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Axioms
, Volume 9 (4) – Oct 18, 2020

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axioms Article On the Bernstein Afﬁne Fractal Interpolation Curved Lines and Surfaces 1,† 2, ,† Nallapu Vijender and Vasileios Drakopoulos * Department of Mathematics, Visvesvaraya National Institute of Technology Nagpur, Nagpur 440006, India; vijendernallapu@gmail.com Department of Computer Science and Biomedical Informatics, University of Thessaly, 35131 Lamia, Greece * Correspondence: vdrakop@uth.gr; Tel.: +30-22310-6-6722 † These authors contributed equally to this work. Received: 31 August 2020; Accepted: 9 October 2020; Published: 18 October 2020 Abstract: In this article, ﬁrstly, an overview of afﬁne fractal interpolation functions using a suitable iterated function system is presented and, secondly, the construction of Bernstein afﬁne fractal interpolation functions in two and three dimensions is introduced. Moreover, the convergence of the proposed Bernstein afﬁne fractal interpolation functions towards the data generating function does not require any condition on the scaling factors. Consequently, the proposed Bernstein afﬁne fractal interpolation functions possess irregularity at any stage of convergence towards the data generating function. Keywords: attractor; Bernstein polynomial; bivariate surfaces; dynamic system; fractal interpolation; iterated function system 1. Introduction Classic interpolation techniques ﬁt an elementary function to the given data in order to render a connected visualisation of a sample. Such elementary functions often imbue the visualisation with a degree of smoothness that may not be consistent with the nature of a prescribed data set. Utilising the theory of an iterated function system ﬁrstly presented in [1] and popularised in [2,3] the concept of a fractal interpolation function was proposed whose graph is the attractor, a fractal set, of an appropriately chosen iterated function system. If this graph has a Hausdorff–Besicovitch dimension between 1 and 2, the resulting attractor is called fractal interpolation curved line or fractal interpolation curve. If this graph has a Hausdorff–Besicovitch dimension between 2 and 3, the resulting attractor is called fractal interpolation surface. In general, fractal interpolation functions arise as ﬁxed points of the Read-Bajraktarevic ´ operator deﬁned on suitable function spaces. Using fractal interpolation methodology, it is possible to construct interpolants, i.e. functions used to generate interpolation, with integer and non-integer dimensions. Fractal interpolation functions have been applied in order to prevent inappropriate smoothing; for instance, see [4]. Various types of fractal interpolation functions have been constructed and some signiﬁcant properties of them, including calculus, dimension, smoothness, stability, perturbation error, etc., have been widely studied (see [5–7]). For real-time applications of FIFs, one may refer [8–11]. This article mainly focuses on afﬁne fractal interpolation and its useful aspects and can be considered complementary to [12,13] in many ways. Firstly, we are discussing a simple procedure for ﬁnding the box-counting dimension of afﬁne fractal interpolation functions studied in [3]. Secondly, for a prescribed set of data points, there exist an inﬁnite number of afﬁne fractal interpolation functions. We discuss the existence of an optimal afﬁne fractal interpolation function close to a traditional (classical) interpolant studied in [14]. Thirdly, for given interpolation data, by exploiting fractal Axioms 2020, 9, 119; doi:10.3390/axioms9040119 www.mdpi.com/journal/axioms Axioms 2020, 9, 119 2 of 14 interpolation theory and classical Bernstein polynomial, we construct a sequence of Bernstein afﬁne fractal interpolation functions in one and two variables that uniformly converges to the data generating function for any choice of the scaling factors. In Navascués [15] approach, afﬁne fractal interpolation functions converge to the data generating function, if the magnitude of the corresponding scaling factors goes to zero, whereas in our approach convergence of Bernstein