Fractal geometry has been applied widely in the analysis of medical images to characterize the irregular complex tissue structures that do not lend themselves to straightforward analysis with traditional Euclidean geometry. In this study, we treat the nonfractal behaviour of medical images over large-scale ranges by considering their box-counting fractal dimension as a scale-dependent parameter...

Throughout this thesis, we observe close correlations between values of the topological growth rates and various other fractal indices. These observations are based on both analytic derivations and numerical computations of the relevant exponents. In this chapter we derive inequalities that relate our topological growth rates to existing scaling indices such as the box-counting dimension and th...

A fractal, as originally described by Mandelbrot, is a set with an irregular and fragmented shape. Many fractals that have been extensively studied, such as self-similar sets, have the same degree of irregularity and fragmentation at all length scales. In contrast to this, an equi-homogeneous set is an irregular and fragmented shape that at each fixed scale is identical at every point. We show ...

We present a general method of generating continuous fractal interpolation surfaces by iterated function systems on an arbitrary data set over rectangular grids and estimate their Box-counting dimension.

We prove an equivalence between the Bryan--Steinberg theory of $\pi$-stable pairs on $Y = \mathcal{A}_{m-1} \times \mathbb{C}$ and quasimaps to $X \mathrm{Hilb}(\mathcal{A}_{m-1})$, in form equality K-theoretic equivariant vertices. In particular, combinatorics both vertices are described explicitly via box counting. Then we apply study implications for sheaf-counting theories $Y$ arising from ...

Previously, we introduced a passengers’ counting algorithm in public rail transport. The main disadvantage of that algorithm is it lacks automatic event detection. In this article, we implement two automatic wavelet-based passengers counting algorithms. The new algorithms employ the spatial-domain Laplacian-of-Gaussian-based wavelet, and the frequency-domain applied Non-Linear Difference of Gau...

Dynamics of nanoscopic arrays of monodomain magnetic elements is simulated by means of the Pardavi-Horvath algorithm. Experimental hysteresis loop is reproduced for the arrays of Ni, with the period 100 nm and the mean coercive field 710 Oe.We investigate the box-counting fractal dimension of a cluster of elements with given orientation of magnetic moments. No fractal behavior is found. Also, t...

Background:Leukemia is cancer of blood and bone marrow cells. In general, there are four types of leukemia: chronic myelogenous leukemia (CML), acute myeloid leukemia (AML), B-cell chronic lymphocytic leukemia (CLL) and acute lymphoblastic leukemia (ALL).  Fractal geometry can be introduced as one of the effective ways to detect this type of cancer. In this study, with introduc...

Detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency Deutsche Aktienindex data. The tail index (α), the Renyi exponents based on the box counting algorithm for the graph (dq) and the generalized Hurst exponents (Hq) are computed in parallel for short and daily return times. The result...

We investigate two published approaches to the problem of estimating fractal dimension: the box-counting approach, and the fractal Brownian function approach. In experiments with synthetic images, we find the fractal Brownian function methods proposed by Pentland and Yokoya to be superior to the box-counting approaches described by Voss, because the former do not require data sampled at equal i...