Research Article | Open Access
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 -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 preserving -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 in . 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.
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  introduced fractal interpolation functions (FIFs) defined on a compact interval in based on the concept of an IFS . 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 . 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 , 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 [7–11]. 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  found a method to determine 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 . Levkovich-Maslyuk  and Berkner  used wavelet analysis to determine the IFS parameters for the reconstruction of a prescribed curve. Guérin et al.  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  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 , Nielson , de Boor , and Pruess . Tension parameters were used to control the shape of an interpolant. All these above-mentioned methods were , 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  introduced a two-pass algorithm for constructing a monotonic piecewise cubic polynomial for a prescribed monotonic data. Fritsch and Butland  proposed a modified technique to simplify the Fritsch-Carlson algorithm in 1984. The above two algorithms are both local and produce continuous curves, even if a global 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 [26–28]. Shape preserving methods based on quadratic spline interpolants have appeared in [29–31] 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 [32–34]. 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 [37–39]. 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 -rational 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 , 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  and the cubic interpolation method of Brodlie and Butt  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 -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 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 -monotonic 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.
2. Fractal Interpolation Functions
2.1. IFS Theory
Suppose is a complete metric space, and . The Hausdorff metric on the space is defined as , where . The space of fractals is a complete metric space. An IFS is a collection of functions defined on the complete metric space . is called a hyperbolic IFS if is a contraction map (say) with contractive factor for . The Hutchinson map  on is defined as for all . Now, is a contraction map on with the contractive factor . By the Banach Fixed Point Theorem, there exists a unique such that for any , 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 . 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 . Based on the IFS theory, a FIF is constructed as the graph of the attractor in the following .
2.2. Construction of FIF
Let be a partition of the real compact interval . Let a data set be given, where is compact in . Set , and let , be contractive homeomorphisms such that for some . Denote . Define the continuous mappings , such that
For the construction of a desired IFS, now define the functions as . The construction of a FIF is based on the following results.
Proposition 1 (see ). The IFS defined above admits a unique attractor . is the graph of a continuous function such that for .
The above function is called a FIF corresponding to the IFS , and the construction of is based on the following discussion.
Suppose . Then is a complete metric space with respect to the metric induced by the uniform norm on . Define the Read-Bajraktarević operator on by
According to (1)-(2), is continuous on the intervals , , and at each of the points . Also is a contraction mapping on the metric space ; that is, where . Since , possesses a unique fixed point, say on such that . According to (3), the FIF satisfies the following functional equation:
The popular IFS to define fractal interpolation functions takes the following form: where , and is a suitable continuous function satisfying (2). In FIF theory, is called the scaling factor of the transformation , and is the scale vector of the IFS. The scaling factors give additional degrees of freedom to the FIF and allow us to modify its shape and properties. In particular, when , the fixed point reduces to some classical piecewise interpolation function, which can be described according to the nature of .
Suppose . The constants and in are calculated using (2) as
From (5), we obtain for , is called a quadratic FIF. By taking different values of scaling vectors , we can construct smooth or nonsmooth FIFs for given data. For instance, we construct Figure 1(a) with , Figure 1(b) with , and Figure 1(c) with . It is easy to observe that the smoothness of the quadratic FIF increases in each subinterval as for all . Now we will discuss the rational FIFs in the following, where is taken as a rational function defined on for .
(a) FIF with
(b) FIF with
(c) FIF with
2.3. Calculus of Rational FIF
We develop the calculus of the rational FIFs using piecewise rational functions as per theory of a polynomial FIF in .
For a rational FIF, we define where is a polynomial of degree and is a polynomial of degree such that for every in (6). Also contains two real parameters that can be evaluated by using (2). Let be a rational FIF related to the IFS . Let the value of integral of be known at (or ) as (or ). If then is a rational fractal function related to the IFS , where is a suitable compact subset of ,
Since for all , the proof of the proposition follows through suitable modifications of the arguments in .
(a) Rational FIF
Similarly, for , interpolates the data (see Figure 2(c)). In this case, the IFS contains the same for , and
Based on Proposition 2, it is clear that the scaling factors are restricted to certain ranges for a differentiable rational FIF. If exists and is continuous for and , that is, , then we have the following theorem based on the join-up conditions for .
