c
World Scientific Publishing Company
ON THE SPATIAL ENTROPY AND PATTERNS OF
TWO-DIMENSIONAL CELLULAR NEURAL NETWORKS
SONG-SUN LIN∗
Department of Applied Mathematics, National Chiao-Tung University, Hsin-Chu 30050, Taiwan
TZI-SHENG YANG
Department of Applied Mathematics, I-Shou University, Kaohsiung 84041, Taiwan
Received April 20, 2000; Revised March 9, 2001
This work investigates binary pattern formations of two-dimensional standard cellular neural networks (CNN) as well as the complexity of the binary patterns. The complexity is measured by the exponential growth rate in which the patterns grow as the size of the lattice increases, i.e. spatial entropy. We propose an algorithm to generate the patterns in the finite lattice for general two-dimensional CNN. For the simplest two-dimensional template, the parameter space is split up into finitely many regions which give rise to different binary patterns. Qualitatively, the global patterns are classified for each region. Quantitatively, the upper bound of the spatial entropy is estimated by computing the number of patterns in the finite lattice, and the lower bound is given by observing a maximal set of patterns of a suitable size which can be adjacent to each other.
1. Introduction
The cellular neural network (CNN), as designed by Chua and Yang [1988a, 1988b], is an array of an identical system of cells that are locally coupled. This work investigates the complexity of stable bi-nary patterns of two-dimensional CNN. Indeed, the state equation of a cell Cij is the set of coupled O.D.E’s ˙ xij(t) =−xij(t) + X |k|,|l|≤d aklyi+k,j+l(t) + z, i, j∈Z2, (1)
with output yij(t) = f (xij(t)). Here f (·) is a piecewise-linear function expressed as
f (x) = 1
2(|x + 1| − |x − 1|) . (2)
The parameter z is a time-invariant bias and d is a positive integer. The coupling parameters akl’s are assumed to be space-invariant, which is ar-ranged in a (2d + 1)× (2d + 1) matrix A called a template.
The stationary solutions x = (xij) of (1) are prerequisite for understanding the CNN dynamics. Indeed, when studying the long-time behavior of a dynamic system, the stationary solutions are the simplest objects to be considered. In the case of CNN with finite cells and symmetric template A, Chua and Yang [1988a], Lin and Shih [1999] demon-strated that every trajectory of (1) tends to an equi-librium as time proceeds. The stationary solutions x = (xij) of (1) satisfy
xij = z + X |k|,|l|≤d
aklf (xi+k,j+l), i, j∈Z. (3) ∗This research was partially supported by the National Science Council of the Republic of China under Contract No. NSC
88-2115-M-259-003.
115 Int. J. Bifurcation Chaos 2002.12:115-128. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/27/14. For personal use only.
A stationary x solution is called a mosaic solution if|xij| > 1 for all i, j ∈Z. The corresponding out-put y = (f (xij)) is called a mosaic pattern. The mosaic solution of (1) is always locally asymptoti-cally stable [Juang & Lin, 1997] (and the references given therein). Among all stationary solutions, the stable mosaic solutions, which have been studied before [Juang & Lin, 1997], are the most funda-mental and important applications (finite cells) in image-processing [Chua, 1988b, 1988]. Those inves-tigations addressed two problems directly related to the mosaic patterns: (a) the directed problem, in which the parameter space is partitioned into finitely many regions so that in each region (1) has the same patterns and (b) the complexity of global patterns.
The complexity of mosaic solutions can be ex-amined according to its entropy. For completeness, the following discussion introduces some definitions and results concerning spatial entropy. Further de-tails can be found in [Chow et al., 1996a]. Denote by{−1, 1}Z2 the set of all y :Z2→ {−1, 1} i.e. the set of all mosaic patterns. Let U be a translation-invariant subset of {−1, 1}Z2 and Zmn = {(i, j) : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. The number of distinct patterns observed among the elements of U when observation is restricted to subset Zmn, is denoted by Γmn(U). The spatial entropy h(U) of U is defined by
h(U) = lim m,n→∞
ln Γmn(U)
mn .
