Estimation of fractal dimension and fractal curvatures from digital images

Most of the known methods for estimating the fractal dimension of fractal sets are based on the evaluation of a single geometric characteristic, e.g. the volume of its parallel sets. We propose a method involving the evaluation of several geometric characteristics, namely all the intrinsic volumes (i.e. volume, surface area, Euler characteristic, etc.) of the parallel sets of a fractal. Motivated by recent results on their limiting behavior, we use these functionals to estimate the fractal dimension of sets from digital images. Simultaneously, we also obtain estimates of the fractal curvatures of these sets, some fractal counterpart of intrinsic volumes, allowing a finer classification of fractal sets than by means of fractal dimension only. We show the consistency of our estimators and test them on some digital images of self-similar sets.     

An improved box-counting method to estimate fractal dimension of images

Fractal dimension (FD) reflects the intrinsic self-similarity of an image and can be used in image classification, image segmentation and texture analysis. The differential box-counting (DBC) method is a common approach to calculating the FD values. This paper proposes an improved DBC-based approach to optimizing the performance of the method in the following ways: reducing fitting errors by decreasing step lengths, considering under-counting boxes on the border of two neighboring box-blocks and making better use of all the pixels in the blocks while not neglecting the middle parts. The experimental results show that the fitting error of the new method can be decreased to 0.012879. The average distance of the FD values is decreased by 16.0/% in the divided images and the average variance of the FD values is decreased by 30/% in the scaled images, compared with other modified methods. The results show that the new method has a better performance in the recognition of the same type of images and the scaled images.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

Noise dependency of algorithms for calculating fractal dimensions in digital images

Fractal properties of real world objects are commonly examined in digital images. Digital images are discrete representations of objects or scenes and are unavoidably contaminated with noise disturbing the representation of the captured objects. We evaluate the noise dependency of frequently applied algorithms for the calculation of the fractal dimension in digital images. Three mathematically defined fractals (Koch Curve, Sierpinski Gasket, Menger Carpet), representative for low, middle and high values of the fractal dimension, together with an experimentally obtained fractal structure were contaminated with well-defined levels of artificial noise. The Box-Counting Dimension, the Correlation Dimension and the rather unknown Tug-of-War Dimension were calculated for the data sets in order to estimate the fractal dimensionality under the presence of accumulated noise. We found that noise has a significant influence on the computed fractal dimensions (relative increases up to 20%) and that the influence is sensitive to the applied algorithm and the space filling characteristics of the investigated fractal structures. The similarities of the effect of noise on experimental and artificial fractals confirm the reliability of the obtained results.     

Fractal dimension for fractal structures: A Hausdorff approach revisited

In this paper, we use fractal structures to study a new approach to the Hausdorff dimension from both continuous and discrete points of view. We show that it is possible to generalize the Hausdorff dimension in the context of Euclidean spaces equipped with their natural fractal structure. To do this, we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties.     

The Minkowski dimension of the bivariate fractal interpolation surfaces

We present a new construction of continuous bivariate fractal interpolation surface for every set of data. Furthermore, we generalize this construction to higher dimensions. Exact values for the Minkowski dimension of the bivariate fractal interpolation surfaces are obtained.     

A method for estimating criteria weights from intuitionistic preference relations

An intuitionistic preference relation is a powerful means to express decision makers’information of intuitionistic preference over criteria in the process of multi-criteria decision making. In this paper, we first define the concept of its consistence and give the equivalent interval fuzzy preference relation of it. Then we develop a method for estimating criteria weights from it, and then extend the method to accommodate group decision making based on them And finally, we use some numerical examples to illustrate the feasibility and validity of the developed method.     

Splitting-integrating method for inverse transformation ofn-dimensional digital images and patterns

The splitting-integrating method is proposed to normalize digital images and patterns inn dimensions under inverse transformation. This method is much simpler than other approaches because no solutions of nonlinear algebraic equations are required. Also, the splitting-integrating method produces images free from superfluous holes and blanks, which often occur in transforming digitized images by other methods.The splitting-integrating method has been applied successfully to pattern recognition and image processing; but no error analysis has been provided so far. Because the image greyness is represented as an integral value, we can derive by numerical analysis error bounds of approximate greyness solutions, to show that when piecewise constant and multi-linear interpolations are used, convergence ratesO(1/N) andO(1/N ²) can be obtained respectively, whereN is a division number such that a pixel in then-dimensional images is split intoN ⁿ subpixels. Moreover, numerical and graphical experiments are carried out for a sample of binary images in two dimensions, to confirm the convergence rates derived.     

A universal dimension formula for complex simple Lie algebras

We present a universal formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra (i.e., a universal formula for the Hilbert functions of homogeneous complex contact manifolds), as well as several other universal formulas. These formulas generalize formulas of Vogel and Deligne and are given in terms of rational functions where both the numerator and denominator decompose into products of linear factors with integer coefficients. We discuss consequences of the formulas including a relation with Scorza varieties.     

