A note on local fractal dimensions of graphs of continuous functions

Patterns of electrical trees in insulators have been investigated numerically using fractal dimension. The complexity of the pattern of the tree increases with applied voltage. The degree of the complexity depends on the polarity of the applied voltage. The change of the pattern of the tree can be indicated by the fractal dimension. However, in high voltages the fractal dimensions of the trees are almost same one in both polarities. This means that the pattern of the tree consist of some fractal patterns. Thus we must focus on the local pattern of the tree which is estimated by local fractal dimension. In this paper, the pattern is analyzed by the local fractal dimension method, i.e. local fractal dimension map and local fractal dimension spectrum. The needle electrode in a silicon rubber has been applied three or four times with impulse voltage. The local fractal dimension maps and its spectra have been compared and then the next growth of the tree or branches has been predicted using the local fractal dimension map. In addition, the change of the local fractal dimension of the tree along the branch has been investigated.

In the paper [2] we introduced and investigated complete orthomodular lattices generated by graphs of continuous functions. A natural question arises: can such a lattice be represented by the lattice of projectors in a Hilbert space (the standard quantum logic)? The answer is no, because the covering law is not satisfied in this case.

There are many research papers dealing with fractal dimension of real-valued fractal functions in the recent literature. The main focus of our paper is to study the fractal dimension of complex-valued functions. This paper also highlights the difference between dimensional results of the complex-valued and real-valued fractal functions. We study the fractal dimension of the graph of complex-valued function $$g(x)+i h(x)$$ , compare its fractal dimension with the graphs of functions $$g(x)+h(x)$$ and (g(x), h(x)) and also obtain some bounds. Moreover, we study the fractal dimension of the graph of complex-valued fractal interpolation function associated with a germ function f, base function b, and scaling functions $$\alpha _k$$ .

Fractal analysis of the martian landscape: A study of kilometre-scale topographic roughness

Bandt and Kravchenko [Nonlinearity 24 (2011), pp. 2717–2728] proved that if a self-similar set spans R m \mathbb {R}^m , then there is no tangent hyperplane at any point of the set. In particular, this indicates that a smooth planar curve is self-similar if and only if it is a straight line. When restricting curves to graphs of continuous functions, we can show that the graph of a continuous function is self-similar if and only if the graph is a straight line, i.e., the underlying function is affine.

In this paper it is shown that there is a difference in local fractal dimension between imaged glandular tissue, supporting tissue and muscle tissue based on an assessment from a mammogram. By estimating the density difference at four different local dimensions (2.06, 2.33, 2.48, 2.70) from 142 mammograms we can define a measure and by using this measure we are able to distinguish between the tissue types. A ROC-analysis gives us an area under the curve-value of 0.9998 for separating glandular tissue from muscular tissue and 0.9405 for separating glandular tissue from supporting tissue. To some extent we can say that the measured difference in fractal properties is due to different fractal properties of the unprojected tissue. For example, to a large extent muscle tissue seems to have different fractal properties than glandular or supportive tissue. However, a large variance in the local dimension densities makes it difficult to make proper use of the proposed measure for segmentation purposes.

A number of important problems in medical imaging can be described as segmentation problems. Previous fractal-based image segmentation algorithms have used either the local fractal dimension alone or the local fractal dimension and the corresponding image intensity as features for subsequent pattern recognition algorithms. An image segmentation algorithm that utilized the local fractal dimension, image intensity, and the correlation coefficient of the local fractal dimension regression analysis computation, to produce a three-dimension feature space that was partitioned to identify specific pixels of dental radiographs as being either bone, teeth, or a boundary between bone and teeth also has been reported. In this work we formulated the segmentation process as a configurational optimization problem and discuss the application of simulated annealing optimization methods to the solution of this specific optimization problem. The configurational optimization method allows information about both, the degree of correspondence between a candidate segment and an assumed textural model, and morphological information about the candidate segment to be used in the segmentation process. To apply this configurational optimization technique with a fractal textural model however, requires the estimation of the fractal dimension of an irregularly shaped candidate segment. The potential utility of a discrete Gerchberg-Papoulis bandlimited extrapolation algorithm to the estimation of the fractal dimension of an irregularly shaped candidate segment is also discussed.

