 −4rL2+3r2+16L1L2−8rL1)
= .
L2(4L1+L2)
L1= 2L L2= L
We make e to simplify the expressions. Thus we have for the previous expression
r2(36l2 −20Lr+3r2)
µ (r)= .
I
9L2
σ2(r)
In an analogous way is given by
[A −µ (L1,L2,r)]2+...+[4A −µ (L1,L2,r)]2 dk
σI2(r) = Rk(cid:16) I I A VII I (cid:17)
k
1
= − (−384L2L12 +576rL12−432r2L1−288L1L22
9L22(4L1+L2)2
+496rL1L2−216L2r2 −48L23 +81r3+124rL22)r5.
Simplifying the above expression we have
r5 9r3−120Lr2 +380rL2−240L3
σ2(r)= − .
I (cid:0) 81L4 (cid:1)
The la
unarity expression in this interval is (cid:28)nally given by
4L2(−300Lr+59r2+324L2)
Λ (r) = .
I (36L2 −20Lr+3r2)2
L2 ≤ r ≤ L2
2.2.2 Interval 2
A (x,y,r) = 2(2rL2)−L22 x∈ [L2−r,r],y ∈[−(L2+2r),0]
I
A (x,y,r) = (2r+y−L2)L2+(L2−y)2r x ∈ [L2−r,r], y ∈ [0,L2]
II
A (x,y,r) = L2×2r x ∈ [L2−r,r], y ∈ [L2,L1+L2−2r]
III
A (x,y,r) = L2(L1+L2−y) x∈ [L2−r,r],
IV
y ∈ [L1+L2−2r,L1+L2−r]
A (x,y,r) = (L1+L2−y)(r+x) x∈ [0,L2−r],
V
y ∈ [L1+L2−2r,L1+L2−r]
A (x,y,r) = 2r(r+x) x∈ [0,L2−r], y ∈ [L2,L1+L2−2r]
VI
A (x,y,r) = 2r(r+x)+(r−x)(L2−y) x∈ [0,L2−r],y ∈ [L2−r,L2]
VII
A (x,y,r) = 4r2−(2r+y−L2)(r−x) x∈ [0,L2−r], y ∈[0,L2−r]
VIII
5
(L1+L2−r)
V IV
(L1+L2−2r)
VI III
L2
VII II
(L2−r)
VIII
0
2
I
−(L2+2r)
L2 ≤r≤L2
Figure 4: The
ross regions to interval 2 .
r2(−20rL+36L2+3r2)
µ (r)=
II
9L2
1
σ2 (r) = (120r7L−12L7r−9r8+1248r5L3+452L5r3
II 81L4
+L8−36L6r2−604r6L2−1128L4r4)
L2 −12L5r−192r5L+452L3r3+L6−36L4r2+12r6+168r4L2
Λ (r)=
II (cid:0) r4(−20rL+36L2 +3r2)2 (cid:1)
L2 ≤ r ≤ L1+L2
2.2.3 Interval 2
A (x,y,r) = 2L2(2r)−L22 x ∈ [0,L2], y ∈ [−L2,0]
I
A (x,y,r) = 4rL2−L22 x ∈[0,L2], y ∈ [0,r−L2]
II
A (x,y,r) = L2(r+y)+(r−y)2r x ∈ [0,L2], y ∈[r−L2,L1−r]
III
A (x,y,r) = (L1L2)+(r−y)2r x ∈ [0,L2], y ∈ [L1−r,r)]
IV
A (x,y,r) = (L1−(y−r))L2 x∈ [0,L2], y ∈ [r,L1]
V
4L 1 2
µ (r) = rL+ L2+ r2
III
3 9 3
6
L1
V
r
IV
(L1−r)
III
(r−L2)
II
0
3
I
−L2
L2≤r≤ (L1+L2)
Figure 5: The
ross regions for theinterval 2 .
76 124 100 196 112
σ2 (r) = − r4− r2L2+ Lr3+ L3r− L4
III 27 9 9 27 81
64r4+320r2L2−348Lr3 −204L3r+37L4
Λ (r) = −3
III (12rL+L2+6r2)2
L1+L2 ≤ r ≤ L1
2.2.4 Interval 2
A (x,y,r) = 2L2(2r)−L22 x∈ [0,L2], y ∈ [−L2,0]
I
A (x,y,r) = 2rL2+(r−y−L2)L2+(y+r)L2 x∈ [0,L2], y ∈ [0,L1−r]
II
A (x,y,r) = L1L2+2rL2+(r−y−L2)L2 x∈ [0,L2], y ∈ [L1−r,r−L2]
III
A (x,y,r) = L1L2+(r−y)2r x ∈ [0,L2], y ∈ [r−L2,r)]
IV
A (x,y,r) = (L1−(y−r))L2 x∈ [0,L2], y ∈ [r,L1]
V
4L 1 2
µ (r)= rL+ L2+ r2
IV
3 9 3
4 68 20 236 212
σ2 (r) = − r4+ r2L2− Lr3− L3r+ L4
IV 9 9 27 27 81
L 256r2L+28r3−228rL2+71L3
Λ (r) =3
IV (cid:0) (12rL+L2+6r2)2 (cid:1)
7
L1
V
r
IV
(r−L2)
III
(L1−r)
II
0
4
I
−L2
(L1+L2) ≤r≤L1
Figure 6: The
ross regions for theinterval 2 .