U is called spatial chaos if the spatial entropy h(U) is positive. Otherwise, U is called pattern formation.
The case of a two-dimensional template markedly differs from that of a one-dimensional template. In the case of the one-dimensional tem-plate, the transition matrix can be used to construct the global pattern; the exact spatial entropy is then computed [Lin & Yang, 2000]. However, the tran-sition matrix cannot be defined in the case of a two-dimensional template. Up to now, no efficient and general method is available for two-dimensional templates. In [Juang & Lin, 1997] and [Shih, 1998], are discussed two-dimensional symmetric and asym-metric square cross templates, respectively. The di-rect problem is completely solved in both cases and the global patterns are classified by building blocks and the corresponding compatible conditions. In addition, the spatial chaos is confirmed to occur
by identifying the blocks which can be adjacent to each other. In contrast to the above developments, this work presents a novel algorithm to generate the patterns in the finite lattice for a general tem-plate, which gives the upper bound of the spatial entropy. Moreover, the lower bound of the spatial entropy is estimated by obtaining the maximal set of the patching blocks of some size (depending on the characteristic of global patterns). This inves-tigation concentrates mainly on the simplest two-dimensional template, L-shaped liked, i.e.
A = 0 r 0 0 p s 0 0 0 .
The tools developed herein can be easily applied to the general templates. For example, the case of a fully connected template
A0 = a1 a2 a3 a4 a5 a6 a7 a8 a9 ,
the reader can go through the process of partition-ing parameter space, determinpartition-ing the feasible local patterns as in Sec. 2, then finding the patterns in a finite lattice by using an algorithm similar to the one in Sec. 3. All the process are very similar to the case of template A except the shape of the local patterns.
Generally, a neural network concerns the re-lationship between input and output patterns. A learning algorithm is used to establish this relation-ship. Such a learning algorithm usually starts with initial weights i.e. coupling parameters of the neural network and then consecutively updates the weights according to the difference between actual output and desired output patterns. Our investigation pro-vides a method (for CNN) to analyze (a) possible output patterns in different regions of parameter space, (b) the choice of the range of coupling pa-rameters in the learning process.
The rest of this paper is organized as follows. Section 2 addresses the direct problem. The method is quite similar to the case of the one-dimensional template [ r p s ][Lin & Yang, 2000]. Section 3 presents a novel algorithm to generate the patterns in the finite lattice and give the upper bound of the spatial entropy. Section 4 describes and classifies the global patterns. For the case of spatial chaos, we find the maximal set of patching blocks of some
size (depending on the characteristic of global pat-terns) and give the lower bound of the spatial en-tropy. Conclusions are finally made in Sec. 5.
2. Partitioning the Parameter Space First, we introduce the notation of local patterns [Juang & Lin, 1997; Hsu et al., 2000] for the tem-plate A = 0 r 0 0 p s 0 0 0 . The state equation (1) now becomes
˙
xij(t) = ryi−1,j(t) + syi,j+1(t) + pyij(t) + z− 1 . (4)
For a mosaic solution x, the output at cell Cij is +1, i.e. xij > 1 if and only if
ryi−1,j+ syi,j+1+ p + z− 1 > 0 , (5) and, similarly, the output at cell Cij is−1, i.e. xij < −1 if and only if
ryi−1,j+ syi,j+1− p + z + 1 < 0 . (6) Denote by X2 = {−1, 1}2 the set of all possible (yi−1,j, yi, j+1). Each (k, m)∈ X2 is naturally asso-ciated with two local patterns Pk,mand Nk,m, which have yij “+1” and yij “−1”, respectively. Visually, the “+1” and “−1” are represented by black and white squares, respectively.
P1,1 = , N1,1 = , P1,−1 = , N1,−1 = , P−1,1 = , N−1,1= , P−1,−1 = , N−1,−1=
Obviously, for a given template A and threshold z, Pk,mand Nk,mare feasible if and only if (7) and (8) hold, respectively.
z + (p− 1) + kr + ms > 0 , (7) z− (p − 1) + kr + ms < 0 . (8)
The set of feasible local patterns, denoted by F(A, z), is defined to be the collection of feasible Pk,m0s and Nk,m0s.