Fractal interpolation is a method of generating functions that pass through given points. We describe the method through an example, illustrated in Figures 1-4. Figure 1 shows four points, through which we will pass a function. Figure 2 shows linear interpolation through the four points; this is the stage in fractal interpolation, and will be referred to as the first To obtain the second iteration, shown in Figure 3, the whole function of Figure 2 is superimposed on each line segment of Figure 2, with the vertical scale multiplied (in this instance) by 0.7. The third iteration is obtained by copying the function of Figure 3 onto each line segment of Figure 2, and so on. Each iteration is thus obtained as three distorted copies of the previous iteration. (This process of distorting and copying is a so-called shear transformation; the heart of the construction is, therefore, three shear transformations.) The function obtained as the limit of infinitely many iterations is the fractal interpolation function. Figure 4 shows the eighth iteration of our example. The limit function is a continuous function that passes through the originally specified points, so it does interpolate. Regarded as a subset of the plane, the graph of the function has, in general, a (fractional) dimension greater than 1 but less than 2, so it is a fractal. The dimension of the graph of the limit function of Figures 1-4 is 1.675. Fractal interpolation is interesting for musical applications because complicated shapes can be specified with relatively little information, namely, the coordinates of the original points, and for each line segment in the iteration (linear interpolation), a number between -1.0 and 1.0 called the displacement for that segment. If we take x values in the range 0 to 1.0 and y values in the range -1.0 to 1.0, this information for the example of Figures 1-4 is just the four points (0, 0), (0.333, 0.5), (0.667, 0.5) and (1.0, 0) and the three displacements 0.7, 0.7, and 0.7. A negative displacement simply means that the waveform of the previous iteration is inverted before being scaled and copied. Figure 5 shows the eighth iteration of the function through the same points as in Figure 1, but with displacements -0.7, -0.7, and -0.7. Figure 6 again shows the same points, but with displacements 0.2, -0.5, and 0.4. Figures 7 and 8 show the and tenth iterations of an extreme example; there are three original points at (0, 0), (0.7853, 0.5), and (1.0, 0), and the two displacements are 0.9 and -0.9.

Objectives: The purpose of this paper is to describe a technique for computing the local fractal dimension of the human cerebral cortex as extracted from high-resolution magnetic resonance imaging scans. Methods: 3D models of the human cerebral cortex were extracted from high resolution magnetic resonance images of 10 healthy adult volunteers using FreeSurfer. The local fractal dimension of the cortex was computed using a custom-written cube-counting algorithm. The effect of constraining the maximum region size on the measured value of local fractal dimension was examined. A proof of principle was demonstrated by comparing an individual with Alzheimer’s disease to a healthy individual. Results: Local values of cortical fractal dimension can be obtained by constraining the size of the region over which the cube counting is performed. Cubic regions of intermediate size (30 × 30 × 30 mm) yielded a profile that demonstrated greater regional variability compared to smaller (15 × 15 × 15 mm) or larger (60 × 60 × 60 mm) region sizes. Conclusions: Local fractal dimension of the cerebral cortex is a novel measure that may yield additional, quantitative insight into the clinical meaning of cortical shape changes.

Patterns of electrical trees in insulators are estimated numerically by fractal dimension. However they are not simple fractal but multifractal. They can be estimated by global spectrum and singularity, which do not have clear local information. A local fractal dimension and its spectrum are introduced. Local fractal dimension maps of impulse trees were calculated and the development considered by it and its spectrum. The change of the pattern of the impulse trees in applying the next impulse voltage can be predicted by considering the difference in the local fractal dimension spectrum in each stage.

We investigate various estimators based on extreme value theory (EVT) for determining the local fractal dimension of chaotic dynamical systems. In the limit of an infinitely long time series of an ergodic system, the average of the local fractal dimension is the system's global attractor dimension. The latter is an important quantity that relates to the number of effective degrees of freedom of the underlying dynamical system, and its estimation has been a central topic in the dynamical systems literature since the 1980s. In this work, we propose a framework that combines phase space recurrence analysis with EVT to estimate the local fractal dimension around a particular state of interest. While the EVT framework allows for the analysis of high-dimensional complex systems, such as the Earth's climate, its effectiveness depends on robust statistical parameter estimation for the assumed extreme value distribution. In this study, we conduct a critical review of several EVT-based local fractal dimension estimators, analyzing and comparing their performance across a range of systems. Our results offer valuable insights for researchers employing the EVT-based estimates of the local fractal dimension, aiding in the selection of an appropriate estimator for their specific applications.