afﬁne fractal interpolation functions does require any condition on the scaling factors. Particularly, in Section 2 we brieﬂy review the theory of iterated function systems. In Section 3, we revisit the fractal interpolation theory and state the prerequisites of the main construction. In Section 4, the afﬁne FIFs are deﬁned and constructed. In Section 5, we introduce the construction of Bernstein afﬁnefractal interpolation functions and study their convergence. The construction of Bernstein afﬁne fractal interpolation surface and its convergence are carried out in Section 6. Finally, Sections 7 and 8 summarize our conclusions and points out areas of future work. 2. Iterated Function System and Scaling The following notation and terminologies will be used throughout the article. The set of real numbers will be denoted by R, whilst the set of natural numbers by N. For a ﬁxed N 2 N, we shall write N for the set of the ﬁrst N natural numbers. Let (X , d ) be a complete metric space and H(X ) N X be the set of all nonempty compact subsets of X . Subsequently, H(X ) is a complete metric space with respect to the Hausdorff metric h, where h is deﬁned as h( A, B) = maxfd ( A, B), d (B, A)g and X X d ( A, B) = max min d (x, y). X X x2 A y2B Let w : X ! X be continuous functions for n 2 N . The collection I = fX ; w , n 2 N g is called n N n N iterated function system or IFS for short. An IFS I is called hyperbolic, if each w , n 2 N is a contraction with corresponding contractivity n N factor a for n 2 N , i.e. ja j k < 1. For any A 2 H(X ), the set valued Hutchinson operator W on n N n H(X ) is deﬁned as W(B) = w (B), B 2 H(X ). n=1 If the IFS I is hyperbolic, then it is well known that W is a contraction map on H(X ) with contractivity factor jaj = maxfja j : n 2 N g. Then by the Banach Fixed Point Theorem, there exists a ﬁxed point ¥ n A 2 H(X ) of W, i.e., W( A) = A. Deﬁnition 1. The ﬁxed point of the Hutchinson operator is called the attractor or deterministic fractal for the IFS I . The ﬁxed point A 2 H(X ) is some times called the invariant or self-referential set of I . In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set S in a Euclidean space R , or more generally in a metric space (X, d). Another notion dealing with the measurement of fractals is the fractal derivative or Hausdorff derivative, which is a non-Newtonian generalisation of the derivative. Fractal derivatives were created for the study of anomalous diffusion, by which traditional approaches fail to factor in the fractal nature of the media. A fractal measure t is scaled according to t . Such a derivative is local, in contrast to the similarly applied fractional derivative. When we study a problem, the scale used is of great importance. This observation leads to a two-scale transformation to convert approximately a fractal space to a continuous partner. The two scale transform, for example, in x-direction, is s = x , where x is for the small scale and s for large scale, a the two-scale dimensions. Using the two-scale transform, the fractional differential equations can be converted into traditional differential ones, which are easy to be solved; see also [16]. Axioms 2020, 9, 119 3 of 14 3. Fractal Interpolation Functions Let x < x < x < x < x , where N > 1, be a partition of the closed interval I = [x , x ], 0 1 2 N1 N 0 N and z , z , . . . , z be a collection of real numbers. Let L , n 2 N be a set of homeomorphisms from I 0 1 N n N to I = [x , x ] satisfying n n n1 L (x ) = x , L (x ) = x . (1) n 0 n1 n N n Let F be a function from I K to K, where K is a suitable compact subset of R), which is continuous in the x-direction and contractive in the y-direction with contractivity or vertical scaling factor ja j k < 1, such that F (x , z ) = z , F (x , z ) = z , n 2 N . (2) n 0 0 n1 n N N n N In the existing constructions, the maps L and F are deﬁned as n n L (x) = a x + b , F (x, z) = a z + q (x), n 2 N , (3) n n n n n n N where q : I ! R are suitable continuous functions such that (2) is satisﬁed. Let G = fg : I ! R j g is continuous, g(x ) = z and g(x ) = z g. We deﬁne a metric on G by 0 0 N N r(h, g) = maxfjh(x) g(x)j : x 2 Ig for h, g 2 G . Then (G , r) is a