Monotonicity Preserving Rational Quadratic Fractal Interpolation Functions

Fractal interpolation is an advanced technique for analysis and synthesis of scientific and engineering data. We introduce the C-rational quadratic fractal interpolation functions (FIFs) through a suitable rational quadratic iterated function system (IFS). The novel notion of shape preserving fractal interpolation without any shape parameter is introduced through the rational fractal interpolation model in the literature for the first time. For a prescribed set of monotonic data, we derive the sufficient conditions by restricting the scaling factors for shape preservingC-rational quadratic FIFs. A local modification pertaining to any subinterval is possible in this model if the scaling factors are chosen appropriately. We establish the convergence results of a monotonic rational quadratic FIF to the original function inC. For given data with derivatives at grids, our approach generates several monotonicity preserving rational quadratic FIFs, whereas this flexibility is not available in the classical approach. Finally, numerical experiments support the importance of the developed rational quadratic IFS scheme through construction of visually pleasing monotonic rational fractal curves including the classical one.


Introduction
The interpolation of smooth curve shape constitutes a major research area for reconstruction and representation problems in medical imaging, computer aided geometric design, robotics, automobile engineering, architecture, and multimedia data representation.In manufacturing science, mathematical models to relate part characteristics with process parameters are typically developed from experimental data where the physics based models are not available.The fractal interpolation is an advance technique in fitting of nonsmooth and smooth data from a physical or experimental set-up.To approximate data that follows some kind of self-similarity under magnification, Barnsley [1] introduced fractal interpolation functions (FIFs) defined on a compact interval in R based on the concept of an IFS [2].These FIFs are not necessarily differentiable, and they differ from classical interpolants in the sense that (i) FIFs obey an implicit functional relation and (ii) FIFs have noninteger fractal dimensions in general.Since FIFs are being able to extrapolate patterns from one scale to all scales, the use of these functions divulge the presence of an underlying determinism in apparently disorganized data.FIFs have been extensively used due to their characterization for either the generation of geometrically complex graphs of continuous functions or fitting of experimental data.The fractal dimension of a FIF is used to measure the complexity of a signal, and in this way it allows an automatic comparison of recordings [3].The power of fractal methodology enables the generalization almost any other interpolation techniques; see, for instance, [4,5].The study of spline FIFs has been initiated by Barnsley and Harrington [6], wherein the construction is based on an algebraic method.Due to restricted boundary conditions in this construction, fractal splines with general boundary conditions have been studied recently [7][8][9][10][11].The studies of IFSs have provided powerful tools for the investigation of fractal sets that are used for approximation of natural or scientific data.The determination of an IFS approximating prescribed data is called "the inverse problem" in the FIF theory.Few works have been reported on this subject based on the FIF model.Strahle [12] found a method to determine 2 Advances in Numerical Analysis the scaling parameters in the description of FIFs for such an inverse problem.A different approach in determining the IFS parameters is proposed by Mazel and Hayes [13].Levkovich-Maslyuk [14] and Berkner [15] used wavelet analysis to determine the IFS parameters for the reconstruction of a prescribed curve.Guérin et al. [16] proposed the projected IFS model to approximate rough curves.The shape preserving capabilities of FIFs are not explored in the literature due to their implicit representations.In the CAGD application, particularly in reverse engineering, where the shape is reconstructed from optical scanned data, any curve/surface design should confirm to the overall shape as described by the data.Fractal interpolation can help in preserving local variation but still confirming to the global shape as described by the data.Some typical examples are modelling biological shapes for computational modelling and analysis of biomedical design and haptic surgical simulation or developing virtual models of architectural monuments for digital preservation.In order to construct the shape preserving fractal interpolants in these areas, the paper initiates the theory of monotonicity preserving smooth curves through the rational quadratic FIF models.
After Schoenberg [17,18] introduced "spline functions" to the mathematical literature, splines have proved to be enormously important in smooth curve representations to discrete data.For smooth curve interpolants, it is crucial to incorporate the inherited features of given data.Data are classified as positive, constrained, monotone, or convex according to their graphs.Schweikert [19] was the first to construct the shape preserving interpolating functions with exponential splines.Shape preserving and cubic spline interpolants with tension parameters were studied in the literature; see, for instance Späth [20], Nielson [21], de Boor [22], and Pruess [23].Tension parameters were used to control the shape of an interpolant.All these abovementioned methods were C 2 , global and interpolatory.Automatic algorithms to evaluate the shape parameters by these methods to control shape and monotonicity were involved.In 1980, Fritsch and Carlson [24] introduced a two-pass algorithm for constructing a monotonic piecewise cubic polynomial for a prescribed monotonic data.Fritsch and Butland [25] proposed a modified technique to simplify the Fritsch-Carlson algorithm in 1984.The above two algorithms are both local and produce C 1 continuous curves, even if a global C 2 solution exists.Furthermore, there is no flexibility in defining the user's desired properties for the resultant spline, for example, the objective function or constraints for an optimization problem.Based on the Fritsch-Carlson algorithm, Costantini proposed several methods to compute shape preserving splines [26][27][28].Shape preserving methods based on quadratic spline interpolants have appeared in [29][30][31] and references therein.The motivation to this work is due to the past work of many authors; for example, the rational quadratic interpolation methodology has been adopted in [32][33][34].Rational interpolants play important role in geometric modeling, computer graphics, and CAD/CAM [35,36].For shape preserving interpolants, the rational splines are preferred over ordinary splines [37][38][39].Taking the fractal interpolation technique in one hand and the rational interpolation in the other, we introduce the rational FIF in the literature for the first time.In particular, we develop the rational quadratic FIF that preserves desired properties like smoothness and monotonicity as required by a prescribed data set.
In this paper, we initiate the construction of the C 1rational quadratic fractal interpolant through piecewise rational functions whose numerators and denominators are quadratic polynomials.When all scaling factors are zero, in particular, we retrieve the classical piecewise rational quadratic interpolant [32], and this shows the power of generalization of fractal methodology.The rational quadratic FIF described in this paper is unique for a given set of data and a fixed set of scaling factors.The developed rational quadratic fractal scheme is extremely useful when the derivative of the original function is nonsmooth in nature.Monotonicity is an important shape preserving property as uric acid levels in patients suffering from gout, erythrocyte sedimentation rate (ESR) in cancer patients, and data generated from stress of a material are few examples of entities which are always monotonic.This paper examines the shape preserving property monotonicity of the prescribed data through the rational quadratic FIF model.Because of the recursive nature of a FIF, the necessary condition for monotonicity on derivatives at knots alone may not ensure the monotonicity of a rational quadratic fractal interpolant for a given set of monotonic data.Therefore, we derive sufficient conditions for monotonicity based on appropriate conditions on the scaling factors   such that these conditions together with the necessary conditions ensure the monotonicity of a rational quadratic fractal interpolant for prescribed monotonic data.Our construction does not need any additional points for the rational quadratic FIFs, whereas the quadratic spline methods of Schumaker [30] and the cubic interpolation method of Brodlie and Butt [40] require the introduction of additional points for shape preserving interpolants.
The content of this paper is organized as follows.The mathematical backgrounds of FIFs based on the IFS theory along with the calculus of rational FIFs are discussed in Section 2. In Section 3, the inverse problem of interpolation with a C 1 -rational quadratic FIF is introduced.We discuss sufficient conditions for these interpolants to be monotonic by deriving appropriate restrictions on the scaling parameters of the associated rational IFS in Section 4, and an upper bound of the uniform error of the monotonic rational quadratic FIF with the original function in C 4 is estimated for the convergence results in Section 4.2.Finally, the application of the shape preserving rational quadratic IFS scheme is illustrated on monotonic data set for visually pleasing C 1monotonic rational fractal interpolants, and the effects of change in the scaling parameters on the rational quadratic FIF and its derivative are demonstrated in Section 5.

