Numerical integration of bivariate fractal interpolation functions on rectangular domains.

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.

Application of fractal theory in analysis of human electroencephalographic signals.

In medical discipline, complexity measure is focused on the analysis of nonlinear patterns in processing waveform signals. The complexity measure of such waveform signals is well performed by fractal dimension technique, which is an index for measuring the complexity of an object. Its applications are found in diverse fields like medical, image and signal processing. Several algorithms have been suggested to compute the fractal dimension of waveforms. We have evaluated the performance of the two famous algorithms namely Higuchi and Katz. They contain some problems of determining the initial and final length of scaling factors and their performance with electroencephalogram (EEG) signals did not give better results. In this paper, fractal dimension is proposed as an effective tool for analyzing and measuring the complexity of nonlinear human EEG signals. We have developed an algorithm based on size measure relationship (SMR) method. The SMR algorithm can be used to detect the brain disorders and it locates the affected brain portions by analyzing the behavior of signals. The efficiency of the algorithm to locate the critical brain sites (recurrent seizure portion) is compared to other fractal dimension algorithms. The K-means clustering algorithm is used for grouping of electrode positions.