complete metric space. Deﬁne the Read-Bajraktarevic ´ operator T on (G , r) by 1 1 T g(x) = F (L (x), g L (x)), x 2 I . (4) a n n n n Using the properties of L and (1)-(2), T g is continuous on the interval I for n 2 N , and at each of n a n N the points x , x , . . . , x . Also, 0 1 N r(T g, T h) jaj r(g, h), a a ¥ where jaj = maxfja j : n 2 N g < 1. Hence, T is a contraction mapping on the complete metric ¥ n N a space (G , r). Therefore, by the Banach ﬁxed point theorem, T possesses a unique ﬁxed point, let’s say f , on G , i.e., (T f )(x) = f (x) for all x 2 I. According to (4), the function f satisﬁes the a a a a functional equation 1 1 f (x) = F (L (x), f L (x)), x 2 I , n 2 N . (5) a n a n N n n Further, using (1)-(2), it is easy to verify that f (x ) = z , i 2 N . By deﬁning a mapping w : I K ! a n i i I K as w (x, z) = (L (x), F (x, z)), (x, z) 2 I K, n 2 N , the graph G( f ) of f satisﬁes n n n n N a a G( f ) = w (G( f )), a n a n2N whereas f is called fractal interpolation function or FIF for short corresponding to the IFS I = f I K, w (x, y) = (L (x), F (x, y)), n 2 N g. n n n Remark 1. The main differences of a fractal interpolant with a traditional interpolant include: (i) the construction via IFS theory that offers a functional equation for the interpolant and it implies a self similarity in small scales; (ii) the construction by iteration of the interpolant instead of using an analytic formula; and, (iii) the usage of scaling factors, which offers ﬂexibility in the choice of interpolant in contrast to the unicity of a speciﬁc traditional interpolant. 4. Afﬁne Fif L (x)x 1 1 0 For n 2 N , if q (x) = qc + (1 q)d , q = , x 2 I in (3), then f is called an afﬁne FIF n n a N n n x x N 0 and it is expressed as L (x) x 1 1 1 n f (x) = a f (L (x)) + c q + d (1 q), q = , x 2 I . (6) a n a n n n n x x N 0 Axioms 2020, 9, 119 4 of 14 1 1 Taking x = x and x = x in (6), we get d = z a z and c = z a z respectively with help n1 n n1 n 0 n n N n n of (2). Finally the afﬁne FIF f takes the following form: For n 2 N , f (x) = a f (L (x)) + (z a z )q + (z a z )(1 q), x 2 I . (7) a n a n n N n1 n 0 n Barnsley [3] studied the box-counting dimension of an afﬁne FIF, where its details are given in the following proposition. Proposition 1. Let f(x , z ), i 2 N g be a set of data ponts. Then the graph G of the afﬁne FIF corresponding i i to the given data points has box-counting dimension D, if ja j > 1 and data points are noncollinear G å dim = (8) n=1 1, otherwise, D1 where D is the solution of ja ja = 1. å n n=1 Example 1. Consider the interpolation data f(1, 1.4), (0, 12.8), (1.7, 28.9), (8, 31), (13.3, 44.5)g. From (7), it is clear that afﬁne FIF is recursive. Hence, one has to use iterative procedure to evaluate the afﬁne FIF at different points of [1, 13.3]. For the above data, after ﬁrst, second, and sixth iteration, afﬁne FIF with the scaling factors a = a = a = 0.9, a = 0.9 is generated respectively in Figure 1a–c. From Figure 1a–c, it is 1 2 3 4 clear that to obtain the values of afﬁne FIF at more points of [1, 13.3], one has to use more number of iterations. Similarly, some more afﬁne FIFs are generated after sixth iteration in Figure 1d–f using different choices of the scaling factors as mentioned in the respective ﬁgure. 4.1. Inscribing Afﬁne Fif in a Rectangle In most of the applications, for instance fractal-based image encoding and compression, we need to interpolate the given data within a given rectangle. The sufﬁcient conditions on the scaling factors which ensure that afﬁne FIF sits within in the given rectangle are studied in [17,18]. The following proposition provides the details. Proposition 2. Let f be an afﬁne FIF associated with the data (x , z ), i 2 N . Let k < minfz : i 2 N g a i i 1 i N N and k > maxfz : i 2 N g. Then graph of afﬁne FIF f contained in the rectangle [x , x ] [k , k ], if the 2 i 0 N 1 2 scaling factors are chosen in the following way: a 2 (t , t ), n 2 N (9) n n n N n o z k z k k z k z n1 1 n 1 2 n1 2 n where t = min 1, , , , , z k z k k z k z 0 1 N 1 2 0 2 N n o (z k ) (z k ) (k z ) (k z ) n n n1 1 1 2 n1 2 t = max 1, , , , . k z k z z k z k 2 0 2 N 0 1 N 1 Axioms 2020, 9, 119 5 of 14 50 60 0 −10 −2 0 2 4 6 8 10 12 14 −2 0 2 4 6 8 10 12 14 (a) An AFIF after ﬁrst iteration with (b) An AFIF after second iteration with a = a = a = 0.9, a = 0.9. a = a = a = 0.9, a = 0.9. 