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Abstract and Applied Analysis
Volume 2015 (2015), Article ID 278313, 7 pages
http://dx.doi.org/10.1155/2015/278313
Research Article

New Approach to Fractal Approximation of Vector-Functions

1Geometry Department, Lobachevskii Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan 420008, Russia
2Kazan (Volga Region) Federal University, IT-Lyceum of Kazan University, Kazan 420008, Russia
3Department of Autonomous Robotic Systems, High School of Information Technologies and Information Systems, Kazan (Volga Region) Federal University, Kazan 420008, Russia

Received 19 August 2014; Revised 24 January 2015; Accepted 8 February 2015

Academic Editor: Poom Kumam

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.

Abstract

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.

1. Introduction

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 [13].

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 [1] and their generalizations [4] instead of canonical smooth functions (polynomials and splines).

This paper is multidimensional generalization of [5]. 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 [6] 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

Figure 1: Fractal interpolation vector-function .

3. Approximation

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 [7]. 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 [8].

Theorem 3. Let be complete metric space and is contractive mapping with contraction coefficient and fixed point . Then for all .

Considering (4) and (6), rewrite (7)where Thus, we minimize the functional

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 [9].
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 .

From Lemma 4, it follows that functional (16) reaches minimum when

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 .

Figure 2: Vector-function and fractal interpolation vector-function completely identical.

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:

Figure 3: Approximation of vector-function by fractal interpolation function with three (a) and four (b) points of interpolation correspondingly.

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:

Figure 4: Approximation of vector-function by fractal interpolation function with three (a) and five (b) points of interpolation correspondingly.

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.

Figure 5: Approximation of vector-function , , by fractal interpolation function with twelve (a) and seventeen (b) points of interpolation correspondingly.

Example 10. Figure 6 shows approximation of vector-function , , by fractal interpolation vector-function with sixteen points of interpolation.

Figure 6: Approximation of vector-function by fractal interpolation function .

Example 11. The example illustrates approximation of graph of Weierstrass function (Figure 7) by fractal interpolation vector-function.
This example is taken from [10], where fractal approximation is used for approximate calculation of box dimension of fractal curves.

Figure 7: Weierstrass function (blue one) and approximating vector-function (red one).

5. Conclusion

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.

Acknowledgments

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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