Construction of recurrent fractal interpolation surfaces(rfiss) on rectangular grids

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1 Construction of recurrent fractal interpolation surfacesrfiss on rectangular grids Wolfgang Metzler a,, Chol Hui Yun b a Faculty of Mathematics University of Kassel, Kassel, F. R. Germany b Faculty of Mathematics and Mechanics Kim Il Sung University, Pyongyang, D. P. R. Korea March, 2008 Abstract A recurrent iterated function system RIFS is a genaralization of an IFS and provides nonself-affine fractal sets which are closer to natural objects. In general, it s attractor is not a continuous surface in R 3. A recurrent fractal interpolation surface RFIS is an attractor of RIFS which is a graph of bivariate continuous interpolation function. We introduce a general method of generating recurrent interpolation surface which are attractors of RIFSs about any data set on a grid. Keywords: Recurrent Iterated function systemrifs; Fractal interpolation functionfif; Bo-counting dimension 1 Introduction Fractal interpolation surfaces FISs are graphs of bivariate fractal interpolation functions FIFs, which have been used in approimation theory, computer graphics, image compression, metallurgy, physics, geography, geology and so on. Barnsley 1986, [3] introduced the idea of a FIF as a function whose graph is an attractor of IFS, which by many scientists Massopust, Elton, Hardin, Geronimo, Zhao, Malysz, Bouboulis etc. has widely been studied and applied. Massopust 1990, [16] presents the construction of self-affine FISs on triangular data sets, where the interpolation points on the boundary data are coplanar, which by Geronimo and Hardin 1993, [13] was generalized to allow more general boundary data and by Zhao 1996, [18] more general vertical scaling factor. Dalla 2002, [10] introduced the construction of a FIS by an IFS in the case where the interpolation points on the boundary data sets on a grid are collinear and Malysz 2006, [15] generalized this method to allow more general data set on the grid, where the free contrativity factor is constant. The FISs generated by the above construction are all self-affine. By Metzler and Yun 2008, [17], the construction of FISs with a free vertical contractivity factor function on grids was presented which generalizes the results in [15]. Corresponding author. address: metzler@mathematik.uni-kassel.de 1

2 There are no objects in nature which have an eact structure of a self-affine set, which is nothing but special model of a fractal for modelling natural objects. That requires more fleible constructions of FISs, for which a recurrent iterated function system RIFS is a good way. Bouboulis etc. 2006, [6], 2007, [8] present more general constructions of nonself-affine FISs on grids by RIFS and apply to image compression, which has a lack because the free vertical constractivity factor is still there constant. In this paper, we introduce a more fleible construction of recurrent fractal interpolation surfaces which are attractors of RIFSs with a free vertical contractivity factor function on grids and consider the construction in R N. 2 Recurrent fractal interpolation surfaces RFISs on rectangular grid In general, a recurrent iterated function system RIFS is defined as a pair of a collection w 1,..., w N of ipschitz mappings in complete metric space i.e. an IFS and a row irreducible stochastic matri P = p ij N N satisfying N j=1 p ij = 1 for all i {1,..., N}, and for any i, j {1,..., N}, there eist i 1,..., i k such that p ii1 p i1i 2... p ik j > 0 Barnsley, Elton and Hardin[4]. The attractor A of a RIFS is computed as follows: By a stochastic matri P, for an initial k 0 {1,..., N} a sequence {k i } 0 is given, which obeys p ab = Pr {k i+1 = b k i = a}. With this, we get a sequence of transformations {w ki } 0 that generates an orbit {q i } 0 for an initial q 0 R 3, that is, k i+1 is chosen with the probability that k i+1 = j equals to p kij and then we have q i+1 = w ki+1 q i. In the matri p kl, p kl shows the possibility of applying the transformation w l to the point in state k, so that the system transits to state l. The attractor A is defined as a limit set of these orbits, which consists of the points whose every neighbourhood contains infinitely many q i for almost all orbits. By Barnsley, Elton and Hardin [4] the eistance, uniqueness and characteristics of this limit set A was proved. Usually, a RIFS is constructed with a given data set in a complete metric space and it s attractor generally is not a graph of a continuous interpolaion function about the data set. The attractor A which is a graph of a bivariate continuous fractal interpolation function of a given data set in R 3 is called a recurrent fractal interpolation surface RFIS. We present a construction of RIFS, whose attractor is a graph of fractal interpolation function of a given data set on a grid. To do this, an IFS and an irreducible row stochastic matri P should be defined. 2.1 ocal IFSs Because an attractor of RIFS depends on the IFS and only a connection matri C = c kl N N defined by the stochastic matri P as follows: { 1 : plk > 0 c kl = 0 : p lk = 0, 1 we consider the IFS and the connection matri C. We construct a local IFS { R 3 ; w ij = ij, F ij, i = 1,..., m, j = 1,..., n } over the grid, whose general definition is as follows [2]An idea of such IFS was introduced by Jacquin [14]. Definition 1 et X, d be a compact metric space. et R be a nonempty subset of X. et w : R X and let s be a real number with 0 s < 1. If d w, w y s d, y for all, y in R, 2

