Some of the modeling for lung geometry uses exponential models, starting with the premise that the lung is a dichotomous branch tree with a parental branch of greater length and diameter and an exponential decrease with every daughter branch due to maximizing entropy. Fractal geometry has been applied to lung branching by considering the lung as a fractal object lacking characteristic scale and having self-similarity. Yamada, in Principles of Tissue Engineering (Fifth Edition), 2020 Geometryįractal geometry has provided a conceptual basis for identifying mechanisms that may underlie the formation of branches in biological settings. The fractal dimension was measured over 8-by-8 pixel blocks across the image, and the result used to segment the image into regions of different texture. This technique was used in Pentland (1984) to produce fractal maps from images. If one wishes to use data in multiple directions, but anisotropy is not of interest, then the 2-D power spectrum can be calculated over the image area of interest, averaged azimuthally to yield an average radial power spectrum, and the fractal dimension calculated in the 1-D manner. The fractal dimension, D, is then related to the slope of the fitted line, as given in Table 4-6. For either the semivariogram or the power spectrum, a log-log plot is calculated, and a linear function is fit to the log-transformed data (equivalent to fitting a power function to the data before taking the logarithms). The fractal dimension is related to either the semivariogram or power spectrum of the transect ( Carr, 1995). Several tools have been developed for estimating the fractal dimension, D, from an image. Most natural objects and surfaces are found to be approximately fractal over a limited range of scales. The logarithm of the measured perimeter is found to obey a linear relationship to the logarithm of the ruler length ( Feder, 1988) the slope of that log-log relation is related to the fractal dimension. Shorter rulers lead to longer values, and vice versa. If the perimeter is measured with rulers of different lengths (scales), different values are obtained. A classic example is measurement of the length of the perimeter of the coastline of Britain ( Mandelbrot, 1967). This means that the object is statistically the same, no matter what scale it is viewed at. Objects that are “fractal” also obey the notion that they are “self-similar” in their statistics. Generally, the higher the fractal dimension, the finer and “rougher” the texture. An object that is “fractal” has an intermediate dimensionality, such as 1.6 for an irregular line or 2.4 for an image “surface”. 9 There are four topological dimensions in traditional Euclidean geometry: 0-D for points, 1-D for straight lines, 2-D for planes, and 3-D for volumetric objects like cubes and spheres. SchowengerdtProfessor, in Remote Sensing (Third edition), 2007 4.6.5 Fractal Geometryįractal geometry is a way to describe the “texture” of a surface.
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