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In recent years, fractal geometries have been explored in various branches of science and engineering. In antenna engineering several of these geometries have been studied due to their purported potential of realizing multi-resonant antennas. Although due to the complex nature of fractals most of these previous studies were experimental, there have been some analytical investigations on the performance of the antennas using them. One such analytical attempt was aimed at quantitatively relating fractal dimension with antenna characteristics within a single fractal set. It is however desirable to have all fractal geometries covered under one framework for antenna design and other similar applications. With this objective as the final goal, we strive in this paper to extend an earlier approach to more generalized situations, by incorporating the lacunarity of fractal geometries as a measure of its spatial distribution. Since the available measure of lacunarity was found to be inconsistent, in this paper we propose to use a new measure to quantize the fractal lacunarity. We also demonstrate the use of this new measure in uniquely explaining the behavior of dipole antennas made of generalized Koch curves and go on to show how fundamental lacunarity is in influencing electromagnetic behavior of fractal antennas. It is expected that this averaged measure of lacunarity may find applications in areas beyond antennas.

Tractography images produced by Magnetic Resonance Imaging scans have been used to calculate the topology of the neuron tracts in the human brain. This technique gives neuroanatomical details, limited by the system resolution properties. In the observed scales the images demonstrated the statistical self-similar structure of the neuron axons and its fractal dimensions were estimated using the classic Box Counting technique. To assess the degree of clustering in the neural tracts network, lacunarity was calculated using the Gliding Box method. The two-dimensional tractography images were taken from four subjects using various angles and different parts in the brain. The results demonstrated that the average estimated fractal dimension of tractography images is approximately D_f = 1.60 with standard deviation 0.11 for healthy human-brain tissues, and it presents statistical self-similarity features similar to many other biological root-like structures.

The average distance between points of a fractal is proposed as a natural measure of the way in which the points of a fractal are distributed. The average distance between points of the Cantor Set is found to be , the average distance between points of the Cantor p-set is , and the average distance between points of the Fat Cantor p-set is . A general formula for computing the average distance between points of a self-similar set satisfying the open set condition is found.

This paper presents the method of grading of cervical cancer images according to cell formation in the tissue. The variation of intensity and texture complexity of cancer cell images are calculated by fractal dimension methods. Box Counting Method (D_B) and HarFA Programme software are used to find out the dimension of the cells. Contact model and Epidemic Model are the two different models used here. The contact model shows the contact of the cells and the formation of the cancer in the part of the organ. The Epidemic model shows the growth of cancer. Lacunarity is formed between the cells. If the Lacunarity increases the dimension also increases. On seeing the growth of the cancer cell the pathologist can determine the grade of the cancer which in turn will help him to diagnose the cervical cancer and recommend appropriate treatment.

Lacunarity (L) is a scale (r)-dependent parameter that was developed for quantifying clustering in fractals and has subsequently been employed to characterize various natural patterns. For multifractals it can be shown analytically that L is related to the correlation dimension, D₂, by: dlog(L)/dlog(r) = D₂ - 2. We empirically tested this equation using two-dimensional multifractal grayscale patterns with known correlation dimensions. These patterns were analyzed for their lacunarity using the gliding-box algorithm. D₂ values computed from the dlog(L)/dlog(r) analysis gave a ~1:1 relationship with the known D₂ values. Lacunarity analysis was also employed in discriminating between multifractal grayscale patterns with the same D₂ values, but different degrees of scale-dependent clustering. For this purpose, a new lacunarity parameter, 〈L〉, was formulated based on the weighted mean of the log-transformed lacunarity values at different scales. This approach was further used to evaluate scale-dependent clustering in soil thin section grayscale images that had previously been classified as multifractals based on standard method of moments box-counting. Our results indicate that lacunarity analysis may be a more sensitive indicator of multifractal behavior in natural grayscale patterns than the standard approach. Thus, multifractal behavior can be checked without having to compute the whole spectrum of non-integer dimensions, D_q(-∞ < q < +∞) that typically characterize a multifractal. The new 〈L〉 parameter should be useful to researchers who want to explore the correlative influence of clustering on flow and transport in grayscale representations of soil aggregates and heterogeneous aquifers.