3 then w is called a local contraction mapping on X, d. The number s is a contractivity factor for w. Definition 2 et X, d be a compact metric space, and let w i : R i X be a local contraction mapping on X, d, with contractivity factor s i, for i = 1, 2,..., N, where N is a finite positive integer. Then {w i : R i X : i = 1, 2,..., N} is called a local iterated function system local IFS or partitioned IFS [12]. The number s = ma {s i : i = 1, 2,..., N} is called the contractivity factor of the local IFS. et the data set on the rectangular grid be S = { i,, z ij R 3 ; i = 0, 1,..., m, j = 0, 1,..., n }, such that 0 < 1 <... < m, y 0 < y 1 <... < y n. et denote N mn = {1,..., m} {1,..., n}, I = [ 0, m ], I y = [y 0, y n ], I i = [ i 1, i ], I yj = [ 1, ], E = I I y E ij = I i I yj which we call the region, for i, j N mn, P α = { α, y l, z αl S; l = 0, 1,..., n }, for α {0,..., m}, P yβ = { k, y β, z kβ S; k = 0, 1,..., m }, for β {0,..., n}. ] et choose an interval Ĩ i to be [, = m i ii 1 ii k=1 I k for i {1,..., m} then, there eist k1, k2 [0, m] such that i i 1 = k1 and i i = k2, which we denote by γ i i 1 and γ ii respectively such that i For any i {1,..., m}, i i 1 < i i i i 1 ii For any i {2,..., m 1}, there eist i 1 α { i 1 i 2, i 1 i 1 }, i+1 β { i+1 i, i+1 i+1 } such that i 1 α = i i 1 and i i = i+1 β, which we denote these points - the endpoints of connecting the intervals Ĩ i - by i 1,i and i,i+1 respectively. An interval Ĩ has the same form for j {1,..., n}. We denote Ẽij = Ĩ i Ĩ and call Ẽij the domain. We define the domain contraction transformations ij : Ẽij E ij, for i, j N mn by ij, y = i, yj y, where i : Ĩ i I i, yj : Ĩ I yj are contractive homeomorphisms with contractivity factors a i, a yj obeying i For any i {1,..., m}, j {1,..., n}, } } i : { ii 1, ii { i 1, i }, yj : {y jj 1, y jj { 1, }, 3

4 I 1 I 2 I 3 I ~ I ~ I ~ I 2 1 ~ I 1 Figure 1: Shape of i ii For any i {1,..., m 1}, j {1,..., n 1}, there eist i,i+1, ỹ j,j+1 such that i+1 i,i+1 = i i,i+1 = i, yj+1 ỹ j,j+1 = yj ỹ j,j+1 = See Figure 1. Denote a ij = Ma { } a i, a yj, for i, j Nmn. Then, the a ij are contractivity factors of the transformations ij. et F ij : Ẽij R R, for i, j N mn be defined by 1 F ij, y, z = d ij, y z g, y + h ij, y, 3 where d, y is a vertical continuous contraction such that d, y < 1 on E, h, y and g, y are continuous ipschitz mappings on E with the ipschitz constants h, g satisfying g iα, y jβ = z γiα,j β, for α, β {i 1, i} {j 1, j}, h i, = z ij, for i, j N mn, where γ i α, j β = γ γ iα, γy y = γ jβ k, y l = k, l, that is, g goes through 4 endpoints of Ẽij, h all data points of E. Then, the F ij satisfy join up conditions, for α {i 1, i}, β {j 1, j}, F ij i α, y j β, z γiα,j β = z σ ij iα,y j β, 4