The parameter space, P4 = {(r, s, p, z)|z, p, r, s∈R}, can be partitioned according to the feasible local patterns. To partition the r-s plane, we first arrange the elements of X2according to the value of kr +ms. Denote by I[i, j] the set of integers are not smaller than i and not larger than j. Given a (r, s), the (r, s)-arrangement of X2 is a function from X2 into I[1, 4], which assigns the integer i to (k0, m0) provided that k0r + m0s is the ith largest among all of kr + ms, (k, m)∈ X2. It can be verified that the following half lines, L1, . . . , L8, denoted by
L1: r > 0, s = 0 , L2 : r > 0, r = s , L3: r = 0, s > 0 , L4 : r > 0, r =−s , L5: r < 0, s = 0 , L6 : r < 0, r = s , L7: r = 0, s < 0 , L8 : r > 0, r =−s ,
Fig. 1.
Fig. 2. Partition of r–s plane according to (r, s)-arrangement.
divide the r − s plane into eight open regions, R1, . . . , R8. Here Ri is the open region bounded by Li and Li+1, for 1 ≤ i ≤ 7 and R8 is the open region bounded by L8 and L1 such that only one (r, s)-arrangement of X2 is induced in each Ri (see Fig. 2). Denote by Ji, the (r, s)-arrangement of X2 induced in Ri.
Figure 2 is illustrated as follows: Each element (k, m) of X2 corresponds to a position in a 2× 2 lattice as shown in Fig. 1. On each Ri, there is also a 2× 2 lattice and the values of Ji(k, m) are put into the position of this lattice, which corresponds to (k, m).
Now, we partition z−p plane when (r, s) in Ri. If Ji(k, m) = j, let `+j and `−j be the lines on z− p plane with equations:
z + (p− 1) + kr + ms = 0 , (9) and
z− (p − 1) + kr + ms = 0 , (10) respectively. Therefore {`+j }4j=1 and {`−j}4j=1 form two sets of parallel lines in the z − p plane with `+j+1 and `−j lying above `+j and `−j+1, respectively. Let [µ, ν]i, 0 ≤ µ, ν ≤ 4, denote the open region bounded by `+µ, `+µ+1, `−5−ν, `−4−ν with `+0, `−0, `+5, `−5 being the empty lines, as observed in Fig. 3. Hence, the z− p plane is partitioned into finitely many dis-joint regions, i.e. [µ, ν]i, 0≤ µ, ν ≤ 4. After parti-tioning the parameter space, Theorem 2.1 indicates
Fig. 3.
how the feasible local patterns are determined for each region [µ, ν]i.
Theorem 2.1. For (z, p)∈ [µ, ν]i, the feasible lo-cal patterns are exactly the union of
[ Ji(k,m)≤µ Pk,m and [ Ji(k,m)≥5−ν Nk,m. Proof. Assume Ji(k1, m1) = µ, Ji(k2, m2) = µ + 1 and (z, p)∈ [µ, ν]i. According to the definition of [µ, ν]i, z and p satisfy
z + (p− 1) + k1r + m1s > 0 , (11) and
z + (p− 1) + k2r + m2s < 0 , (12) which implies that
z + (p− 1) + kr + ms > 0 (13) if Ji(k, m)≤ Ji(k1, m1) = µ , and z + (p− 1) + kr + ms < 0 (14) if Ji(k, m)≥ Ji(k2, m2) = µ + 1 .
In addition, there is no (k, m) such that µ < Ji(k, m) < µ + 1. Hence, SJi(k,m)≤µPk,m are exactly the feasible local patterns whose left-lower cell is +1. A similar argument shows that S
Ji(k,m)≥5−νNk,mare exactly the feasible local pat-terns whose left-lower cell is −1. The proof is complete.
Example 2.1. If r = 2, p = 3/2, s = 1 and z = 0, a mosaic solution (xij) satisfies
xij = 3
2yij + 2yi−1,j+ yi,j+1, (i, j)∈Z 2.
