Applications of topology in computer algorithms

The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on advances in computing hardware, the major intellectual achievements are the algorithms. The applications of general topology in other branches of mathematics are not discussed, since they are not applications of mathematics outside of mathematics.Comment: This paper is based on the invited lecture at International Conference on Topology and Applications held in August 23--27, 1999, at Kanagawa University in Yokohama, Japa

Fractals Study and Its Application

The overall of this paper is a review of fractal in many areas of application. The review exposes fractal definition, analysis, and its application. Most applications discussed are based on analysis from geometric and image processing studies. Patterns of some fractals will be discussed. Some simulation results are supplied to illustrate the discussion. Simulation resulted are from various software and tools. Some principles of fractals with informative patterns have been simulated. Whereas the simulations could support some recommendations for prospective purposes and applications. The prospective application may help in predictive pattern of many fields. The predictive pattern will lead to pattern control and pattern disruptions

Fractal analyses of some natural systems

Fractal dimensions are estimated by the box-counting method for real world data sets and for mathematical models of three natural systems. 1 he natural systems are nearshore sea wave profiles, the topography of Shei-pa National Park in Taiwan, and the normalised difference vegetation index (NDV1) image of a fresh fern. I he mathematical models which represent the natural systems utilise multi-frequency sinusoids for the sea waves, a synthetic digital elevation model constructed by the mid-point displacement method for the topography and the Iterated Function System (IFS) codes for the fern leaf. The results show that similar fractal dimensions are obtained for discrete sub-sections of the real and synthetic one-dimensional wave data, whilst different fractal dimensions are obtained for discrete sections of the real and synthetic topographical and fern data. The similarities and differences are interpreted in the context of system evolution which was introduced by Mandelbrot (1977). Finally, the results for the fern images show that use of fractal dimensions can successfully separate void and filled elements of the two-dimensional series

Evo-devo and the search for homology ("sameness”) in biological systems

Developmental biology and evolutionary studies have merged into evolutionary developmental biology ("evo-devo”). This synthesis already influenced and still continues to change the conceptual framework of structural biology. One of the cornerstones of structural biology is the concept of homology. But the search for homology ("sameness”) of biological structures depends on our favourite perspectives (axioms, paradigms). Five levels of homology ("sameness”) can be identified in the literature, although they overlap to some degree: (i) serial homology (homonomy) within modular organisms, (ii) historical homology (synapomorphy), which is taken as the only acceptable homology by many biologists, (iii) underlying homology (i.e., parallelism) in closely related taxa, (iv) deep evolutionary homology due to the "same” master genes in distantly related phyla, and (v) molecular homology exclusively at gene level. The following essay gives emphasis on the heuristic advantages of seemingly opposing perspectives in structural biology, with examples mainly from comparative plant morphology. The organization of the plant body in the majority of angiosperms led to the recognition of the classical root-shoot model. In some lineages bauplan rules were transcended during evolution and development. This resulted in morphological misfits such as the Podostemaceae, peculiar eudicots adapted to submerged river rocks. Their transformed "roots” and "shoots” fit only to a limited degree into the classical model which is based on either-or thinking. It has to be widened into a continuum model by taking over elements of fuzzy logic and fractal geometry to accommodate for lineages such as the Podostemacea

Dendritic Bloom

University of Michiganhttps://deepblue.lib.umich.edu/bitstream/2027.42/144744/1/DendriticBloomThesis_Allen-Wickler_Olivia.pd

Evaluating High-Resolution Aerial Photography Acquired by Unmanned Aerial Systems for Use in Mapping Everglades Wetland Plant Associations

Mapping of vegetation patterns over large extents using remote sensing methods requires field sample collections for two different purposes: (1) the establishment of plant association classification systems from samples of relative abundance estimates; and (2) training for supervised image classification and accuracy assessment of satellite data derived maps. One challenge for both procedures is the establishment of confidence in results and the analysis across multiple spatial scales. Continuous data sets that enable cross-scale studies are very time consuming and expensive to acquire and such extensive field sampling can be invasive. The use of high resolution aerial photography (hrAP) offers an alternative to extensive, invasive, field sampling and can provide large volume, spatially continuous, reference information that can meet the challenges of confidence building and multi-scale analysis

All the Inbent Fractals of Connection

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Patch Autocorrelation Features: A translation and rotation invariant approach for image classification.

The autocorrelation is often used in signal processing as a tool for finding repeating patterns in a signal. In image processing, there are various image analysis techniques that use the autocorrelation of an image in a broad range of applications from texture analysis to grain density estimation. This paper provides an extensive review of two recently introduced and related frameworks for image representation based on autocorrelation, namely Patch Autocorrelation Features (PAF) and Translation and Rotation Invariant Patch Autocorrelation Features (TRIPAF). The PAF approach stores a set of features obtained by comparing pairs of patches from an image. More precisely, each feature is the euclidean distance between a particular pair of patches. The proposed approach is successfully evaluated in a series of handwritten digit recognition experiments on the popular MNIST data set. However, the PAF approach has limited applications, because it is not invariant to affine transformations. More recently, the PAF approach was extended to become invariant to image transformations, including (but not limited to) translation and rotation changes. In the TRIPAF framework, several features are extracted from each image patch. Based on these features, a vector of similarity values is computed between each pair of patches. Then, the similarity vectors are clustered together such that the spatial offset between the patches of each pair is roughly the same. Finally, the mean and the standard deviation of each similarity value are computed for each group of similarity vectors. These statistics are concatenated to obtain the TRIPAF feature vector. The TRIPAF vector essentially records information about the repeating patterns within an image at various spatial offsets. After presenting the two approaches, several optical character recognition and texture classification experiments are conducted to evaluate the two approaches. Results are reported on the MNIST (98.93%), the Brodatz (96.51%), and the UIUCTex (98.31%) data sets. Both PAF and TRIPAF are fast to compute and produce compact representations in practice, while reaching accuracy levels similar to other state-of-the-art methods