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Konstantin Igudesman, Marsel Davletbaev, Gleb Shabernev, "New Approach to Fractal Approximation of Vector-Functions", Abstract and Applied Analysis, vol. 2015, Article ID 278313, 7 pages, 2015. https://doi.org/10.1155/2015/278313
New Approach to Fractal Approximation of Vector-Functions
This paper introduces new approach to approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions which are multidimensional generalization of fractal interpolation functions. Best values of fractal interpolation vector-functions parameters are found. We give schemes of approximation of some sets of data and consider examples of approximation of smooth curves with different conditions.
It is well known that interpolation and approximation are an important tool for interpretation of some complicated data. But there are multitudes of interpolation methods using several families of functions: polynomial, exponential, rational, trigonometric, and splines to name a few. Still it should be noted that all these conventional nonrecursive methods produce interpolants that are differentiable a number of times except possibly at a finite set of points. But, in many situations, we deal with irregular forms, which can not be approximate with desired precision. Fractal approximation became a suitable tool for that purpose. This tool was developed and studied in [1–3].
We know that such curves as coastlines, price graphs, encephalograms, and many others are fractals since their Hausdorff-Besicovitch dimension is greater than unity. To approximate them, we use fractal interpolation curves  and their generalizations  instead of canonical smooth functions (polynomials and splines).
This paper is multidimensional generalization of . In Section 2, we consider fractal interpolation vector-functions which depend on several matrices of parameters. Example of such functions is given. In Section 3, we set the optimization problem for approximation of vector-function from by fractal approximation vector-functions. We find best values of matrix parameters by means of matrix differential calculus. Section 4 illustrates some examples.
2. Fractal Interpolation Vector-Functions
Let be a nonempty interval; let and be the interpolation points. For all , consider affine transformationHenceforth, small bold letters denote columns (rows) of length and big bold letters denote matrices of .
Require that for all the following conditions hold true:Then, Solving the system, we havewhere matrices are considered as parameters.
Remark 1. Notice that .
Also notice that for all operator takes straight segment between and to straight segment which connects points of interpolation and .
Let be a space of nonempty compact subsets of , with Hausdorff metric. Define the Hutchinson operator  By the condition (2) Hutchinson operator takes a graph of any continuous vector-function on segment to a graph of a continuous vector-function on the same segment. Thus, can be treated as operator on the space of continuous vector-functions .
For all , denoteIn (1), substitute to vector-function . We have that acts on according to
Suppose that we consider all matrices as linear operators on . Furthermore, they are contractive mappings; that is, constant exists such that for all and we haveThen, from (7), it follows that operator is contraction with contraction coefficient on Banach space , where . By the fixed-point theorem, there exists unique vector-function such that and for all we have Function is called fractal interpolation vector-function. It is easy to notice that if , , and , then passes through points of interpolation. In this case functions are called prefractal interpolation vector-functions of order .
Example 2. Figure 1 shows fractal interpolation vector-function of plane. Here , , and and , , and . Values of matrices and are
Henceforth, we assume that for all linear operator is contractive mapping with contraction coefficient . We approximate vector-function by fractal interpolation vector-function constructed on points of interpolation . Thus, we need to fit matrix parameters to minimize the distance between and .
We use methods that have been developed for fractal image compression . Denote Banach space of square integrated vector-functions on segment as , where norm defines Then from (7) and (8) and Remark 1 it follows that for all Thus, is a contractive operator and is its fixed point.
Instead of minimizing we minimize that makes the problem of optimization much easier. The collage theorem provides validity of such approach .
Theorem 3. Let be complete metric space and is contractive mapping with contraction coefficient and fixed point . Then for all .
Lemma 4. Let be square integrated vector-functions. Suppose that matrix is nondegenerated. Matrix integration is implied to be componentwise. Then, the functional reaches its minimum in .
Proof. To prove it, we use matrix differential calculus .
ConsiderNecessary condition of existence of functional extremum is for all . Since there is -linearity of functional , it is sufficient to prove only for matrices that consist of zeros and one unity. Therefore, we have expressions for finding coefficients of matrix . In matrix form these expressions are as follows: from which
Hence, and then functional is convex one. Thus, the value is absolute minimum of .
Example 5. Let us approximate vector-function on segment by the fractal interpolation vector-function constructed on values of in points , , and and , , and (see Figure 2). Then,
Calculate , according to formula (22) as follows:
Apply affine transformations from (1) to vector Thus, and .
4. Discretization and Results
In this section, we approximate discrete data , by fractal interpolation vector-function constructed on points of interpolation , , . Assume that . We fit matrix parameters to minimize functional
It is necessary to use results of previous section. Approximate by constant piecewise vector-function . More precisely , where is the nearest approximation neighbor of . By substituting integrals in (22) to discretization points sums we obtainIt is sufficient to apply (1) for constructing fractal interpolation vector-function after we find .
Consider several examples of approximation of discrete data.
Example 6. Let us approximate vector-function , where . Figure 3 shows the results. Here, we have two pictures; the first one illustrates initial vector-function and its approximation with 3 points and the second one with 4 points, where two functions are nearly identical.
In this case affine transformations (1) have the following form:
Remark 7. Vectors in matrices of affine transformations (1) equal (like in previous example). It means that fractal interpolation vector-function can be treated as attractor of classical affine IFS in .
Example 8. Next example is devoted to a circle , . Figure 4 shows the results. Here we also have two pictures; the first one illustrates initial vector-function and its approximation with 3 points and the second one with 5 points.
In this case affine transformations (1) have the following form:
Example 9. Spiral of Archimedes , , where the scheme is equal to the examples above, but here we use far more points of interpolation, as illustrated in Figure 5.
Example 10. Figure 6 shows approximation of vector-function , , by fractal interpolation vector-function with sixteen points of interpolation.
Example 11. The example illustrates approximation of graph of Weierstrass function (Figure 7) by fractal interpolation vector-function.
This example is taken from , where fractal approximation is used for approximate calculation of box dimension of fractal curves.
In this paper, we have introduced new effective method of approximation of continuous vector-functions and vector sequences by fractal interpolation vector-functions, which are affine transformations with matrix parameters. Parameter fitting was a crucial part of approximation process. We have found appropriate parameter values of fractal interpolation vector-functions and illustrate it with several examples of different types of discrete data.
We assume that fractal approximation is highly promising computational tool for different types of data and it can be used in many ways, even in interdisciplinary fields, with a quite high precision that allows us to apply fractal approximation methods to a wide variety of curves, smooth and nonsmooth alike.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work was performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. The authors greatly acknowledge the anonymous reviewer for carefully reading the paper and providing constructive comments that have led to an improved paper.
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Copyright © 2015 Konstantin Igudesman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.