To consider which local patterns are feasible, (a) yij = 1 implies xij = 3/2 + 2yi−1,j + yi,j+1> 1, hence
and
are feasible. (b) yij =−1, implies xij = 3/2 + 2yi−1,j+ yi,j+1<−1, hence
and are feasible.
On the other hand, we can also induce the feasible local patterns by Theorem 2.1 as follows. (r, s) = (2, 1)∈ R1 implies
`+1 : z + (p− 1) + 3 = 0, ; `−1 : z− (p − 1) + 3 = 0, ; `+2 : z + (p− 1) + 1 = 0, ; `−2 : z− (p − 1) + 1 = 0, ; `+3 : z + (p− 1) − 1 = 0, ; `−3 : z− (p − 1) − 1 = 0, ; `+4 : z + (p− 1) − 3 = 0, ; `−4 : z− (p − 1) − 3 = 0, .
Fig. 4. Partition of z− p plane when (r, s) = (2, 1).
Figure 4 shows that (z, p) ∈ [2, 2], according to Theorem 2.1, which implies the local patterns asso-ciated with `+1, `+2, `−3 and `−4 are feasible i.e.
, , , .
3. An Algorithm to Generate the Patterns in the Finite Lattice
Herein, the patterns in the finite lattice can be con-structed based on the feasible local patterns. In-deed, assume that we have the set of feasible local patterns F(A, z) = {U1, . . . , Uk} where Ui has the
shape . We dilate Ui to the minimal rectangu-lar block containing Ui by adding a black or white block in the upper right corner. By doing so, we obtain the set of patterns of 2× 2 size correspond-ing to F(A, z), say G = {V1, . . . , V2k}. Starting fromG, we use the technique of [Lin & Yang, 2000] by gluing right or below to obtain the patterns of larger size [Lin & Yang, 2000]. The patterns in finite lattice can be obtained by executing the following algorithm.
Algorithm
Initially set G = {V1, . . . , V2k}. Repeat executing (A) and (B) in suitable order. The patterns in the lattice of any finite size can be attained.
(A) One step horizontal extension.
(i) Suppose that each pattern inG is of size m × n. From G, take all pairs (Vi, Vj) such that Vi can glue to Vj on the right with overlapping m− 1 columns, i.e. the last m− 1 columns of Vi must equal to the first m−1 columns of Vj, respectively. Such a pair is referred to herein as a hor-izontal compatible pair of patterns. (ii) RefreshG as the set of patterns obtained
by gluing the horizontal compatible pairs of patterns in (i) together with overlap-ping m− 1 columns. Then, G is now the set of patterns of size m× (n + 1). (B) One step vertical extension.
(i) Suppose that each pattern inG is of size m × n. From G, take all pairs (Vi, Vj) Int. J. Bifurcation Chaos 2002.12:115-128. Downloaded from www.worldscientific.com by NATIONAL CHIAO TUNG UNIVERSITY on 04/27/14. For personal use only.
such that Vi can glue to Vj below with overlapping m−1 rows, i.e. the last m−1 rows of Vi are equal to the first m− 1 rows of Vj, respectively. Such a pair are referred to herein as a vertical compatible pair of patterns.
(ii) Refresh G as the set of the patterns ob-tained by gluing the vertical compatible pairs of patterns in (i) together with over-lapping m− 1 rows. Then, G is now the set of patterns of size (m + 1)× n. For fixed A and z, let hm,n = (ln Γ(m, n))/(mn), which can be obtained by executing (A) m times and (B) n times. Proposition 3.1 shows the proper-ties of hm,n.
Proposition 3.1. {h2km,2kn}∞k=1 is a decreasing sequence and converge to the spatial entropy of mosaic patterns.
Proof. First, we claim that h2m,2n ≤ hm,n. In a 2m× 2n lattice, there are 2 × 2 disjoint blocks of size m× n. When constructing the patterns in the lattice, each block has at most Γ(m, n) choices i.e. Γ(2m, 2m) ≤ (Γ(m, n))2×2. Hence h2m, 2n ≤ hm,n follows immediately. The standard induc-tion argument shows that{h2km,2kn}kis decreasing and its limit follows from the definition of the spa-tial entropy of the mosaic patterns. The proof is complete.