1 2 3 4 1 2 3 4 -50 -50 -2 0 2 4 6 8 10 12 14 -2 0 2 4 6 8 10 12 14 (d) An AFIF after sixth iteration with (c) An AFIF after sixth iteration with n+1 a = (1) 0.9, n = 1, 2, 3, 4. a = a = a = 0.9, a = 0.9. 1 2 3 4 n 100 45 50 30 0 15 -50 0 -2 0 2 4 6 8 10 12 14 -2 0 2 4 6 8 10 12 14 (e) An AFIF after sixth iteration with (f) An AFIF after sixth iteration with n n+1 a = (1) 0.9, n = 1, 2, 3, 4. a = (1) 0.1, n = 1, 2, 3, 4. n n Figure 1. Afﬁne FIFs. Example 2. We now illustrate the importance of Proposition 2 by constructing the examples of afﬁne FIFs that interpolate the data set f(3, 4), (6, 1), (11, 9), (15, 2)g. Suppose that for some reason it is required to inscribe the graph of the interpolant in the rectangle [2, 16] [0, 11]. To obtain it, with respect to (9), the scaling vector is chosen as a = (0.27, 0.6, 0.55,0.5). The corresponding afﬁne FIF inscribed in the rectangle Axioms 2020, 9, 119 6 of 14 [2, 16] [0, 11] is generated in Figure 2a. Note in particular that the interpolating curve is non-negative, thus solving a constrained interpolation problem for the given data. Similarly, another afﬁne FIF is generated in Figure 2b with the scaling vector a = (0.8, 0.6, 0.6, 0.6). It is easy to verify that scaling vector a = (0.8, 0.6, 0.6, 0.6) does not obey (9). As a result, the afﬁne FIF generated in Figure 2b is not inscribed in the rectangle [2, 16] [0, 11] and also afﬁne FIF taking negative values although the given data is positive. From this experiment, we can understand imporatnce of (9) for obtaining the afﬁne FIFs inscribed in the given rectangle. 16 20 10 10 4 0 -5 -2 -10 -5 0 5 10 15 20 -5 0 5 10 15 20 (a) Afﬁne FIF inscribed in the rectangle (b) Afﬁne FIF not inscribed in the rectangle [2, 16] [0, 11]. [2, 16] [0, 11]. Figure 2. Afﬁne FIFs inscribed in a rectangle. 4.2. Existence of Optimal Afﬁne Fif It can be easily seen from (6) that the afﬁne FIF depends on the choice of the vertical scaling factors and there are inﬁnite number of ways to select them. As a result, there exist an inﬁnite number of afﬁne FIFs corresponding to a given set of data. Hence, one can ask whether an optimal afﬁne FIF exists. As an answer to this question, it is proved in [14] that there exists an afﬁne FIF close to some classical interpolant f that interpolates the data points (x , z ), i 2 N . Now, it is i i easy to verify that the afﬁne FIF f is the ﬁxed point of the Read-Bajraktarevic ´ operator (T h)(x) = a a g(x) + a [h(x) b(x)] L (x), x 2 I , where g is the piecewise linear function that interpolates n n (x , z ), i 2 N , and b is the line joining (x , z ) and (x , z ). Further, jaj is contractivity factor of T . i i 0 0 N N ¥ a Let us recall the Collage theorem [19], [18] which serves as a prelude to our analysis. Theorem 1 (Collage Theorem). Let f 2 C( I) and U : C( I) ! C( I) be a contraction map with contractivity ˜ ˜ factor c 2 [0, 1). If kU f fk #, then k f fk , where f is ﬁxed point of U. ¥ ¥ 1c The previous theorem states that, if f 2 C( I) is given and a is such that kT f fk #, then a ¥ k f f k = k f T f k . a ¥ a a ¥ 1ja j Owing to the above reason, we have to search for a 2 S = fa 2 R : jaj m < 1g such that k f T fk is minimum. a ¥ Proposition 3. Let f 2 C( I). Then the map h : S ! C( I) deﬁned by h(a) = T is Lipschitz continuous. Consequently, the map F : S ! R [f0g deﬁned by F(a) = kT f fk is continuous. a ¥ Axioms 2020, 9, 119 7 of 14 Proof. Let a, b 2 S. From (4) and (5), we have 1 1 T f = g(x) + a f (L (x)) b(L (x) , x 2 I , a n n n n 1 1 T f = g(x) + b f (L (x)) b(L (x) , x 2 I . n n b n n Consequently, jT f T fj ja b jk f bk . a n n ¥ Hence, we have kT f T fk ja bj k f bk ,ja bj = maxfja b j : n 2 N g. a ¥ ¥ ¥ ¥ n n b N That is,kh(a) h(b)k ja bj k f bk . Therefore, h is a Lipschitz continuous map with Lipschitz ¥ ¥ ¥ constant k f bk . Finally, continuity of F follows from the result that the sum and composition of continuous functions are continuous. Corollary 1. There exists an optimal scaling vector a 2 S for which the function deﬁned by F(a) = kT