L1 ≤ r ≤ L1+ L2
2.2.5 Interval 2
(r−L1) (L1+L2−r)
L1
IV III
(r−L2)
V II
0
VI
−(r−L1)
5
I
−[L2−(r−L1)]
L1≤r≤L1+ L2
Figure 7: The
ross regions for the interval 2 .
8
A (x,y,r) = 2(2rL2)−L22 x∈ [r−L1,L1+L2−r],
I
y ∈[−L1−L2+r,−r+L1]
A (x,y,r) = L1L2+2rL2+(r−y−L2)L2 x∈ [r−L1,L2+L1−r],
II
y ∈[−r+L1,r−L2]
A (x,y,r) = L1L2)+(r−y)2r x ∈ [r−L1,L2+L1−r], y ∈ [r−L2,L1]
III
A (x,y,r) = L1L2+(r−y)(r+x+L1) x∈ [0,r−L1], y ∈ [r−L2,L1]
IV
A (x,y,r) = L1L2+(r+x+L1)L2+(r−y−L2)L2 x∈ [0,r−L1],
V
y ∈[0,r−L2]
A (x,y,r) = L1L2+(r+x+L1)L2+(r−y−L2)L2 x∈ [0,r−L1],
VI
y ∈[−(r−L1),0]
−100L3r+90r2L2−24Lr3+2r4+49L4
µ (r) =
V
9L2
2
σ2(r) = − (398L8 +6704L6r2+5717L4r4+448r6L2
V 81L4
−2144r5L3−8516L5r3−2542L7r−48r7L+2r8)
L2 1605L6 +5412L4r2+1662r4L2+40r6−432r5L−3320L3r3−4716L5r
Λ (r)=
V (cid:0) (−100L3r+90r2L2−24Lr3+2r4+49L4)2 (cid:1)
L1+ L2 ≤ r ≤ L1+L2
2.2.6 Interval 2
A (x,y,r) = 4L1L2+L22 x ∈ [L1+L2−r,r−L1],
I
y ∈ [−(r−L1),−(L1+L2−r)]
A (x,y,r) = L22+3L1L2+(r−y−L2)L2 x∈ [−(L2+L1−r),r−L1],
II
y ∈ [−(L1+L2−r),r−L2]
A (x,y,r) = L1L2+(r−y)(2L1+L2) x ∈[L2+L1−r,r−L1],
III
y ∈ [r−L2,L1]
A (x,y,r) = L1L2+(L1+r+x)(r−y) x∈ [0,L2+L1−r], y ∈ [r−L2,L1]
IV
A (x,y,r) = L1L2+(r−y−L2)L2+(L1+r+x)L2 x ∈[0,L2+L1−r],
V
y ∈ [0,r−L2]
A (x,y,r) = L1L2+(r−y−L2)L2+(L1+r+x)L2 x ∈[0,L2+L1−r],
VI
y ∈ [−(L1+L2−r),0]
9
L1
IV III
(r−L2)
V II
0
VI
−(L1+L2−r)
6
I
−(r−L1)
L1+ L2 ≤r≤L1+L2
Figure 8: The
ross regions for theinterval 2 .
−100L3r+90r2L2−24Lr3+2r4+49L4
µ (r) =
VI
9L2
2
σ2 (r) = − (11804L6r2+6881L4r4+464r6L2−2360r5L3
VI 81L4
−11766L5r3−7042L7r−48r7L+2r8+2273L8)
L2 4788L4r2+666r4L2−8r6−3180L3r3−4284L5r+2145L6
Λ (r)= −
VI (cid:0) (−100L3r+90r2L2−24Lr3+2r4+49L4)2 (cid:1)
L1+L2≤ r ≤ 2L1+L2
2.2.7 Interval
A (x,y,r) = 4L1L2+L22 x∈ [0,L2], y ∈ [−L2,0]
I
A (x,y,r) = 4L1L2+L22 x∈ [0,L2], y ∈ [0,r−L1−L2]
II
A (x,y,r) = 3L1L2+L22+(r−y−L2)L2 x∈ [0,L2], y ∈ [r−L1−L2,L1]
III
31 20 2
µ (r)= L2+ rL− r2
VII
9 9 9
4(2L−r)(5L−r)3
σ2 (r)= −
VII 81
10