Proposition 3.1 suggests the upper bound of the spatial entropy hm,n. However, the computation of Γ(m, n) consumes considerable amounts of time and memory as the spatial chaos occurs. Therefore we need a procedure to confirm the spatial chaos for some regions.
4. Global Patterns and Spatial Chaos In this section, we construct the global patterns for each region [µ, ν]i of the parameter space. Accord-ing to the feasible local patterns for each region, we induce and describe the class of global patterns. However, once spatial chaos occurs, the number of restrictions of the global patterns in the finite lat-tice grows too fast to compute even though the class of the global patterns can be characterized. To con-firm spatial chaos, we give the maximal set of the patching blocks of some size (depending on the char-acteristic of the global patterns). Restated, the set
contains the largest number of blocks of some size, which can be adjacent to each other. Through this process, we give the lower bound of the spatial en-tropy.
First, we introduce some special subsets of Z2, which are necessary to describe global patterns.
Definition 4.1. [Juang & Lin, 1997] A horizontal (resp. vertical) edge of length k inZ2 is the subset defined U by U = {(i, j0) : i ∈ [i1, i2]} for some i1, i2 ∈ Z (U = {(i0, j) : j ∈ [j1, j2]} for some j1, j2 ∈ Z respectively). A edge of infinite length is called a line. The solid edge ˜U of U inR2 is defined by ˜U ={(x, j0) : i1 ≤ x ≤ i2} (vertical case is sim-ilarly defined). A union of edges T in Z2 is called a path if (i) T contains no 2×2 lattice (i.e. width one) and (ii) the solid path ˜T of T , i.e. the union of the corresponding solid edges, is connected. A path T inZ2is nonincreasing (resp. nondecreasing) if (i, j) belongs to T implies (i+1, j +1) (resp. (i+1, j−1)) does not belong to T .
Comment. A horizontal or vertical edge is both nonincreasing and nondecreasing paths according to Definition 4.1.
Since the patterns for [µ, ν]i and [ν, µ]i have opposite colors, it is sufficient to discuss only the cases of [ν, µ]i with µ ≥ ν and 1 ≤ i ≤ 4. Herein, we first state the result of this section.
Theorem 4.1. Equation (4) is spatial chaos if and only if r, s, p and z belong to the following regions of P4: 1. [4, 4]i, 1≤ i ≤ 8. 2. [4, 3]i, 1≤ i ≤ 8. 3. [4, 2]i, i = 4, 5, 6, 7. 4. [4, 1]i, i = 5, 6. 5. [3, 3]i, 1≤ i ≤ 8. 6. [3, 2]i, i = 4, 5, 6, 7.
Furthermore, we observe that [µ, ν]i with min(µ, ν)≥ 3 is always spatial chaos for each i.
The proof of Theorem 4.1 proceeds as follows. The class of global patterns for each case of The-orem 4.1 is first described. The lower bound of the spatial entropy is then obtained by finding the maximal set of patching blocks of minimal size. Next, for the remaining regions of the parameter space, only finite types of global patterns appear. Therefore, it is easily seen that they are pattern formation.
Case 1. [4, 4]i, 1≤ i ≤ 8.
All possible local patterns are feasible and, there-fore, each position can have two choices: or . Hence, the spatial entropy is ln 2.
Case 2a. [4, 3]1,2.
Only is infeasible. The global patterns are ob-tained by allowing paths of infinite length to be white and the remainders are black.
Patching blocks of size 2× 2:
, , .
Hence, on the 2n× 2n lattice, we have n2 disjoint blocks of size 2× 2 and each block have three choices. Therefore, h[4, 3]1,2 ≥ limm,n→∞ ln 3n·n 2n· 2n = ln 3 4 . Case 2b. [4, 3]3,4.
