Vassileios Drakopoulos

Curve fitting by fractal interpolation

Fractal interpolation provides an efficient way to describe data that have an irregular or self-similar structure. Fractal interpolation literature focuses mainly on functions, i.e. on data points linearly ordered with respect to their abscissa. In practice, however, it is often useful to model curves as well as functions using fractal intepolation techniques. After reviewing existing methods for curve fitting using fractal interpolation, we introduce a new method that provides a more economical representation of curves than the existing ones. Comparative results show that the proposed method provides smaller errors or better compression ratios.

Parameter identification of 1D fractal interpolation functions using bounding volumes

Fractal interpolation functions are very useful in capturing data that exhibit an irregular (non-smooth) structure. Two new methods to identify the vertical scaling factors of such functions are presented. In particular, they minimize the area of the symmetric difference between the bounding volumes of the data points and their transformed images. Comparative results with existing methods are given that establish the proposed ones as attractive alternatives. In general, they outperform existing methods for both low and high compression ratios. Moreover, lower and upper bounds for the vertical scaling factors that are computed by the first method are presented.

Efficient computation of the Hutchinson metric between digitized images

The Hutchinson metric is a natural measure of the discrepancy between two images for use in fractal image processing. An efficient solution to the problem of computing the Hutchinson metric between two arbitrary digitized images is considered. The technique proposed here, based on the shape of the objects as projected on the digitized screen, can be used as an effective way to establish the error between the original and the, possibly compressed, decoded image. To test the performance of our method, we apply it to compare pairs of fractal objects, as well as to compare real-world images with the corresponding reconstructed ones.

Fractal active shape models

Active Shape Models often require a considerable number of training samples and landmark points on each sample, in order to be efficient in practice. We introduce the Fractal Active Shape Models, an extension of Active Shape Models using fractal interpolation, in order to surmount these limitations. They require a considerably smaller number of landmark points to be determined and a smaller number of variables for describing a shape, especially for irregular ones. Moreover, they are shown to be efficient when few training samples are available.

Parameter Identification of 1D Recurrent Fractal Interpolation Functions with Applications to Imaging and Signal Processing

Recurrent fractal interpolation functions are very useful in modelling irregular (non-smooth) data. Two methods that use bounding volumes and one that uses the concept of box-counting dimension are introduced for the identification of the vertical scaling factors of such functions. The first two minimize the area of the symmetric difference between the bounding volumes of the data points and their transformed images, while the latter aims at achieving the same box-counting dimension between the original and the reconstructed data. Comparative results with existing methods in imaging applications are given, indicating that the proposed ones are competitive alternatives for both low and high compression ratios.

Effective Representation of 2D and 3D Data Using Fractal Interpolation

Methods for representing curves in \mathbb{R}^2 and \mathbb{R}^3 using fractal interpolation techniques are presented. We show that such representations are both effective and convenient for irregular or complicated data. Experiments in various datasets, including geographical and medical data, verify the practical usefulness of these methods.

An overview of parallel visualisation methods for Mandelbrot and Julia sets

Abstract We present a comparative study of simple parallelisation schemes for the most widely used methods for the graphical representation of Mandelbrot and Julia sets. The compared methods render the actual attractor or its complement.

BIVARIATE FRACTAL INTERPOLATION SURFACES: THEORY AND APPLICATIONS

We consider the theory and applications of bivariate fractal interpolation surfaces constructed as attractors of iterated function systems. Specifically, such kind of surfaces constructed on rectangular domains have been used to demonstrate their efficiency in computer graphics and image processing. The methodology followed is based on the labeling used for the vertices of the rectangular domain rather than on the constraints satisfied by the contractivity factors or the boundary data.

Sequential visualisation methods for the Mandelbrot set

Sequential visualisation methods for the most widely used methods for the graphical representation of the Mandelbrot set are compared. Two groups of methods are presented. In the first, the Mandelbrot set (or its border) is rendered and, in the second, its complement is rendered. Examples of two-dimensional images obtained using these methods are also given.

Volume Data Visualization Using Fractal Interpolation Surfaces

Visualization of medical or experimental data is often achieved by extracting an intermediate geometric representation of the data. One such popular method for extracting an isosurface from volume data is the Marching Cubes (MC) algorithm, which creates a polygon mesh by sampling the data at the vertices of the cubes of a 3D grid. We present a method for isosurface approxima- tion that uses the vertex extraction phase of the MC algorithm, but subsequently represents the data by fractal interpolation surfaces instead of a polygon mesh. The proposed method is appropriate for isosurfaces that are not locally flat, such as natural structures. Another advantage is that a coarser grid resolution can typically be used, since fractal interpolation surfaces are particularly good at representing detailed, irregular or self-similar structures. Thus for many cases the resulting isosurface is more accurate and more compact. The multiresolution extension of the method is also straightforward. Experimental data verify the practical usefulness of the proposed method. Index Terms—Volume data visualization, marching cubes, fractal interpolation surfaces, isosurfaces.