5 y j j Domain E ~ ij y j j 1 W ij y j y j 1 i i 1 ii i 1 Region E ij i Figure 2: Mapping the domain to the region where σ ij, y = σ iα jβ a, y b = a, b {i 1, i} {j 1, j}. By 2, 3 we have on the common borders { i } [ 1, ] for i, j { 1,..., m 1} {1,..., n} [ i 1, i ] { } for i, j {1,..., m} {1,..., n 1} F i+1 j i,i+1, y, z = F ij i,i+1, y, z F i j+1, ỹ j,j+1, z = F ij, ỹ j,j+1, z, where i,i+1, i, ỹ j,j+1, obey 2. Hence, for i, j N mn the transformations w ij coincide on common borders. Fourthermore, there eists some metric ρ that is equivalent to the Euclidean metric on R 3 such that the w ij are contractions for all i, j N mn with respect to ρ, which is given on R 3 for, y, z,, y, z R 3 by where θ = ρ, y, z,, y, z = + y y + θ z z, 1 Ma { a i, a yj ; i = 1,..., m, j = 1,..., n } 2Ma { d ma g + h a i, d ma g + h a yj ; i = 1,..., m, j = 1,..., n } and d ma = Ma E d, y, d min = Min E d, y. A contractivity of w ij is Ma {a, d ma }, where a = 1 + Ma { a i, a yj ; i = 1,..., m, j = 1,..., n } < 1. 2 Remark 1. To be simple, let the endpoints of interval [ 0, m ] be 0 and 1. If Ẽij = E for any i, j N mn, then the above i can take two ways of corresponding endpoints of 5

6 Figure 3: Shapes of 1 i, 2 i intervals as follows: { 1 i 1+α i : odd i α = i 1+1 α i : even or { 2 i 1+1 α i : odd i α = i 1+α i : even for α {0, 1}. 1, 2 are the same forms. Thus, for ij there are 4 casessee Figure 3: 1 i, 1, 1 i, 2, 2 i, 1, 2 i, 2. In the paper[15], ij has the form 1 i, 1, where 1 i, 1 are linear mappings. Remark 2. d ij, y is the vertical contraction factor function on the region E ij. In 3, d ij, y can be replaced by d u i, v y, where u i = { θ i1 i1 θiu iu u Z + u = 0 4 and θ ik { 1, 1}, i k {1,..., m}, k = 1,..., u. v y is of the same form. This can improve more fleibility of the IFS, but normally the transformations w ij, can not coincide on common borders, ecepting the case where in 3 d is given by d i, yj y and d, y from Row stochastic matri P We enumerate the set N nm = {1,..., m} {1,..., n} by a injective mapping τ : {1,..., m} {1,..., n} {1,..., m n} and denote M = τ N mn. For simplicity, we denote i, j = τ 1 k by k and m n by N. 6

7 E 1 E 2 C= E E E E 4 E E 3 Figure 4: Connection matri and it s directed graph On the basis of the above construction of local IFS, we define the connection matri C = c kl N N as follows: c kl = { 1 : El Ẽk 0 : otherwise The connection matri C shows that the transformation w k can follow the transformation w l iff c kl = 1. Because this matri C hase to be irreducible, in the above construction of the IFS, the relation between the domain Ẽi and the region E j for i, j {1,..., N} should be taken so that this condition is satisfied. For eample, for any i {2,..., N}, E i 1 Ẽi or E i Ẽi 1 and E N Ẽ1 or E 1 ẼN See Figure 4. Then we have a row irreducible stochastic matri P = p kl N N by 1. A pair consisting of the above local IFS and the row irreducible stochastic matri P is the RIFS. The eistence, uniqueness and characteristic of the attractor A of this RIFS has been proved by Barnsley Elton and Hardin [4]. 2.3 Attractor of RIFS In this section, we review some of the work of Barnsley, Elton and Hardin [4] on the attractor A of RIFS. et K i, d i be compact metric spaces for i {1,..., N}, H i, h i denote the associated metric spaces of nonempty compact subsets which use the Hausdorff metrics and H N = H 1 H N. The transformation W : H N H N is defined by c 11 w 1 c 12 w 1... c 1N w 1 c 21 w 2 c 22 w 2... c 2N w 2 W =......, c N1 w N c N2 w N... c NN w N 7