Only is not feasible. The global patterns are obtained by allowing some nonincreasing paths U to be white, whose each downward step is at most one, i.e. if (i, j) and (i− 1, j) belong to U then (i, j + 1) must belong to U . For example,
· · · . And the remainders are black. Patching blocks of size 2× 2:
, , , . Hence, a similar argument shows that
h[4, 3]3,4≥ ln 4
4 . Case 2c. [4, 3]5, 6.
In this case, only is not feasible. The global pattern is obtained by letting some paths ofZ2, con-taining no , to be white and the remainders to be black.
Patching blocks of size 2× 2: , , , , . Spatial entropy: h[4, 3]5,6 ≥ ln 5 4 . Case 2d. [4, 3]7,8.
Only is not feasible. It is the reflection of the
, the only infeasible local pattern for [4, 3]1,2, about the axis {(i, i) : i ∈ Z}. Hence, the global patterns are obtained by reflecting those of the Case 2a with respect to the axis {(i, i) : i ∈Z}. Patching blocks of size 2× 2:
, , , . Spatial entropy: h[4, 3]7,8≥ ln 4 4 . Case 3a. [4, 2]4,5.
and are infeasible, i.e. more than one con-secutively vertical white cells are forbidden. Hence, the global patterns are obtained by letting some horizontal edges to be white and the remainders to be black.
Patching blocks of size 2× 2: , , , . Spatial entropy: h[4, 2]4,5≥ ln 4 4 . Case 3b. [4, 2]6,7.
and are not feasible, i.e. more than one consecutively horizontal white cells are forbidden. Hence, the global patterns are obtained by allowing some vertical edges to be white and the remainders are black.
Patching blocks of size 2× 2: , , , . Spatial entropy: h[4, 2]6,7 ≥ ln 4 4 .
Case 4. [4, 1]5,6. n
, , o
are not feasible. This is the union of the infeasible local patterns of Cases 2d and 3a. Hence the global patterns are the intersec-tion of those of Cases 2d and 3a. Indeed, the global patterns are obtained by allowing some nonincreas-ing paths whose each downward and rightward step is at most one.
Patching blocks of size 2× 2:
, , , .
In each case [3, 3]i, 1 ≤ i ≤ 8, the set of feasible local patterns has the property: if a
lo-cal pattern is feasible then its color-reversed pat-tern is also feasible. Therefore we say that the global pattern for [µ, ν]i has the symmetric prop-erty in [µ, ν]i if its color-reversed pattern is also for [µ, ν]i. Obviously, every global pattern of [3, 3]i in-herits the symmetric property from its feasible local patterns.
Case 5a. [3, 3]1,2
and are not feasible, due to one more infeasible local pattern than in Case 2a. The global patterns are obtained by taking those from Case 2a, which have the symmetric property in [4, 3]1,2.
Patching block of size 4× 5:
Spatial entropy: h[3, 3]1,2 ≥ ln 20 20 . Case 5b. [3, 3]3,4.
and are not feasible, due to one more infeasible local pattern than in Case 2b. The global patterns are obtained by taking those from Case 2b, which have the symmetric property in [4, 3]3,4. Patching block: Patching blocks of size 3× 4:
Spatial entropy: h[3, 3]3,4 ≥ ln 9 12 . Case 5c. [3, 3]5,6.
and are not feasible, due to one more infeasible local pattern than in the Case 2c. The global patterns are obtained by taking those from Case 2c, which have the symmetric property in [4, 3]5,6. Patching block of size 3× 3:
Spatial entropy: h[3, 3]5,6 ≥ ln 9 9 . Case 5d. [3, 3]7,8.
and are not feasible, due to one more infeasible local pattern than in Case 2d. The global patterns are obtained by taking Case 2d those of which have the symmetric property in [4, 3]7,8.
Patching blocks of size 4× 3:
Spatial entropy: h[3, 3]7,8 ≥ ln 9 20 . Case 6a. [3, 2]4. n , , o
are not feasible. This case has one more infeasible local pattern than in Case 5b. Therefore, the global patterns are obtained by taking those of Case 5b, which contain no .