Fractal Interpolation Functions
The basics of IFS theory are discussed in Section 2.1, and the construction of a FIF from a suitable IFS is presented in Section 2.2.The calculus of the rational FIFs is given in Section 2.3.
2.1.IFS Theory.Suppose (X,  X ) is a complete metric space, and H(X) = { ⊂ X :  ̸ = ,  is compact}.The Hausdorff metric ℎ( X ) on the space H(X) is defined as ℎ(, ) = max{(, ), (, )}, where (, ) = max ∈ min ∈  X (, ).The space of fractals (H(X), ℎ( X )) is a complete metric space.An IFS I = {X;   ,  = 1, 2, . . .,  − 1} is a collection of  − 1 functions defined on the complete metric space (X,  X ).I is called a hyperbolic IFS if   is a contraction map (say) with contractive factor |  | < 1 for  = 1, 2, . . .,  − 1.The Hutchinson map [2] on H(X) is defined as () = ∪ −1 =1   () for all  ∈ H(X).Now,  is a contraction map on (H(X), ℎ( X )) with the contractive factor  = max{|  | :  = 1, 2, . . .,  − 1}.By the Banach Fixed Point Theorem, there exists a unique  ∈ H(X) such that lim  → ∞   () =  for any  ∈ H(X), and this fixed point  is known as an attractor or a deterministic fractal of the hyperbolic IFS.In the inverse problem,  is the object to be approximated by a suitable IFS I. Since an image can be interpreted by its IFS code, the fractal image compression is one of the popular applications in the current research of fractal theory [41].Based on the IFS theory, a FIF is constructed as the graph of the attractor in the following [1].(
The above function ℎ * is called a FIF corresponding to the IFS {;   ,  = 1, 2, . . .,  − 1}, and the construction of ℎ * is based on the following discussion.
Suppose   () :=  2 +    +   .The constants   and   in   are calculated using (2) as From (5), we obtain for  ∈ [1,9], ℎ * is called a quadratic FIF.By taking different values of scaling vectors  = ( 1 ,  2 ,  3 ), we can construct smooth or nonsmooth FIFs for given data.For instance, we construct  It is easy to observe that the smoothness of the quadratic FIF increases in each subinterval as |  | → 0 for all .Now we will discuss the rational FIFs in the following, where   () is taken as a rational function defined on  for  = 1, 2, . . .,  − 1.

