HOLDER PROPERTY OF FRACTAL INTERPOLATION FUNCTION

The purpose of this paper is to prove a Holder property about the fractal interpolation function L(x), ω(L,δ)=O(δ~α), and an approximate estimate |f-L|≤2{α(h)+||f||/1-h^(2-D)·h^(2-D)}, where D is a fractal dimension of L(x).

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the short- coming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a gl-cubic Hermite in- terpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global G2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an al- ternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15（1） （2007）, pp. 41-53].

HAAR EXPANSIONS OF A CLASS OF FRACTAL INTERPOLATION FUNCTIONS AND THEIR LOGICAL DERIVATIVES

In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α.

Fractal interpolation:a sequential approach

Fractal interpolation is a modern technique to fit and analyze scientific data.We develop a new class of fractal interpolation functions which converge to a data generating(original)function for any choice of the scaling factors.Consequently,our method offers an alternative to the existing fractal interpolation functions(FIFs).We construct a sequence of-FIFs using a suitable sequence of iterated function systems(IFSs).Without imposing any condition on the scaling vector,we establish constrained interpolation by using fractal functions.In particular,the constrained interpolation discussed herein includes a method to obtain fractal functions that preserve positivity inherent in the given data.The existence of Cr--FIFs is investigated.We identify suitable conditions on the associated scaling factors so that-FIFs preserve r-convexity in addition to the Cr-smoothness of original function.

Energy and Laplacian of fractal interpolation functions

Abstract. In this paper, we first characterize the finiteness of fractal interpolation functions （FIFs） on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on the Sierpinski gasket （SG）. As an application, we prove that the solution of the following Dirichlet problem on SG is a FIF with uniform vertical scaling factor 1/5 ：△=0 on SG ／ {q1, q2, q3}, and u（qi）=ai, i = 1, 2, 3, where qi, i=1, 2, 3, are boundary points of SG.

THE NON-DIFFERENTIABILITY OF A CLASS OFFRACTAL INTERPOLATION FUNCTIONS

Hardin and Massopust([1]) introduced a class of fractal interpolation functions and calculated their Bouligand dimensions. This paper deals with the non-differentiability of these functions and shows some conditions under which they are nowhere differentiable. The basic technique here is based on the presentation the author obtains.

A new approach for high fidelity seismic data recovery by fractal interpolation

Recovering accurate data is important for both earthquake and exploration seismology studies when data are sparsely sampled or partially missing. We present a method that allows for precise and accurate recovery of seismic data using a localized fractal recovery method. This method requires that the data are self- similar on local and global spatial scales. We present examples that show that the intrinsic structure associated with seismic data can be easily and accurately recovered by using this approach. This result, in turn, indicates that seismic data are indeed self-similar on local and global scales. This method is applicable not only for seismic studies, but also for any field studies that require accurate recovery of data from sparsely sampled datasets with partially missing data. Our ability to recover the missing data with high fidelity and accuracy will qualitatively improve the images of seismic tomography.

Three-Dimensional Modeling of the Retinal Vascular Tree via Fractal Interpolation

In recent years,the three dimensional reconstruction of vascular structures in the field of medical research has been extensively developed.Several studies describe the various numerical methods to numerical modeling of vascular structures in near-reality.However,the current approaches remain too expensive in terms of storage capacity.Therefore,it is necessary to find the right balance between the relevance of information and storage space.This article adopts two sets of human retinal blood vessel data in 3D to proceed with data reduction in the first part and then via 3D fractal reconstruction,recreate them in a second part.The results show that the reduction rate obtained is between 66%and 95%as a function of the tolerance rate.Depending on the number of iterations used,the 3D blood vessel model is successful at reconstruction with an average error of 0.19 to 5.73 percent between the original picture and the reconstructed image.

Graph-Directed Coalescence Hidden Variable Fractal Interpolation Functions

Fractal interpolation function (FIF) is a special type of continuous function which interpolates certain data set and the attractor of the Iterated Function System (IFS) corresponding to a data set is the graph of the FIF. Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is both self-affine and non self-affine in nature depending on the free variables and constrained free variables for a generalized IFS. In this article, graph directed iterated function system for a finite number of generalized data sets is considered and it is shown that the projection of the attractors on is the graph of the CHFIFs interpolating the corresponding data sets.