Patching block of size 3× 4:
Spatial entropy: h[4, 3]4 ≥ ln 3 12 . Case 6b. [3, 2]5. n , , o
are not feasible. This case has one more infeasible local pattern than in Case 5c, so the global patterns are obtained by taking those from Case 5c, which contain no .
Patching blocks of size 7× 3: Spatial entropy: h[3, 2]5 ≥ ln 8 21 . Case 6c. [3, 2]6. n , , o
are not feasible. The global patterns are obtained by reflecting those patterns for [3, 2]5 about the axis{(i, i) : i ∈Z}.
Patching block of size 3× 7:
Spatial entropy: h[3, 2]6 ≥ ln 8 21 . Case 6d. [3, 2]7. n , , o
are not feasible. The global patterns are obtained by reflecting those patterns for [3, 2]4 about the axis{(i, i) : i ∈Z}.
Patching block of size 4× 3:
Spatial entropy:
h[3, 2]7 ≥ ln 3
15 .
The remaining regions of the parameter space are all pattern formation. Only finite types of pat-terns appear. Therefore Table 1 lists the results for each case, where× means no global patterns exist. The Appendix illustrates the type of patterns which Pis0 represent.
5. Discussion
In Sec. 4, we use the maximal set of patching block of minimal size to estimate the lower bound of the
spatial entropy. The reason for using the size is that it requires less time to find them. However, the set of patching blocks of a larger size will give a more precise estimate of lower bound. Indeed, denote by B(m, n) the number of elements in the maximal set of the patching blocks of size m× n. Define
hm,n = ln B(m, n)
mn ,
the lower bound of the spatial entropy given by the maximal set of patching blocks. Proposition 5.1 indicates the superiority of the patching blocks of large size.
Table 1. The regions of pattern formation and the types of patterns appear in each region.
R1 R2 R3 R4 R5 R6 R7 R8
[4, 2] P1, P2 P1, P2, P1, P2, chaos chaos chaos chaos P1, P2,
P5, P11 P6, P12 P6, P12 P5, P11
[4, 1] P1, P2, P1, P2, P1, P8, P1, P8, chaos chaos P1, P7, P1, P7,
P3, P4, P3, P4, P14 P14 P13 P13
P17 P17
[4, 0] P1 P1 P1 P1 P1 P1 P1 P1
[3, 2] P1, P2, P1, P2, P1, P6, chaos chaos chaos chaos P1, P2,
P4, P5, P4, P6, P18, P22 P3, P19, P15, P20 P16, P21 P23 [3, 1] P1, P2, P1, P2, P1, P8 P1, P8 P24, P25 P24, P25 P1, P7 P1, P7 P3, P4 P3, P4 [3, 0] P1 P1 P1 P1 × × P1 × [2, 2] P1, P2, P1, P2, P1, P2, P10, P25 P10, P25 P9, P25 P9, P25 P1, P5 P5 P5 P6 [2, 1] P1, P2, P1, P2, P1, P8 P25 P25 P25 P9 P1, P7 P3 P4 [2, 0] P1 P1 P1 P1 × × × × [1, 1] P1, P2 P1, P2 P10 P10 P24 P24 P9 P9 [1, 0] P1 P1 × × × × × × Proposition 5.1. {h2km,2n}∞k=1 is an increasing sequence.
Proof. First, we claim that h2m,2n ≤ hm,n In a 2m× 2n lattice, there are 2× 2 disjoint blocks of size m× n. When constructing the patching blocks in the lattice, each block has at least B(m, n) choices i.e. B(m, m)≤ (B(m, n))2×2. Hence h2m,2n ≤ hm,n follows immediately. The standard induction ar-gument shows that{h2km,2kn}k is decreasing. The proof is complete.
Combining the Propositions 3.1 and 5.1 we get the inequality:
hm,n≤ h2m,2n ≤ h22m,22n≤ · · · ≤ h22m,22n≤ h2m,2n ≤ hm,n.
References
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Appendix
P1: pattern of all black. P2: pattern of all white.
P3: Z2 is divided into two parts — all black and all white. The boundary of the two parts is a vertical line.