f fk is minimum. a ¥ Proof. Since the function F : S ! R [f0g is continuous and the set S is compact, the existence of an optimal scaling vector a such that F(a ) = min F(a) = min kT f fk a ¥ a2S a2S follows from the result that a continuous real function on a compact metric space attains its maximum and minimum. Having established the existence, now the following result provides a tool to ﬁnd a . Proposition 4. The function F : S ! R [f0g deﬁned by F(a) = kT f fk is convex. a ¥ Proof. Let a, b 2 S and l 2 [0, 1]. It follows that F((1 l)a + lb) = maxfjT f (x) f (x)j : x 2 Ig (1l)a+lb 1 1 =max maxfj[(1 l)a + lb ] f (L (x)) b(L (x) + g(x) b(x)jg n n n n n x2 I 1 1 (1 l)max maxfja f (L (x)) b(L (x) + g(x) f (x)jg n n x2 I (10) 1 1 + lmax maxfjb f (L (x)) b(L (x) + g(x) f (x)jg n n x2 I =(1 l)kT f fk + lkT f fk a ¥ b ¥ =(1 l)F(a) + lF(b). It is straight forward to see that S is a convex subset of R . Consequently, from the previous proposition, it follows that the problem of ﬁnding a 2 S such that F(a ) = min F(a) is a constrained a2S convex optimization problem. Following the Collage theorem, If a is the optimum scaling vector, F(a ) then the expression provides an upper bound for the uniform distance k f f k , where f is a ¥ 1ja j a classical interpolant and f is the afﬁne FIF close to f . a Axioms 2020, 9, 119 8 of 14 4.3. Convergence of Afﬁne Fif Theorem 2. (Navascués and Sebastián, 2007) Let y 2 C( I). Let f be the afﬁne FIF associated with the data set f(x , z ) 2 R : i 2 N g, where y(x ) = z . Let g be the piecewise linear function that interpolates i i i i xx n1 (x , z ), i 2 N , that is, g(x) = z q + z (1 q), and b(x) = z q + z (1 q), q = , x 2 I . Then n n i i n1 N 0 N x x n n1 jaj k f yk 2w (h) + kg bk , (11) a ¥ y ¥ 1jaj where w (h) is the modulus of continuity of y deﬁned by w (h) = sup jy(x) y(x )j and h is the norm y y jxx jh of the partition deﬁned by h = maxfh : n 2 N g, where h = x x . n n n N n1 Proof. We rewrite (7) in terms of g and b as 1 1 f (x) g(x) = a f (L (x)) a b(L (x)), a n a n n n 1 1 1 1 = a f (L (x)) g(L (x)) + a g(L (x)) a b(L (x)). n a n n n n n n Therefore, for x 2 I , 1 1 1 1 f (x) g(x) a f (L (x)) g(L (x)) + a jg(L (x))j b(L (x)) a n a n n n n n jaj k f gk +jaj kg bk . ¥ a ¥ ¥ ¥ Since the above inequality is valid for each I , n 2 N , we have n N k f gk jaj k f gk +jaj kg bk , a ¥ ¥ a ¥ ¥ ¥ and hence jaj kg bk ¥ ¥ k f gk . (12) a ¥ 1jaj Noting that g(x) y(x) = z q + z (1 q) y(x) n n1 = (z z )q + z y(x) n1 n1 = (z z )q + y(x ) y(x) n n1 n1 = y(x ) y(x ) q + y(x ) y(x). n1 n1 we ﬁnd that kg yk 2w (h). (13) ¥ y Consider the triangle inequality k f yk k f gk +kg yk . (14) a ¥ a ¥ ¥ Combining (13) and (12) with (14), we settle (11). Since y 2 C( I) is uniformly continuous, w (h) ! 0 as h ! 0. Therefore, from Theorem 2, we assert that f converges to y as h ! 0 and jaj ! 0. a ¥ 5. Bernstein Afﬁne Fif Let the data set f(x , z ) 2 R : i 2 N g be obtained from the function y 2 C[x , x ]. Let i i N 1 N h = maxfh : i 2 N g, where h = x x . Let g be the piecewise linear function that interpolates i N1 i i+1 i xx (x , z ), i 2 N , that is, g(x) = z q + z (1 q), q = , x 2 I . In the previous section (Theorem i i N i+1 i i x x i+1 i Axioms 2020, 9, 119 9 of 14 2), it is seen that the afﬁne FIF f associated with the data set f(x , z ) 2 R : i 2 N g converges to a i i N the data generating function if h ! 0 and jaj ! 0. In the present section, we develop a sequence of Bernstein afﬁne FIFs corresponding the data set f(x , z ) 2 R : i 2 N g that converges uniformly to y i i N if h ! 0. In (3), we choose the q as q (x) = g(L (x)) a B (g, x), i i i where B (g, x) is the Bernstein polynomial [20] of g, i.e., 1 n k(x x ) N 1 k nk B (g, x) = (x x ) (x x) g x + ,8x 2 I,8n 2 N. n å 1 N 1 (x x ) k n N 1 k=0 It is easy to verify that B (g, x ) = g(x ), B (g, x ) = g(x ) for all n 2 N. In this case, the afﬁne FIF n 1 1 n N N f = f , n 2 N is called the Bernstein afﬁne FIF corresponding to f(x , z ) 2 R : i 2 N g and a n,a i i N 1 1 f (x) = a f (L (x)) + g(x) a B (g, L (x)), x 2 I , i 2 N . (15) n,a n,a n i i i N1 i i From the construction of fractal functions (see previous section), it can be veriﬁed that for every n 2 N, the