On Non-Tensor Product Bivariate Fractal Interpolation Surfaces on Rectangular Grids

Drakopoulos, Vasileios; Manousopoulos, Polychronis

doi:10.3390/math8040525

Cited by 7 publications

(4 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Without restricting the interpolation points, continuous fractal surfaces are constructed in [ 13 ] over polygonal regions. Redefining the IFS, Drakopoulos et al [ 12 ] solved the problem of continuity. Another attempt by Ruan et al [ 25 ] includes modifying the endpoint conditions of the contraction mappings and adding suitable criteria to the IFS.…”

Section: Introductionmentioning

confidence: 99%

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Aparna

Paramanathan

2023

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

This paper primarily focuses on the derivation of fractal numerical integration for the data sets corresponding to two variable signals defined over a rectangular region. Evaluating numerical integration results through the fractal method helps achieve accurate results with minimum computation effort. The formulation of the fractal numerical integration is achieved by considering the recursive relation satisfied by the bivariate fractal interpolation functions for the given data set. The points in the data set have been used to evaluate the coefficients of the iterated function systems. The derivation of these coefficients considering the index of the subrectangles, and the integration formula has been proposed using these coefficients. The bivariate fractal interpolation functions constructed using these coefficients are then correlated with the bilinear interpolation functions. Also, this paper derives a formula for the freely chosen vertical scaling factor that has been used in reducing the approximation error. The obtained formula of the vertical scaling factor is then used in establishing the convergence of the proposed method of integration to the traditional double integration technique through a collection of lemmas and theorems. Finally, the paper concludes with an illustration of the proposed method of integration and the analysis of the numerical integral results obtained for the data sets corresponding to four benchmark functions.

show abstract

Section: Introductionmentioning

confidence: 99%

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Aparna

Paramanathan

2023

Eur. Phys. J. Spec. Top.

View full text Add to dashboard Cite

show abstract

“…Hong-Yong Wang [8] used a wide class of three-dimensional IFS and proved that their attractors are a class of fractal interpolation surfaces. A method based on bivariable functions on rectangular grids for generating fractal interpolation surfaces is presented by V. Drakopoulos and P. Manousopoulos in [9]. Can every attractor of an iterated function system be the graph of a continuous bivariable fractal function?…”

Section: Introductionmentioning

confidence: 99%

Parameter Identification of Bivariate Fractal Interpolation Surfaces by Using Convex Hulls

Drakopoulos,

Matthes,

Sgourdos

et al. 2023

Mathematics

Self Cite

View full text Add to dashboard Cite

The scope of this article is to identify the parameters of bivariate fractal interpolation surfaces by using convex hulls as bounding volumes of appropriately chosen data points so that the resulting fractal (graph of) function provides a closer fit, with respect to some metric, to the original data points. In this way, when the parameters are appropriately chosen, one can approximate the shape of every rough surface. To achieve this, we first find the convex hull of each subset of data points in every subdomain of the original lattice, calculate the volume of each convex polyhedron and find the pairwise intersections between two convex polyhedra, i.e., the convex hull of the subdomain and the transformed one within this subdomain. Then, based on the proposed methodology for parameter identification, we minimise the symmetric difference between bounding volumes of an appropriately selected set of points. A methodology for constructing continuous fractal interpolation surfaces by using iterated function systems is also presented.

show abstract

“…In the case of multivariate maps, this operator can no longer be applied to get necessary functions, and some additional tools are required. While it is true that fractal approximation is an active field of research currently, and there is an abundant bibliography about multivariate fractal interpolation functions (see, e.g, [6][7][8][9][10][11][12][13][14][15][16][17]), our approach has some specificities. One of them is that the functions proposed are products of perturbations of classical maps, and consequently they can be as close to them as desired.…”

Section: Introductionmentioning

confidence: 99%

Fractal Frames of Functions on the Rectangle

2021

View full text Add to dashboard Cite

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

show abstract

On Non-Tensor Product Bivariate Fractal Interpolation Surfaces on Rectangular Grids

Cited by 7 publications

References 27 publications

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Numerical integration of bivariate fractal interpolation functions on rectangular domains

Parameter Identification of Bivariate Fractal Interpolation Surfaces by Using Convex Hulls

Fractal Frames of Functions on the Rectangle

Contact Info

Product

Resources

About