P4: Z2 is divided into two parts — all black and all white. The boundary of the two parts is a
horizontal line.
P5: Each vertical line is either all black or all white.
P6: Each horizontal line is either all black or all white.
P7: Each vertical line is either all black or all white and the vertical white line is adjacent to only black lines.
P8: Each horizontal line is either all black or all white and the horizontal white line is adjacent to only black lines.
P9: Each vertical line is either all black or all white and every two adjacent vertical lines have different colors.
P10: Each horizontal line is either all black or all white and every two adjacent horizontal lines have
different colors.
P11: The vertical lines are of only two types: (a) whole black or white and (b) upper half white and lower half black.
P12: The horizontal lines are of only two types: (a) whole black or white and (b) left half black and right half white.
P13: The vertical lines are of two types: (a) all black or white. (b) upper half white and lower half black. And (b) is adjacent to vertical black lines.
P14: The horizontal lines are of two types: (a) all black or white. (b) upper half black and lower half black. And (b) is adjacent to the horizontal white lines.
P15: Z2 is divided into two parts: the left part is entirely white and the right part is entirely black. The boundary of the two parts is a nondecreasing
path.
P16: Z2is divided into two parts: the left part is all black and the right part is all white. The boundary of the two parts is a nondecreasing path.
P17: Z2is divided into two parts. The left part is all black and the right part is all white. The boundary of the two parts is a nonincreasing path.
P18: Z2is divided into two parts. The left part is all white and the right part is all black. The boundary of the two parts is a strictly decreasing path whose each downward step is at most one.
P19: Z2is divided into two parts. The left part is all white and the right part is all black. The boundary of the two parts is a strictly decreasing path whose each rightward step is at most one.
P20: This is a mixed type of P15 and P5. Indeed, the pattern consists of (a) P15 with finite width and (b) vertical lines of all black or all white. And (a) is
adjacent to a vertical white line on the right and a white or black vertical line on the left, respectively.
P21: This is a mixed type of P16 and P6. Indeed, the pattern consists of (a) P16 with finite height and (b) horizontal lines of all black or all white. And (a) is adjacent above to a horizontal black line and below a horizontal white line, respectively.
P22: This is a mixed type of P17 and P6. Indeed, the pattern consists of (a) P17 with finite height and (b) horizontal lines of all black or all white. And (a) is adjacent above to a horizontal white line and below a white or black horizontal line, respectively.
P23: This is a mixed type of P18 and P5. Indeed, the pattern consists of (a) P18 with finite width and (b) vertical lines of all black or all white. And (a) is adjacent to vertical lines.
P24: Checkerboard. P25:
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2. WEN-GUEI HU, SONG-SUN LIN. 2009. ZETA FUNCTIONS FOR HIGHER-DIMENSIONAL SHIFTS OF FINITE TYPE. International Journal of Bifurcation and Chaos 19:11, 3671-3689. [Abstract] [References] [PDF] [PDF Plus]
3. JUNG-CHAO BAN, CHIH-HUNG CHANG. 2008. ON THE DENSE ENTROPY OF TWO-DIMENSIONAL INHOMOGENEOUS CELLULAR NEURAL NETWORKS. International Journal of Bifurcation and Chaos 18:11, 3221-3231. [Abstract] [References] [PDF] [PDF Plus]
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International Journal of Bifurcation and Chaos 18:02, 375-390. [Abstract] [References] [PDF] [PDF Plus]
5. YUN-QUAN KE, FENG-YAN ZHOU. 2006. EXISTENCE ANALYSIS OF MOSAIC SOLUTIONS FOR ONE-DIMENSIONAL CELLULAR NEURAL NETWORKS. International Journal of Bifurcation and Chaos 16:12, 3669-3677. [Abstract] [References] [PDF] [PDF Plus]
6. Zbigniew Galias, Maciej OgorzaŁek. 2006. Design of Coupled Nonlinear Systems for Storage of Prescribed Binary Patterns.
Nonlinear Dynamics 44:1-4, 63-72. [CrossRef]
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