Bernstein afﬁne FIF f is obtained via the IFS deﬁned by n,a I = f I K, L (x), F (x, z) , i 2 N g, (16) i n,i N1 where F (x, z) = a z + g(L (x)) a B (g, x). From (15), it is easy to notice that for a given f 2 C( I) n,i i i i there exists a sequence f f g of Bernstein afﬁne FIFs. The following theorem addresses the n,a n=1 convergence of the sequence f f g towards the data generating function y 2 C( I). n,a n=1 Theorem 3. Let y 2 C( I). Let f(x , z ) 2 R : i 2 N g be a data set, where y(x ) = z . Let g(x) = i i N i i xx z q + z (1 q), q = , x 2 I . Let a = (a , a , . . . , a ). Then, for every scaling vector a, the 1 2 N1 i+1 i i x x i+1 i sequence I of IFSs determine a sequence f f g of Bernstein afﬁne FIFs that converges uniformly to n n,a n=1 n=1 the data generating function y. Proof. From (15), it is easy to deduce that k f gk jaj k f B (g, .)k , n,a ¥ ¥ n,a n ¥ jaj [k f gk +kg B (g, .)k ]. ¥ n,a ¥ n ¥ Hence we obtain jaj k f gk kg B (g, .)k . (17) n,a ¥ n ¥ 1jaj Substituting (13) and (17) in (14), we get jaj k f yk 2w (h) + kg B (g, .)k , (18) n,a ¥ y n ¥ 1jaj where w (h) is the modulus of continuity of y. Since y 2 C( I) is uniformly continuous, w (h) ! 0 as y y h ! 0 and from the approximation theory [20], it follows that kg B (g, .)k ! 0 and n ! ¥. As a n ¥ result, from (18), it follows that f ! y uniformly if h ! 0 and n ! ¥. n,a 6. Bernstein Afﬁne Fis Let us consider the surface data set placed on the rectangular grid D : x < x < < grid 1 2 x < x , y < y < < y < y , be given by 4 = f(x , y , z ) 2 R j i 2 N , j 2 N g. Let N1 N 1 2 M1 M 1 i j i,j N M Axioms 2020, 9, 119 10 of 14 I = [x , x ], I = [x , x ], J = [y , y ], J = [y , y ], and D = I J, and D = I J . We construct 1 N i i i+1 1 M j j j+1 i,j i j the Bernstein afﬁne fractal surface F , n 2 N as a blending of the Bernstein afﬁne FIFs constructed along the grid lines of the interpolation domain D so that F : D ! R, F (x , y ) = z , i 2 N , j 2 N . n n i j i,j N M Now for above surface data, T = f(x , z ) : i 2 N g is the interpolation data along the jth, j 2 N N M j i i,j grid line parallel to x-axis. Similarly, T = f(y , z ) : j 2 N g is the interpolation data along the j i,j M xx ith, i 2 N grid line parallel to y-axis. For j 2 N , let g (x) = z q + z (1 q), q = , x 2 I . N M j i+1,j i,j i x x i+1 i yy Similarly, for i 2 N , let g (y) = z f + z (1 f), f = , y 2 J . Suppose Y (x, a ) and N i,j+1 i,j j n,j i,j i y y j+1 j Y (y, a ) are the Bernstein afﬁne FIFs interpolating the data sets T and T respectively. By utilizing n,i i,j i the functional equation of the Bernstein afﬁne FIF f (cf. (15)), we obtain the functional equations of n,a Y (x, a ), j 2 N and Y (y, a ), i 2 N respectively as M N n,j i,j n,i i,j L (x) x 1 1 Y (x, a ) = a Y (L (x), a ) + g (x) a B (g , L (x)), q = , x 2 I , (19) n,j i,j i,j n,j i,j j i,j j i i i x x N 1 L (y) y 1 1 Y (y, a ) = a Y (L (y), a ) + +g (y) a B (g , L (y)), f = , y 2 J , (20) n j n,i i,j i,j n,i i,j i i,j i j j y y M 1 a and a are the scaling factors in x-direction and y-direction respectively satisfying ja j < 1 i,j i,j i,j (y y )y y y y y j+1 j M j 1 j+1 and ja j < 1, L (y) = c y + d = + : J ! J is a homeomorphism such that j j j i,j j y y y y N 1 M 1 L (y ) = y , L (y ) = y , j 2 N . Now we deﬁne the Bernstein afﬁne FIS F as blending of the 1 j M j+1 M1 n j j above afﬁne FIFs Y , j 2 N and Y , i 2 N . In the present construction, we use the following n,j M N n,i choice of blending functions: L (x)x 1 xx 2 2 i i a (q) = (1 q) (1 + 2q), a (q) = q (3 2q), q = = , x 2 I , x,0 x,1 x x x x N 1 i+1 i L (y)y 1 yy j j 2 2 b (f) = (1 f) (1 + 2f), b (f) = f (3 2f), f = = , y 2 J . y,0 y,1 j y y y y M 1 j+1 j The boundary of the sub-rectangle D is taken as the union of four boundary lines I y , I y , i,j i j i j+1 x J , and x J . We deﬁne F , n 2 N over D , i 2 N , j 2 N , as n N1 M1 i j i+1 j i,j F (x, y) = M U (x, y) M , (x, y) 2 D , (21) n 1 n i,j 2 3 0 Y (x, a ) Y (x, a ) n,j i,j n,j+1 i,j+1 6 7 Y (y, a ) z z where U (x, y) = , M = [1 a (q) a (q)], n 4 i,j i,j+1 5 n,i i,j 1 x,0 x,1 Y (y, a ) z z i+1,j i+1,j+1 n,i+1 i+1,j and M = [1 b (f) b (f)]. From (21), it easy to verify that F (x , y ) = z , F (x , y ) = z , 2 y,0 y,1 n i j i,j n i+1 