Theorem 3. Let be a given data set. Suppose that , ,, are suitably chosen polynomials in of degree , , respectively, and for every . Suppose for some integer , , . Let where represents the th derivative of , If ,, ; then determines a rational FIF , and is the rational FIF determined by for is a suitable compact subset of .
The proof of the above theorem follows from the inductive arguments based on Proposition 2.
3. Rational Quadratic FIFs
The principle of construction and evaluation of a -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 are the rational quadratic functions, where both the numerators and denominators are polynomials in of degree ; that is, .
3.1. Principle of Construction of -Rational Quadratic FIFs
Theorem 4. Let be a given data set, where , and let be the derivative values at the knots. Consider the IFS , where satisfies (1), contains four real parameters, and are polynomials of degree 2, , and . Let , where represents the derivative of with respect to . If, for , then the attractor of the IFS is the graph of a -rational quadratic FIF.
Proof. Suppose that . Let be the metric on induced by the -norm on . Then is a complete metric space. Define the Read-Bajraktarević operator on as Since , the conditions and (19) give that is a contractive operator on . The fixed point 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 and in (20), we get two equations involving and , respectively, as Since satisfies the functional equation Since , and for , it is easy to verify that is a fractal function. Substituting and in (22), we have two equations involving and , 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 , it can be shown that the IFS has a unique attractor, and it is the graph of the rational quadratic FIF .
Remark 5. The function 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: .
3.2. Evaluation of -Rational Quadratic FIFs
Denote , , . Assume that where are suitable real parameters. By considering Theorem 4 with the above and , we get the following functional equation:
The parameters are evaluated using the following interpolatory properties; namely,
If , the rational quadratic FIF reduces to a constant in the subinterval with as in (25). Otherwise we proceed as follows.
By taking , (25) gives that
Similarly, in (25) gives that
Using in (25), we have the coupled equations
If , then . 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, need not be nonzero for all . 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 for all . This is explained in Section 4.
In most applications, the derivatives 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 -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, . We use the following approximations for the shape preserving rational quadratic FIFs.
3.2.1. 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
3.2.2. Geometric Mean Method
The nonlinear approximation method is used to approximate the derivatives at the intermediate nodes as and at the end points, we have where .
For a given bounded data set, the above derivative approximations are bounded.
Remark 6. Let be a given data set, where , and let be the derivative values given or approximated (by the above-mentioned methods) at the knots. If , for all , , then the rational quadratic FIF (cf. (25)) is in and is unique for fixed , such that and .
Examples 3. We construct six different -rational quadratic fractal interpolants for the data set , , , , of sine function with six different sets of scaling factors, where it is given that , , , , and . Using (6) in our examples, , , , and . Depending on the scaling factors (see Table 1), the corresponding parameters , , , , , are determined according to the procedure described in Section 3.2. Using the IFS the rational quadratic FIFs are generated iteratively (see Figures 3(a) and 3(f)). If we assume , then it is observed that , , and the rational quadratic FIFs of sine curve have 2-fold symmetry about and (see Figures 3(a) and 3(b)). When , , then , and the rational quadratic FIFs are shifted 1-line of horizontal symmetry about (see Figures 3(c) and 3(d)). If are alternatively the same, that is, , , then there is no relation between any part of the curve of a rational quadratic FIF (see Figure 3(e)), and this case is similar to the behavior of the rational quadratic FIFs when all 's are unequal. When all scaling factors , the rational quadratic FIF reduces to the classical quadratic spline that is generated by using the IFS (37) given in Figure 3(f). These examples illustrate the importance of the scaling factors in the modeling of periodic curves, where different types of symmetries can be explored depending on various scientific and engineering applications.
4. 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 are deduced in Section 4.2.
4.1. Sufficient Condition for Monotonic Rational Quadratic FIF
The -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 be a given monotonic data set. Let the derivative values satisfy the necessary conditions for the monotonicity; namely,
If the scaling factors , are chosen in the following way: where , then the -rational quadratic FIF in (31) is monotonic over .
Proof. From elementary calculus, the rational quadratic FIF is monotonic on if and only if either or for every . Therefore, differentiating (31) with respect to , and after some rigorous calculations, we get
Due to the recursive nature of and the coefficients involved in (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 -rational quadratic FIF as follows.
Case I (monotonically decreasing data). In this case , and hence for . It follows from (40) that is a monotonically decreasing rational quadratic FIF for a monotonically decreasing data if and for , . For on , the sufficient conditions are for . Since