This paper analyzes landcover in the Amazon using local fractal dimensions in Synthetic Aperture Radar (SAR) data compared to SAR and Thematic Mapper (TM) brightness information. To improve SAR data classification accuracy, we apply local fractal dimensions to classifying water areas in the Amazon.SAR, unlike TM, cannot distinguish water types. Local fractal dimensions in SAR data classify two landcover classes-varzea and others-that SAR brightness information cannot classify. Local fractal dimensions also extract detailed landcover at data acquisition more accurately than TM brightness information. The estimation ratio of the water-1 & water-2 in two study sites is about 34.2% using local fractal dimensions in SAR data.

The title means, of course, one and the same set of points be both the graph of a continuous function and the graph of a discontinuous function? If S is a set of points in a plane and we can set up axes in such a way that no two points of S are in the same vertical line, then S is the graph of some function. If we can do this in two ways, S is the graph of two functions. The question is, then, can one of these functions be continuous and the other discontinuous? The answer is yes. Let us start by quoting the definition of continuity.

Trees in insulators have been investigated using fractal geometry. The complexity of the tree is estimated by only one number of the fractal dimension. In some cases that the local shape of the branches of the trees is different in another branches, it is not useful to analyze whole tree. Tree should be multifractal. Therefore multifractal analysis is done and then a global spectrum is obtained. It is, however, difficult to understand the curve of the global spectrum (Global dimension and singularity) because both numbers mean dimension. We sometimes need a local information on the tree. Multifractal, however, does not tell us it because the local information is lost when it is calculated. We suggested new numerical method for trees; local fractal dimension. Fractal dimension at each point on the tree is calculated. The correlation function is used for estimating local fractal dimension. We furthermore suggest wavelet analysis instead of correlation function. For the correlation function, the area calculated is determined by someone. In the case of wavelet function, the area is, however, automatically determined, because the wavelet function changes their supported region automatically. It is easy to point out the change of the tree and it is useful to estimate the development of the tree using these methods.

To analyze microvascular and structural changes in radiation maculopathy and their influence on visual acuity (VA), using optical coherence tomography (OCT) and OCT angiography (OCTA). This was a retrospective analysis of consecutive patients with radiation maculopathy, 12 months or more after proton-beam irradiation for uveal melanoma, imaged with fluorescein angiography, OCT, and OCTA. Clinical parameters potentially affecting VA were recorded, including OCTA-derived metrics: foveal avascular zone (FAZ) area, vascular density, and local fractal dimension of the superficial (SCP) and deep capillary plexuses (DCP). Nonirradiated fellow eyes served as controls. Ninety-three patients were included. FAZ was larger, while SCP/DCP capillary density and local fractal dimension were lower in the 35 irradiated than in the 35 fellow eyes (P < 0.0001). Microvascular alterations graded on fluorescein angiography (minimally damaged/disrupted/disorganized) were correlated to FAZ area and SCP/DCP density on OCTA (P < 0.01). By univariate analysis, worse VA was associated to macular detachment at presentation (P = 0.024), total macular irradiation (P = 0.0008), higher central macular thickness (CMT) (P = 0.019), higher absolute CMT variation (P < 0.0001), cystoid edema (P = 0.030), ellipsoid zone disruption (P = 0.002), larger FAZ (P < 0.0001), lower SCP (P = 0.001) and DCP capillary density (P < 0.0001), and lower SCP (P = 0.009) and DCP local fractal dimension (P < 0.0001). Two multivariate models with either capillary density or fractal dimension as covariate showed that younger age (P = 0.014/0.017), ellipsoid zone disruption (P = 0.034/0.019), larger FAZ (P = 0.0006/0.002), and lower DCP density (P = 0.008) or DCP fractal dimension (P = 0.012), respectively, were associated with worse VA. VA of eyes with radiation maculopathy is influenced by structural and microvascular factors identified with OCTA, including FAZ area and DCP integrity.