j i+1,j F (x , y ) = z , F (x , y ) = z , i 2 N , j 2 N . Thus F interpolates 4 at n i j+1 i,j+1 n i+1 j+1 i+1,j+1 N1 M1 n 1 the grid points of the interpolation domain D. We invite the readers to check that F (x , y) = Y (y, a ), F (x , y) = Y (y, a ), F (x, y ) = Y (x, a ), F (x, y ) = Y (x, a ). In n i+1 n j j i,j n j+1 n,j+1 i,j+1 n,i i,j n,i+1 i+1,j other words, along the boundaries I y , I y , x J , and x J of D , the fractal surface i j i j+1 i j i+1 j i,j F reduces to Bernstein afﬁne FIFs Y (x, a ), Y (x, a ), Y (y, a ), and Y (y, a ) n n,j i,j n,j+1 i,j+1 n,i i,j n,i+1 i+1,j respectively. Similarly, using (21), the fractal surface F over the sub-rectangle D is deﬁned i,j+1 as a blending of the Bernstein afﬁne FIFs Y (x, a ), Y (x, a ), Y (y, a ), y 2 J , n,j+1 i,j+1 n,j+2 i,j+2 j+1 n,i i,j+1 and Y (y, a ), y 2 J . Along the boundary line I J , the fractal surface F reduces to j+1 i j+1 n,i+1 i+1,j+1 Y (x, a ), and hence F is continuous over the the domains D [ D , i 2 N , j 2 N . n,j+1 i,j+1 n i,j i,j+1 N1 M2 A similar type of arguments gives that F is continuous over the the domain D [ D , i 2 N , n N2 i,j i+1,j j 2 N . From the above discussion, we conclude that the fractal surface F is continuous over the M1 n interpolation domain D. Since we have used only Bernstein afﬁne FIFs in the construction of F , we refer it as Bernstein afﬁne FIS. The scaling factors involved in the Bernstein afﬁne FIFs Y , j 2 N , n,j M and Y , i 2 N are put in matrices a = [a ] , and a = [a ] respectively. i,j (N1) M N( M1) n,i i,j Axioms 2020, 9, 119 11 of 14 Remark 2. If a = [0] and a = [0] , then we get the classical afﬁne surface interpolant as (N1) M N( M1) S (x, y) =b (f)Y (x, 0) + b (f)Y (x, 0) + a (q)Y (y, 0) n y,0 y,1 x,0 n,j n,j+1 n,i + a (q)Y (y, 0) a (q)b (f)z a (q)b (f)z (22) x,1 x,0 y,0 i,j x,0 y,1 i,j+1 n,i+1 a (q)b (f)z a (q)b (f)z , x,1 y,0 x,1 y,1 i+1,j i+1,j+1 where Y (x, 0) = g (x) and Y (y, 0) = g (y) are the classical afﬁne interpolants for the data sets T and T n,j j j n,i i i respectively. 6.1. Convergence of Bernstein afﬁne FIS Theorem 4. For n 2 N, let F be the Bernstein afﬁne FIS with respect to the surface data f(x , y , z ) : i j i,j i 2 N , j 2 N g generated from the function F 2 C(D). Then, the sequence fF g of Bernstein afﬁne N M n n=1 FISs converges uniformly to F 2 C(D) if h ! 0 and k ! 0, where h = maxfx x : i 2 N g and i+1 i N1 k = maxfy y : j 2 N g. j+1 j M1 Proof. From (21) and Remark 2, we have jF (x, y) S (x, y)j b (f)jY (x, a ) Y (x, 0)j n n y,0 n,j i,j n,j + b (f)jY (x, a ) Y (x, 0)j y,1 n,j+1 i,j+1 n,j+1 (23) + a (q)jY (y, a ) Y (y, 0)j x,0 n,i i,j n,i + a (q)jY (y, a ) Y (y, 0)j. x,1 n,i+1 i,j n,i+1 Since Y (x, 0) = g (x) and Y (y, 0) = g (y), using (17), we obtain n,j j n,i i ja j jY (x, a ) Y (x, 0)j kg B (g , .)k , j 2 N , n ¥ M n,j i,j n,j j j 1ja j j ¥ (24) ja j Y (y, a ) Y (y, 0)j kg B (g , .)k , i 2 N , n ¥ N n,i i,j n,i i i 1ja j ja j = maxfja j : i 2 N g, and ja j = maxfja j : j 2 N g. Also it is easy to calculate that j ¥ i,j N1 ¥ M1 i i,j a 1, a 1, b 1, b 1. (25) x,0 x,1 y,0 y,1 ja j jF (x, y) S (x, y)j kg B (g , .)k n n n ¥ j j 1ja j ja j j+1 ¥ + kg B (g , .)k j+1 n j+1 ¥ 1ja j j+1 ¥ (26) ja j + kg B (g , .)k n ¥ i i 1ja j ja j i+1 + kg B (g , .)k . n ¥ i+1 i+1 1ja j i+1 Axioms 2020, 9, 119 12 of 14 Since the above inequality is true for every (x, y) 2 D , i 2 N , j 2 N , we get i,j N1 M1 ja j kF S k kg B (g , .)k n n ¥ j n j ¥ 1ja j j ¥ ja j j+1 ¥ + kg B (g , .)k j+1 n j+1 ¥ 1ja j j+1 (27) ja j + kg B (g , .)k n ¥ i i 1ja j ja j i+1 + kg B (g , .)k . n ¥ i+1 i+1 1ja j i+1 Applying the procedure which is similar to the procedure used in obtaining (13), we get kF S k e(w + w ), (28) F (h) F (k) j i where e is a suitable constant, w is the modulus of continuity of F(x, y ), and w is the modulus F (h) F (k) j i of continuity of F(x , y). Consider the triangle inequality kF Fk kF S k +kS Fk . (29) n ¥ n n ¥ n ¥ Combining (27) and (28) with (29), we obtain ja j kF Fk kg B (g , .)k n ¥ n ¥ j j 1ja j ja j j+1 ¥ + kg B (g , .)k j+1 n j+1 ¥ 1ja j j+1 ¥ (30) ja j + kg B (g , .)k n ¥ i i 1ja j ja j i+1 + kg B (g , .)k + e(w + w ). n ¥ i+1 i+1 F (h) F (k) j i 1ja j i+1 Now, it is easy to verify that (i) kg B (g , .)k ! 