Calculus of Rational FIF.
We develop the calculus of the rational FIFs using piecewise rational functions as per theory of a polynomial FIF in [6].

Rational Quadratic FIFs
The principle of construction and evaluation of a C 1 -rational quadratic FIF are described in Sections 3.1 and 3.2, respectively.In our construction of the rational quadratic FIFs, it is assumed that   ( = 1, 2, . . .,  − 1) are the rational quadratic functions, where both the numerators and denominators are polynomials in  of degree 2; that is,  =  = 2.
The fixed point  ∈ C 1 () of  * is a fractal function that satisfies the functional relation The four parameters in the rational quadratic function   are evaluated by using the interpolation conditions (18) of  as follows.Substituting  =  1 and  =   in (20), we get two equations involving   and  +1 , respectively, as Since  ∈ C 1 (),  (1) satisfies the functional equation Since |  |/  < 1, and  ,2 () ̸ = 0 ∀ ∈  for  = 1, 2, . . .,  − 1, it is easy to verify that  (1) is a fractal function.Substituting  =  1 and  =   in (22), we have two equations involving   and  +1 , respectively, as When the four parameters of   are determined from ( 21) and ( 23), then the rational quadratic FIF exists.By using similar arguments as in [1], it can be shown that the IFS I * has a unique attractor, and it is the graph of the rational quadratic FIF  ∈ C 1 ().
Remark 5.The function  ∈ C 1 () is called a fractal function because of (i) the presence of the scaling vector  in ( 20), (ii) the derivative   is a typically fractal function, and (iii) the graph of , say   , satisfies the equation:   = ∪ −1 =1   (  ).
By taking  =  1 , (25) gives that Similarly,  =   in (25) gives that Using  (1) (  ) =   ,  (1) ( +1 ) =  +1 in (25), we have the coupled equations Assume that  +1 −   −   (  −  1 ) ̸ = 0.The solution of the system (29) gives that where Substituting the values of   ,   ,   ,   in (25), the desired rational quadratic FIF is finally obtained as If Δ  = 0, then  ,2 () = 0.In order to avoid singularity in the expression of the rational quadratic FIF , we define  as a constant function with the value   in the subinterval   .In general,  ,2 () need not be nonzero for all  ∈ [0, 1].But the objective of this paper is to obtain monotonic curves though the rational quadratic FIF  for given monotonic data, and when we impose the sufficient conditions for monotonicity based on the derivative values and scaling factors, automatically we get  ,2 () ̸ = 0 for all  ∈ [0, 1].This is explained in Section 4.
In most applications, the derivatives   ( = 1, 2, . . ., ) are not given and hence must be calculated either from the given data or some numerical methods.In this paper, they are computed from the given data in such a way that the C 1 -smoothness of the fractal interpolant (31) is retained.These methods are different type of approximations based on the various mathematical theories in the literature; see, for instance, [42].We use the following approximations for the shape preserving rational quadratic FIFs.

Arithmetic Mean Method.
The three-point difference method is used to approximate the derivatives at the intermediate nodes as and at the end points, we have

Monotonicity Preserving Fractal Interpolation
The sufficient conditions are derived to preserve the monotonicity of a given monotonic data set in Section 4.1.The convergence results of a monotonic rational quadratic FIF to the original function  ∈ C 4 () are deduced in Section 4.2.