A Method Determining Vertical Scaling Parameters of Fractal Interpolation

A method determining vertical scaling parameters of fractal interpolation is given in this paper. By computer experiments, it is clear that this method is very effective.

Multi-Dimensional Piece-Wise Self-Affine Fractal Interpolation Model

Iterated function system （IFS） models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 （R is the set of real numbers）. In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn （n is an integer greater than 3）, which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a ＂mapping partial derivative＂, and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in R^n. Therefore, the result is very general. The class of functions obtained is much more diverse because their values depend continuously on all of the variables, with all the coefficients of the possible multi-dimensional affine maps determining the functions.

Analysis of Multi-Scale Fractal Dimension for Image Interpolation

This article presents a novel image interpolation based on rational fractal fimction. The rational function has a simple and explicit expression. At the same time, the fi＇actal interpolation surface can be defined by proper parameters. In this paper, we used the method of ＇covering blanket＇ combined with multi-scale analysis; the threshold is selected based on the multi-scale analysis. Selecting different parameters in the rational function model, the texture regions and smooth regions are interpolated by rational fractal interpolation and rational interpolation respectively. Experimental results on benchmark test images demonstrate that the proposed method achieves very competitive performance compared with the state-of-the-art interpolation algorithms, especially in image details and texture features.

Fractal Characteristics and Prediction of Backsilting Quantity in Yangtze Estuary Deepwater Channel

Fractal interpolation has been an important method applied to engineering in recent years. It can not only be used to fit smooth curve and stationary data but also show its unique superiorities in the fatting of non-smooth curve and non-stationary data. Through analyzing such characteristic values as average value, standard deviations, skewness and kurtosis of measured backsilting quantities in the Yangtze Estuary 12.5 m Deepwater Channel during2011–2017, the fractal interpolation method can be used to study the backsilting quantity distribution with time.According to the fractal interpolation made on the channel backsilting quantities from January 2011 to December2017, there was a good corresponding relationship between the annual(monthly) siltation quantities and the vertical scaling factor. On this basis, a calculation formula for prediction of the backsilting quantity in the Yangtze Estuary Deepwater Channel was constructed. With the relationship between the predicted annual backsilting quantities and the vertical scaling factor, the monthly backsilting quantities can be obtained. Thus, it provides a new method for estimating the backsilting quantity of the Yangtze Estuary Deepwater Channel.

A new fractal algorithm to model discrete sequences

Employing the properties of the affine mappings, a very novel fractal model scheme based on the iterative function system is proposed. We obtain the vertical scaling factors by a set of the middle points in each affine transform, solving the difficulty in determining the vertical scaling factors, one of the most difficult challenges faced by the fractal interpolation. The proposed method is carried out by interpolating the known attractor and the real discrete sequences from seismic data. The results show that a great accuracy in reconstruction of the known attractor and seismic profile is found, leading to a significant improvement over other fractal interpolation schemes.

Enhancing PIV image and fractal descriptor for velocity and shear stresses propagation around a circular pier

In this study,the fractal dimensions of velocity fluctuations and the Reynolds shear stresses propagation for flow around a circular bridge pier are presented.In the study reported herein,the fractal dimension of velocity fluctuations（u′,v′,w′） and the Reynolds shear stresses（u′v′ and u′w′） of flow around a bridge pier were computed using a Fractal Interpolation Function（FIF） algorithm.The velocity fluctuations of flow along a horizontal plane above the bed were measured using Acoustic Doppler Velocity meter（ADV）and Particle Image Velocimetry（P1V）.The PIV is a powerful technique which enables us to attain high resolution spatial and temporal information of turbulent flow using instantaneous time snapshots.In this study,PIV was used for detection of high resolution fractal scaling around a bridge pier.The results showed that the fractal dimension of flow fluctuated significantly in the longitudinal and transverse directions in the vicinity of the pier.It was also found that the fractal dimension of velocity fluctuations and shear stresses increased rapidly at vicinity of pier at downstream whereas it remained approximately unchanged far downstream of the pier.The higher value of fractal dimension was found at a distance equal to one times of the pier diameter in the back of the pier.Furthermore,the average fractal dimension for the streamwise and transverse velocity fluctuations decreased from the centreline to the side wall of the flume.Finally,the results from ADV measurement were consistent with the result from PIV,therefore,the ADV enables to detect turbulent characteristics of flow around a circular bridge pier.