0, j 2 N and kg B (g , .)k ! 0, i 2 N as n ¥ M n ¥ N j j i i n ! ¥, (ii) w ! 0, j 2 N and w ! 0, i 2 N , as h, k ! 0. Consequently, we get the desired M N F (h) F (k) j i result from (30). Example 3. The Bernstein afﬁne FISs F and F in Figure 3a,b are constructed with respect to the surface 1 26 data given in Table 1 and the scaling matrices a = [0.99] and a = [0.99] . 34 43 Table 1. Surface data. # y/x ! 4 3 2 1 0.1 2 12 9 7 0.2 7 3 1 2 0.3 8 3 9 8 0.8 2 3 6 9 Axioms 2020, 9, 119 13 of 14 10 10 −10 5 −20 −30 0 0.8 0.8 0.6 −1 0.6 −1 0.4 −2 0.4 −2 0.2 −3 0.2 −3 0 −4 0 −4 (a) The Bernstein Afﬁne FIS F with (b) The Bernstein Afﬁne FIS F with a = [0.99] and a = [0.99] . a = [0.99] and a = [0.99] . 34 43 34 43 Figure 3. Bernstein Afﬁne FISs 7. Discussion If the magnitude of the scaling factors goes to zero, then the corresponding existing afﬁne FIFs converge to the data generating function. In this case, the scaling factors may not fulﬁl condition (8). Consequently, the box-counting dimension of the existing afﬁne FIFs would be one. In this article, using the Bernstein polynomials and the theory of IFSs, we have presented Bernstein afﬁne FIFs as a comprehensive tool to analyse the data that originated from an irregular phenomenon. In our approach, the convergence of Bernstein FIFs towards the original function does not demand any condition on the scaling factors. As a result, we can fulﬁl the condition (8) and the box-counting dimension of the Bernstein afﬁne FIFs must lie between one and two. In this work, we have also introduced the Bernstein afﬁne FIS for the data arranged on the rectangular grid. The convergence of the afﬁne FISs studied in [21] demand a condition on the scaling factors whereas our Bernstein afﬁne FIS does not need any such condition. Because the shapes of the Bernstein afﬁne FISs can be adjusted by using different choices of the scaling factors, our scheme offers a large ﬂexibility for simulation or modelling of irregular objects. The optimal approximation of the Bernstein afﬁne FIS for a given surface is under investigation using a genetic algorithm. 8. Materials and Methods In the present article, we have used a sequential approach for obtaining a new class of afﬁne FIFs, namely, Bernstein afﬁne FIFs. Owing to the sequential technique, the convergence of the proposed Bernstein afﬁne FIFs or FISs does depend on the choice of the scaling factors. A three dimensional problem can be approximated by either a two-dimensional or one-dimensional case, but some information will be lost. Two-scale mathematics is needed in order to reveal the lost information due to the lower dimensional approach. Generally, one scale is established by usage where traditional calculus works, and the other scale is for revealing the lost information where the continuum assumption might be forbidden, and fractional calculus or fractal calculus has to be used. Additionally, we have exploited the blending technique [22] for the construction of Bernstein afﬁne FISs. Author Contributions: Conceptualization, V.D.; methodology, N.V. and V.D.; software, N.V.; supervision, V.D.; visualization, N.V.; writing–original draft preparation, N.V.; writing–review and editing, V.D. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Axioms 2020, 9, 119 14 of 14 Acknowledgments: The authors would like to thank the reviewers for their efforts towards improving our manuscript. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Hutchinson, J.E. Fractals and self similarity. Indiana Univ. Math. J. 1981, 30, 713–747. 2. Barnsley, M.F. Fractal functions and interpolation. Constr. Approx. 1986, 2, 303–329, doi:10.1007/BF01893434. 3. Barnsley, M.F. Fractals Everywhere, 3rd ed.; Dover Publications, Inc.: New York, NY, USA, 2012. 4. Maragos, P. Fractal aspects of speech signals: dimension and interpolation. In Proceedings of the ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing, Toronto, ON, Canada, 14–17 April 1991; Volume 1, pp. 417–420. 5. Chand, A.K.B.; Kapoor, G.P. Generalized cubic spline fractal interpolation functions. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Oct 18, 2020

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