Sufficient Condition for Monotonic Rational Quadratic
FIF.The C 1 -rational quadratic FIF  may not preserve the monotonicity property of given monotonic data if we simply assume all interpolatory conditions.To achieve the shape preserving property, namely, monotonicity, by the rational quadratic FIF  for given monotonic data, we need some mathematical treatment that is based on the recursive nature of a FIF.The following theorem addresses this issue.
Theorem 8. Let {(  ,   ,   ),  = 1, 2, . . ., } be a given monotonic data set.Let the derivative values satisfy the necessary conditions for the monotonicity; namely, If the scaling factors   ,  = 1, 2, . . .,  − 1, are chosen in the following way: where Proof.From elementary calculus, the rational quadratic FIF  is monotonic on  if and only if either  (1) () ≥ 0 or  (1) () ≤ 0 for every  ∈ .Therefore, differentiating (31) with respect to , and after some rigorous calculations, we get where Due to the recursive nature of  (1) and the coefficients involved in Ω  (cf.( 40)), the necessary conditions (38) may not be sufficient to ensure the monotonicity of the rational quadratic FIF .Therefore, we put restrictions on the scaling factors   so that these conditions together with the above necessary conditions give the monotonicity nature to our C 1rational quadratic FIF  as follows.
Again suppose that Δ  > 0, using similar arguments as in Case I; From (44),  ,1 ≥ 0,  ,2 > 0,  ,3 ≥ 0 whenever (43) holds.In this case, now it is easy to see that if the scaling factors   ,  = 1, 2, . . .,  − 1, are chosen according to (39), then the C 1 -rational quadratic FIF  is monotonically increasing over .Therefore, using the results from Case I and Case II, we conclude that the selection of the scaling factors according to (39) is sufficient to obtain the monotonic rational quadratic FIF  for a given set of monotonic data.This completes the proof of Theorem 8.
In order to preserve the increasing nature of given increasing data, the restrictions on the scaling factors according to (39) are given by  1 = 0,  2 ∈ [0, 0.025),  3 ∈ [0, 0.0715),  4 ∈ [0, 0.2061),  5 ∈ [0, 0.1476), and  6 ∈ [0, 0.2023).A standard rational quadratic FIF (Figure 4(a)) is constructed with a suitable choice of scaling factors (see Table 2).Now, we modify the scaling parameter  4 as 0.010 (see Table 2), and the corresponding quadratic FIF is generated (see Figure 4   negligible.Similarly, we modify  6 (see Table 2) with respect to Figure 4(a), and the corresponding rational quadratic FIF is generated in Figure 4(c).In comparison with Figure 4(a), it is observed that the graph of the rational quadratic FIF in the sixth sub-interval [    3).We observe that deviations in the rational quadratic FIF in Figure 4(a) due to  4 are more than that of the scaling factor  6 .Also we have calculated the uniform distance between their derivatives (see Table 3).The effects of the scaling factors   the classical interpolant is a piecewise smooth in the interval [ 1 ,  7 ] (see Figure 5(f)).Because of this reason, if the original function is  1 -smooth and monotonic but its derivative is a continuous nowhere differentiable function, then our Advances in Numerical Analysis rational quadratic FIF  is an ideal tool to approximate such function instead of the classical rational quadratic interpolant, whose derivative is a piece-wise smooth function.As the approximation of an original function like Figure 4(a) and its derivative is concerned, numerical data given in Table 3 reveals that proposed rational quadratic FIF  is a better approximant over the classical rational quadratic interpolant .

Conclusion
In the present paper, the notion of rational fractal interpolation without shape parameter is introduced in the literature for the first time.In particular, the C 1 -rational quadratic FIFs that contain parameters, namely, the scaling factors, are developed for representation of a given real data set.The derivative values are approximated at grids by the Arithmetic Mean Method and Geometric Mean Method so that the C 1 -smoothness of quadratic FIF is maintained.A rational quadratic FIF need not be monotonic for arbitrary choice of scaling factors for given monotonic data in general.To preserve the monotonicity feature of a data set, the sufficient conditions are prescribed through the scaling factors on the associated IFS.Therefore, the data dependent scaling factors are introduced in this work for shape preserving rational FIFs.
A uniform error bound is deduced between the monotonic rational quadratic FIF and an original function.From this, it is possible to achieve (ℎ  ) ( = 1, 2, 3, 4) convergence, when accurate derivative values of order (ℎ −1  ) are available at grids, and the scaling factors are taken as 0 ≤   <    for  = 1, 2, . . .,  − 1.The role of scaling factors in local control of shape of rational quadratic FIFs is demonstrated through suitable examples.The effects of scaling factors in the derivative of rational quadratic FIF may be local, moderately local, or global, and it varies according to the interpolation data.The proper choice of scaling factors, for which the uniform errors ‖ − ‖ ∞ and ‖  −   ‖ ∞ are minimum, is an open problem in the FIF optimization technique, where the original function  ∈ C 1 is such that its derivative   is similar to a continuous no-where-differentiable function.Since the construction of a rational FIF depends on the choice of scaling factors, one can construct different shapes of monotonicity preserving C 1 -rational quadratic FIFs for the same data by varying its scale vector.Due to this added flexibility, our rational quadratic FIFs will have huge number of applications in computer visualization, computational geometry, object recognition, shape abstraction and modeling, CAD, reverse engineering, and other scientific applications as compared to the classical rational quadratic splines.

