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Tissue Spectroscopy
Published in Vadim Backman, Adam Wax, Hao F. Zhang, A Laboratory Manual in Biophotonics, 2018
Vadim Backman, Adam Wax, Hao F. Zhang
A fractal is defined as a system that exhibits self-similarity at different length scales [4]. A weak form of this definition is statistical self-similarity, which applies to random fractals. The autocorrelation function of a random fractal follows a power law. Conceptually, this is because a power law is scale invariant. In other words, the medium looks statistically similar at different scales or “zoom levels.” Autocorrelation functions of real materials can follow a power law only over some finite range of length scales from a small inner to a large outer scale [4]. Obviously, the refractive index correlation cannot continue beyond the length scale of the body, nor does it have meaning at the length scale of atoms, but a power-law correlation over some finite range represents an approximate fractal. When the value of parameter D is less than the Euclidean dimension (in 3D space, DE = 3), the value D represents the mass fractal dimension, and the autocorrelation Bn(r) is proportional to rD−3 for values of r much less than Ln. As r approaches Ln, the function falls off more quickly than the power law, and Ln can be considered the outer scale of the fractal (Figure 5.2).
Analysis techniques for optimising production systems
Published in D.R. Moore, D.J. Hague, Building Production Management Techniques, 2014
Some of the relevant issues concerning prioritisation were covered briefly in Chapter 2 when the scheduling of resources was introduced. This is useful in that one aspect of identifying activities which have the potential to exhibit uncontrolled rates of change is to look for the extent of self similarity within the activity (Moore 1998a). Self similarity is a feature used within chaos theory to differentiate between activities which are only complex and those which are actually chaotic. Chaotic activities are of most importance in this context in that they are prone to rates of change which cannot be readily predicted (Addison 1997). They are therefore the activities within a project which represent the largest risk of that project going out of control and consequently failing.
Fractal Analysis
Published in Shyam S. Sablani, M. Shafiur Rahman, Ashim K. Datta, Arun S. Mujumdar, Handbook of Food and Bioprocess Modeling Techniques, 2006
Fractal analysis is mainly applied when other methods fail or become tedious to solve complex or chaotic problems. Many natural patterns are either irregular or fragmented to such an extreme degree that Euclidian or classical geometry could not describe their form (Mandelbrot 1977, 1987). Any shape can be characterized by whether or not it has a characteristic length (Takayasu 1990). For example, a sphere has a characteristic length defined as the diameter. Shapes with characteristic lengths have an important common property of smoothness of surface. A shape having no characteristic length is called self-similar. Self-similarity is also known as scale-invariance, because selfsimilar shapes do not change their shape under a change of observational scale. This important symmetry gives a clue to understanding complicated shapes, which have no characteristic length, such as the Koch curve or clouds (Takayasu 1990). The idea of a fractal is based on the lack of characteristic length or on self-similarity. The word fractal is a new term introduced by Mandelbrot (1977) to represent shapes or phenomena having no characteristic length. The origin of this word is the Latin adjective fractus meaning broken. The English words “fractional” and “fracture” are derived from this Latin word fractus, which means the state of broken pieces being gathered together irregularly.
Fractality in water distribution networks: application to criticality analysis and optimal rehabilitation
Published in Urban Water Journal, 2021
Kegong Diao, David Butler, Bogumil Ulanicki
Fractals have been identified as one of the most general features of many natural and artificial networks that exhibit self-similarity of the topological patterns, i.e. different parts of the system have similar structures to each other as well as to the whole system (Mandelbrot 1982; Song, Havlin, and Makse 2005, 2006). Understanding fractals is a critical aspect of decoding complex systems, as the pattern of such large systems (with an enormous number of components) can be revealed by identifying only a small part of the system (e.g. as shown in Figure 1(a), the smaller leaves in red and blue boxes have the same structure as the whole leave). Furthermore, the identification of existing features of such systems can start at larger spatial resolution levels, which have fewer details of the system under scrutiny, before proceeding to a more detailed analysis at finer levels (e.g. as shown in Figure 1(b)).
A Novel Infrared Image Enhancement Based on Correlation Measurement of Visible Image for Urban Traffic Surveillance Systems
Published in Journal of Intelligent Transportation Systems, 2020
Jingyue Chen, Xiaomin Yang, Lu Lu, Qilei Li, Zuoyong Li, Wei Wu
Image self-similarity is important because it forms the basis for many imaging techniques such as non-local means denoising and fractal image coding. The feature of a patch can be represented by the relationship of its self-similarity. In an image, similar patches existed for arbitrary image patches, and its attributes can be expressed by self-similarity (Benabdelkader, Cutler, Nanda, & Davis, 2001; Deselaers & Ferrari, 2010; Shechtman & Irani, 2008). Gray information can characterize the objective nature of images, with accurate, stable and other features. In the proposed method, the Euclidean distance (Elmore & Richman, 2001; Li & Lu, 2009) is employed to measure the similarity of image patches. In Eq. (2), d(i, j) is the distance between (i, j) and surrounding points. Figure 6 shows the schematic of self-similarity measurement.
Introducing BAT inspired algorithm to improve fractal image compression
Published in International Journal of Computers and Applications, 2020
Rafik Menassel, Idriss Gaba, Khalil Titi
FIC is one of the new techniques for lossy image compression [10]. It was first introduced by Hutson [11] and Barnsley [12]. This technique identifies possible self-similarity within the image that can be used to reduce the amount of data required to reproduce the image. Traditionally these methods have been time-consuming. They search self-similarities between different isolated image areas [13,10] and store only the parameters of contract transform instead of the image pixels. Though this technique is branded by an asymmetric process, it uses so much time in the encoding process, that consists in the exploration of the best match block, frequently on large size images. So, it is best recommended for textures and low-resolution shapes.