8 for B = B 1,..., B N H N, c 11 w 1 B 1 c 12 w 1 B 2... c 1N w 1 B N c 21 w 2 B 1 c 22 w 2 B 2... c 2N w 2 B N W B =. c N1 w N B 1 c N2 w N B 2... c NN w N B N j Λ1 w 1 B j j Λ2 w 2 B j =., j ΛN w N B j where Λ i = {j : c ij = 1} for i {1,..., N}. For eample, in the case where P = C = w1 0, W =, w 2 w , W B = w1 B 1 w 2 B 1 w 2 B 2 Then, an unique invariant set A of this transformation W eists, which is called an attractor of the RIFS : W A = A i.e. there eist the unique nonempty compact sets A 1,..., A N such that A i = w i A k, k Λi. A = N A i = i=1 k Λi 2.4 Recurrent fractal interpolation surfaces w i A k. 5 The following theorem shows that this attractor A is a recurrent fractal interpolation surface. Theorem 1 et the set A be the attractor of the above RIFS of data set S. Then, there eists a continuous interpolation function of data set S whose graph is the attractor A. Proof et denote C E = { ϕ C 0 E : ϕ i, = z ij, i = 0,..., m, j = 0, 1..., n }, F E = {f f : E R}. Defining an operator T : C E F E by T ϕ, y = F ij 1 i, 1 y, ϕ 1 i, 1 y, for, y E ij, it can be easily proved that the operator T has the following properties: 8

9 Table 1: The interpolation points of RIFS. y Table 2: The interpolation points of the free vertical contractivity function d, y. y i T ϕ C E ii T is contractive in the sup-norm with contractivity factor d ma. Thus, according to the fied point theorem in the complete metric space C E, the operator T has a unique fied point f C E with f i, y = F i, y, f, y, for, y Ẽi, i {1,..., N}. This means that Gr f = N i=1 k Λi w i Gr f Ek. 6 by 5, 6, we have Gr f = A. Eample. Figure 5 shows the RFIS of the data set in Table 1 with the vertical contractivity factor function d, y which is given by a polynomial interpolation of a data set in Table 2. The transformation i, yj are linear one, the function g, h are defined by the polynomial interpolation. Here d, y, g, h can be defined by the fractal interpolation. 3 Bo-counting dimension of RFIS In general, it is not easy to calculate a fractal dimension of the attractor of a RIFS. In the case where the attractor is a RFIS, we estimate the bo-counting dimension of a RFIS, which is denoted by dim B A. 9

10 RFIS2.nb Figure 5: RFIS of data set in Table 1 with the vertical contractivity factor function d, y which is a polynomial interpolation function of a data set in Table 2 Theorem 2 In th section 2, let the transformation i be similitude with scaling a for i {1,..., N}, d, y = d 0 and A be the RFIS. If there eist k {1,..., N} and α {0,..., m} or β {0,..., n} such that the points of P α Ẽk or P yβ Ẽk are noncolinear, and then dim B A is the unique D such that ρ diag d 0 a C > 1, ρ diag d 0 a D 1 C = 1, otherwise dim B A = 2. Here ρ B denotes the spectral radius of the matri B. Proof The proof is similar to Theorem 4.2 in [4], THEOREM 4.1 in [13]. 4 Construction in R N We consider the construction of RIFSs with the data set on grid in R N. The idea is the same as that in R 3. Therefore, we present the results. et the data set be denoted by S = { 1,i1, 2,i2,..., M,iM, z i1,i 2,...,i M R M+1 ; i k = 0, 1,..., m k, m k N, k = 1,..., M }, where k,0 < k,1 <... < k,mk for k {1,..., M}, M, m k N, and denote P k,ik = { 1,i1,..., k,ik,..., M,iM, z i1,...,i k,...,i M P; i l = 0,..., m l, l = 1,..., k 1, k + 1,..., M}, 10