Figure 2 :
Figure 2: Rational FIF and its integrals
Increasing rational quadratic FIF r (b) Effects of  4 in (a) Increasing rational quadratic FIF r (c) Effects of  6 in (a) Increasing rational quadratic FIF r (d) Effects of  3 and  5 in (a) Increasing rational quadratic FIF r (e) Effects of  4 ,  5 , and  6 in (a) S Increasing rational quadratic interpolant S (f) Classical rational quadratic interpolant
(b)).In comparison with Figure 4(a), we observe that there are visually pleasing changes in the graph of the rational quadratic FIF in the fourth sub-interval [ 4 ,  5 ] of Figure 4(b), and the changes in other intervals are

Figure 5 :
Figure 5: Derivatives of the rational quadratic FIFs and the classical rational quadratic interpolant.

Figures 5 (
Figures 5(a)-5(e) are typical fractal functions close to continuous but nowhere differentiable function.By taking the rational quadratic FIF in Figure4(a) as the original function, we have calculated the uniform distance between this original function and the rational quadratic FIFs in Figures4(b) and 4(f) (see Table3).We observe that deviations in the rational quadratic FIF in Figure4(a) due to  4 are more than that of the scaling factor  6 .Also we have calculated the uniform distance between their derivatives (see Table3).The effects of the scaling factors  4 and  6 are very much prominent in the fourth and sixth subintervals of Figures 4(b)-4(c), respectively, in comparison with Figure 4(a), but they render major effects in their derivatives (see Figures 5(b)-5(c) and Table 3).The rational FIFs in Figures 5(a)-5(e) are irregular in nature over the interval [ 1 ,  7 ], whereas the derivative of Figures 5(a)-5(e) are typical fractal functions close to continuous but nowhere differentiable function.By taking the rational quadratic FIF in Figure4(a) as the original function, we have calculated the uniform distance between this original function and the rational quadratic FIFs in Figures4(b) and 4(f) (see Table3).We observe that deviations in the rational quadratic FIF in Figure4(a) due to  4 are more than that of the scaling factor  6 .Also we have calculated the uniform distance between their derivatives (see Table3).The effects of the scaling factors  4 and  6 are very much prominent in the fourth and sixth subintervals of Figures 4(b)-4(c), respectively, in comparison with Figure 4(a), but they render major effects in their derivatives (see Figures 5(b)-5(c) and Table 3).The rational FIFs in Figures 5(a)-5(e) are irregular in nature over the interval [ 1 ,  7 ], whereas the derivative of

Table 1 :
Parameters of the rational quadratic IFSs for sine function.

Table 2 :
Scaling factors used in the construction of the rational quadratic FIFs.
6,  7 ] converging to the straight line between ( 6 ,  6 ) and ( 7 ,  7 ) as  6 → 0 + and variations in the other sub-intervals are not noticeable.Similarly we modify  3 and  5 (see Table2) with respect to Figure4(a), and the appropriate rational quadratic FIF is constructed in Figure4(d).By analyzing Figure4(d) with respect to Figure4(a), the perceptible effects are observed in the third and fifth sub-intervals.In particular, the graph of the rational quadratic FIF in the third sub-interval [ 3 ,  4 ] is converging to a concave shape as  3 → 0 + .Again a simultaneous modification of  4 ,  5 , and  6 (see Table2) yields the rational quadratic FIF in Figure4(e), and we observe that the individual effects of  4 ,  6 , and  5 , respectively, from Figures4(b)-4(d) are reflected in Figure4(e).From the above discussion, it is interesting to see that although the rational quadratic FIF  is a global interpolant, the effects of the scaling factors  3 ,  4 ,  5 , and  6 are very much local in In this example, the interpolation data set for   is {(0, 0), (2.5, 9.7058), (3, 10.0251), (6, 1.4401), (11, 1.8054), (15, 1.4133), (20, 1.7467)}.The derivative functions of the rational quadratic FIFs in Figures4(a