Image interpolation based on Wavelet Transform

Image interpolation is widely studied and used in digital image processing. In this paper, a method of image magnification according to the properties of fi＇actal interpolation and wavelet transformation are presented. We focus the development of edge forming methods to be applied as a post process of standard image zooming methods for grayscale images, with the hope of retaining edges. Experiments make sure it valid.

THE THEORY OF FRACTAL INTERPOLATED SURFACE AND ITS APPLICATIONS

In this paper, the principle of construction of a fractal surface is introduced, interpolation functions for a fractal interpolated surface are discussed, the theorem of the uniqueness of an iterated function system of fractal interpolated surface is proved, the theorem of fractal dimension of fractal interpolated surface is derived, and the case that practical data are used to interpolate fractal surface is studied.

Proppant transport in rough fractures of unconventional oil and gas reservoirs

A method to generate fractures with rough surfaces was proposed according to the fractal interpolation theory.Considering the particle-particle,particle-wall and particle-fluid interactions,a proppant-fracturing fluid two-phase flow model based on computational fluid dynamics(CFD)-discrete element method(DEM)coupling was established.The simulation results were verified with relevant experimental data.It was proved that the model can match transport and accumulation of proppants in rough fractures well.Several cases of numerical simulations were carried out.Compared with proppant transport in smooth flat fractures,bulge on the rough fracture wall affects transport and settlement of proppants significantly in proppant transportation in rough fractures.The higher the roughness of fracture,the faster the settlement of proppant particles near the fracture inlet,the shorter the horizontal transport distance,and the more likely to accumulate near the fracture inlet to form a sand plugging in a short time.Fracture wall roughness could control the migration path of fracturing fluid to a certain degree and change the path of proppant filling in the fracture.On the one hand,the rough wall bulge raises the proppant transport path and the proppants flow out of the fracture,reducing the proppant sweep area.On the other hand,the sand-carrying fluid is prone to change flow direction near the contact point of bulge,thus expanding the proppant sweep area.

A time-driven transmission method for well logging networks

Long delays and poor real-time transmission are disadvantageous to well logging networks consisting of multiple subnets. In this paper, we proposed a time-driven transmission method （TDTM） to improve the efficiency and precision of logging networks. Using TDTM, we obtained well logging curves by fusing the depth acquired on the surface, and the data acquired in downhole instruments based on the synchronization timestamp. For the TDTM, the precision of time synchronization and the data fusion algorithm were two main factors influencing system errors. A piecewise fractal interpolation was proposed to fast fuse data in each interval of the logging curves. Intervals with similar characteristics in curves were extracted based on the change in the histogram of the interval. The TDTM is evaluated with a sonic curve, as an example. Experimental results showed that the fused data had little error, and the TDTM was effective and suitable for the logging networks.

Fractal characteristic evaluation and interpolation reconstruction for surface topography of drilled composite hole wall

In this paper,an improved fractal interpolation model is proposed to reconstruct the surface topography of composite hole wall.This model adopts the maximum positive deviations and maximum negative deviations between the measured values and trend values to determine the contraction factors.Hole profiles in 24 directions are measured.Fractal parameters are calculated to evaluate the measured surface profiles.The maximum and minimum fractal dimension of the hole wall are 1.36 and 1.07,whereas the maximum and minimum fractal roughness are 4.05 x 10-5 and 4.36 x 10-10 m,respectively.Based on the two-dimensional evaluation results,three-dimensional surface topographies in five typical angles(0°,45°,90°,135°,and 165°)are reconstructed using the improved model.Fractal parameter Ds and statistical parameters Sa9 Sq,and Sz are used to evaluate the reconstructed surfaces.Average error of Ds,Sa,Sq,and Sz between the measured surfaces and the reconstructed surfaces are 1.53%,3.60%,5.60%,and 9.47%,respectively.Compared with the model in published literature,the proposed model has equal reconstruction effect in relatively smooth surface and is more advanced in relatively rough surface.Comparative results prove that the proposed model for calculating contraction factors is more reasonable.