A method for estimating interpolating polynomials

The question of obtaining a lower bound for some interpolating polynomials is considered. Under specific conditions it is proved that these bounds are sharp. As a corollary of the general theorem, under specific restrictions on the points of interpolation, lower bounds for Goncharov interpolation polynomials are obtained which coincide with known upper bounds.Translated from Matematicheskie Zametki, Vol. 17, No. 4, pp. 555–561, April, 1975.     

ROI-based procedures for progressive transmission of digital images: A comparison

Nowadays, problems arise when handling large-sized images (i.e. medical image such as Computed Tomographies or satellite images) of 10, 50, 100 or more Megabytes, due to the amount of time required for transmitting and displaying, this time being even worse when a narrow bandwidth transmission medium is involved (i.e. dial-up or mobile network), because the receiver must wait until the entire image has arrived. To solve this issue, progressive transmission schemes are used. These schemes allow the image sender to encode the image data in such a way that it is possible for the receiver to perform a reconstruction of the original image from the very beginning of transmission. Despite this reconstruction being, of course, partial, it is possible to improve the reconstruction on the fly, as more and more data of the original image are received. There are many progressive transmission methods available, such as it planes, TSVQ, DPCM, and, more recently, matrix polynomial interpolation, Discrete Cosine Transform (DCT, used in JPEG) and wavelets (used in JPEG 2000). However, none of them is well suited, or perform poorly, when, in addition to progressive transmission, we want to include also ROIs (Region Of Interest) handling. In the progressive transmission of ROIs, we want not only to reconstruct the image as we receive image data, but also to be able to select which part or parts of the emerging image we think are relevant and want to receive first, and which part or parts are of no interest. In this context we present an algorithm for lossy adaptive encoding based on singular value decomposition (SVD). This algorithm turns out to be well suited for progressive transmission and ROI selection of 2D and 3D images, as it is able to avoid redundancy in data transmission and does not require any sort of data recodification, even if we select arbitrary ROIs on the fly. We compare the performing of SVD with DCT and wavelets and show the results.     

The classification problem for simple unital finite rank dimension groups

The Borel complexity of the isomorphism problem for finite-rank unital simple dimension groups increases with rank. This implies that the isomorphism problems for the corresponding classes of Bratteli diagrams and LDA-groups also increase with rank.     

The bose distribution without bose condensate: Dependence of the chemical potential on fractal dimension

We construct an isotherm on the P,Z plane, where P is the pressure and Z is the compressibility factor, for a problem of number theory, i.e., for an ideal thermodynamic gas. Further, we describe how this isotherm is changed when the interaction between the particles is taken into account.     

The threshold method for estimating total rainfall

The threshold method estimates the total rainfall F _G in a region G using the area B _G of the subregion where rainfall intensity exceeds a certain threshold value c. We model the rainfall in a region by a marked spatial point process and derive a correlation formula between F _G and B _G. This correlation depends not only on the rainfall distribution but also on the variation of number of raining sites, showing the importance of taking account of the spatial character of rainfall. In the extreme case where the variation of number of raining sites is dominant, the threshold method may work regardless of rainfall distributions and even regardless of threshold values. We use the lattice gas model from statistical physics to model raining sites and show a huge variation in the number of raining sites is theoretically possible if a phase transition occurs, that is, physically different states coexist. Also, we show by radar observation datasets that there are huge variations of raining sites actually.     

ON THE BOX DIMENSION FOR A CLASS OF NONAFFINE FRACTAL INTERPOLATION FUNCTIONS

We present lower and upper bounds for the box dimension of the graphs of certain nonaffine fractal interpolation functions by generalizing the results that hold for the affine case.     

Model based method for estimating an attractor dimension from uni/multivariate chaotic time series with application to Bremen climatic dynamics

In this paper, a method for estimating an attractor embedding dimension based on polynomial models and its application in investigating the dimension of Bremen climatic dynamics are presented. The attractor embedding dimension provides the primary knowledge for analyzing the invariant characteristics of the attractor and determines the number of necessary variables to model the dynamics. Therefore, the optimality of this dimension has an important role in computational efforts, analysis of the Lyapunov exponents, and efficiency of modeling and prediction. The smoothness property of the reconstructed map implies that, there is no self-intersection in the reconstructed attractor. The method of this paper relies on testing this property by locally fitting a general polynomial autoregressive model to the given data and evaluating the normalized one step ahead prediction error. The corresponding algorithms are developed in uni/multivariate form and some probable advantages of using information from other time series are discussed. The effectiveness of the proposed method is shown by simulation results of its application to some well-known chaotic benchmark systems. Finally, the proposed methodology is applied to two major dynamic components of the climate data of the Bremen city to estimate the related minimum attractor embedding dimension.     

A very simple algorithm for estimating the number of k-colorings of a low-degree graph

A fully polynomial randomized approximation scheme is presented for estimating the number of (vertex) k-colorings of a graph of maximum degree Δ, when k ≥ 2Δ + 1. © 1995 John Wiley & Sons, Inc.