11 for k {1,..., M}, i k {0,..., m k }. We denote I k = [ k,0, k,mk ], I k,l = [ k,l 1, k,l ], E i = I 1,i1... I M,iM which we call the region, Ω = {1, i 1,..., M, i M ; i k = 1,..., m k, k = 1,..., M}, 7 where i Ω, i k {1,..., m k }, k {1,..., M}. Then E = I 1... I M = i E i, I k = m k ] l=1 I k,l. et choose an interval Ĩk,l to be [, = k,ll 1 k,ll j I k,j for k {1,..., M} then there eist k,l1, k,l2 [0, m k ] such that k,l l 1 = k,l1 and k,l l = k,l2, which we denote by γ k k,l l 1 and γ k k,l l respectively such that i For any l {1,..., m k }, k,l k,l 1 < k,l l k,l l 1 } ii For any l {2,..., m k 1}, there eist k,l 1 α {,, k,l 1l 2 k,l 1l 1 k,l+1 β } {, such that k,l+1l k,l+1l+1 k,l 1 α = k,l l 1 and k,l l = k,l+1 β i.e. the endpoints of connecting the intervals Ĩk,l, which we denote by k,l 1,l and k,l,l+1 respectively, generally k,j,j+1 for j {1,..., m k 1} which connects the interval Ĩk,j and the interval Ĩ k,j+1. We denote Ẽi = Ĩ1,i 1... ĨM,i M for i Ω, which we call the domain. Then Ẽi contains some of regions E i for i Ω. We construct a local IFS { R M+1 ; W i = i, F i ; i Ω }. The contraction transformations from the domain to the region i : Ẽi E i with contractivity factors a i are defined by i 1,..., M = 1,i1 1,..., M,iM M, where k,ik : Ĩk,i k I k,ik, for i k {1,..., m k }, k {1,..., M} are contractive homeomorphisms with the contractivity factors a k,ik satisfying } i k,ik : {, k,ikik 1 { k,ikik k,ik 1, k,ik }, i k {1,..., m k }, ii For any k,ik { k,1,..., k,mk 1}, there eist k,ikik +1 such that k,ik +1 k,ikik = +1 k,ik k,ikik = +1 k,ik, 8 and a i = Ma {a 1,i1,..., a M,iM }. We define the vertical contraction functions F i : Ẽi R R by F i, z = d i z g + h i, for, z Ẽi R, 9 where d obeys d < 1 and g, h are continuous ipschitz mappings on E with ipschitz constants h, g satisfying, for α 1,..., α M {i 1 1, i 1 }... {i M 1, i M }, i 1,..., i M {0,..., m 1 }... {0,..., m M }, g 1,i1α1,..., M,iM αm = z γi1α1,...,i M αm, h 1,i1,..., M,iM = z i1,...,i M, 11

12 where γ i 1α1,..., i M αm = γ γ 1,..., γ 1,i1α1 M M,i M αm = 1,β1,..., M,βM =β 1,..., β M. Then, the F i satisfy join-up conditions F i 1,i 1α1,..., M,i M αm, z γi1α1,...,i M αm = z, σ i 1,i,..., 1α1 M,i M αm where for α 1,..., α M {i 1 1, i 1 }... {i M 1, i M }, σ i,..., 1,i1α1 M,iM αm = 1,j1,..., M,jM = j 1,..., j M {i 1 1, i 1 }... {i M 1, i M }. Consequently, W i are contractive transformations for all i Ω with respect to some metric which is equivalent to Euclidean metric on R M+1. et m 1... m M be denoted by m 1 M and enumerate the set Ω by a injective mapping τ : Ω {1,..., m 1 M }. We defined a connection matri C = c kl m1 M by c kl = { 1 : Eτ 1 l Ẽτ 1 k 0 : otherwise. In the construction of local IFS, the relation between E i and Ẽi so that the connection matri defined above should be irreducible. Then we get a RIFS, whose attractor A is a graph of continuous fractal interpolation function of the data set S. The following theorem gives the Bo-counting dimension of the attractor A. Theorem 3 et τ 1 i be similitude with scaling a for i {1,..., m 1 M }, d, y = d 0 and A be the RIFS. If there eist k {1,..., m 1 M } and i α {0,..., m α } such that the points of P α,iα Ẽτ 1 k are not in M-1 dimension space and then dim B A is the unique D such that otherwise dim B A = M. References ρ diag d 0 a C > 1, ρ diag d 0 a D 1 C = 1, [1] M.F. Barnsley, Fractals Everywhere, Academic Press, New York, [2] M.F. Barnsley,.P. Hurd, Fractal Image Compression, AK Peters, Wellesley, [3] M.F. Barnsley, Fractal function and interpolation, Constr. Appro [4] M.F. Barnsley, J.H. Elton, D.P. Hardin, Recurrent iterated function systems, Constr. Appro [5] M.F. Barnsley, S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. Soc. ondon Ser. A

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