Estimation of instrinsic dimension via clustering

The problem of estimating the intrinsic dimension of a set of points in high dimensional space is a critical issue for a wide range of disciplines, including genomics, finance, and networking. Current estimation techniques are dependent on either the ambient or intrinsic dimension in terms of computational complexity, which may cause these methods to become intractable for large data sets. In this paper, we present a clustering-based methodology that exploits the inherent self-similarity of data to efficiently estimate the intrinsic dimension of a set of points. When the data satisfies a specified general clustering condition, we prove that the estimated dimension approaches the true Hausdorff dimension. Experiments show that the clustering-based approach allows for more efficient and accurate intrinsic dimension estimation compared with all prior techniques, even when the data does not conform to obvious self-similarity structure. Finally, we present empirical results which show the clustering-based estimation allows for a natural partitioning of the data points that lie on separate manifolds of varying intrinsic dimension

Fractal Analysis for Cancer Research: Case Study and Simulation of Fractals

2010 Mathematics Subject Classification: 65D18.This paper discusses the possibilities of application of fractal geometry for cancer research. Fractal geometry is a new tool that can be extremely useful for many problems in almost every scientific field. The studies recently done in medicine show fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Cancer diagnosis can be done via determination of fractal dimension. A likelihood ratio test for the Hausdorff dimension is employed in [7] We empirically checked the obtained tests on Sierpinski Carpet and on cancer data. However, several issues arisen, especially those related to simulation of fractals which may mimic tissues. These are discussed in the present paper

Use of intensity - and spatial-based image descriptors to characterise and quantify neoplastic lesions in positron emission tomography

Intra-tumour biological heterogeneity is a characteristic shared by all cancers and is thought to contribute to treatment failure. Within-lesion spatial heterogeneity can be qualitatively visualised in Positron Emission Tomography (PET) imaging. Quantifying the variability of the biological processes and the complexity of the signal being measured in PET oncology is essential. The aim of this thesis was to develop and validate intensity- and spatial-based metrics to quantitatively account for the complexity of radiotracer uptake and to annotate intra-tumour PET heterogeneity. Texture analysis was employed to characterise the in vivo tumour heterogeneity of cell proliferation in breast tumours using 18F-fluorothymidine (18F-FLT) PET. The repeatability of the feature measurements was assessed in patients who had two PET scans prior to therapy. Associations between features at baseline and clinical response measured after three cycles of chemotherapy were explored. Associations between feature changes at one week after the start of chemotherapy and clinical response were also explored. Furthermore, the influence of analysis parameters and imaging protocols were studied. A subset of textural features produced reliable measurements and were associated with treatment response. A technique based on multifractal analysis was also developed for characterising the space-filling properties of an object of interest in PET imaging. The derived spatial index was further combined with intensity metrics and the technique was shown to correct for partial volume effects. The method was illustrated on mathematical objects, validated on test-retest 18F-FLT PET clinical data and applied to realistic PET simulations. This work contributes to the demonstration that intensity- and spatial-based image analysis methods can supplement existing methods in PET quantification studies. These techniques provide some improvements on existing methods to derive classical quantitative PET indices and permit extraction of additional information to further characterise patient populations in the clinical setting and in relation to therapy.Open Acces

Interactive evolutionary 3D fractal modeling.

Pang, Wenjun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 83-88).Abstracts in English and Chinese.ACKNOWLEDGEMENTS --- p.iiABSTRACT --- p.iv摘要 --- p.vCONTENTS --- p.viList of Tables --- p.viiiList of Figures --- p.ixChapter 1. --- INTRODUCTION --- p.1Chapter 1.1 --- Recent research work --- p.4Chapter 1.2 --- Objectives --- p.8Chapter 1.3 --- Thesis Organization --- p.10Chapter 2. --- FRACTAL MODELING --- p.12Chapter 2.1 --- Fractal and Fractal Art --- p.12Chapter 2.2 --- Fractal Geometry --- p.15Chapter 2.3 --- Construction of Fractals --- p.21Chapter 2.4 --- Fractal Measurement and Aesthetics --- p.27Chapter 3. --- OVERVIEW OF EVOLUTIONARY DESIGN --- p.30Chapter 3.1 --- Initialization --- p.33Chapter 3.2 --- Selection --- p.33Chapter 3.3 --- Reproduction --- p.34Chapter 3.4 --- Termination --- p.36Chapter 4. --- EVOLUTIONARY 3D FRACTAL MODELING --- p.38Chapter 4.1 --- Fractal Construction --- p.38Chapter 4.1.1 --- Self-similar Condition of Fractal --- p.38Chapter 4.1.2 --- Fractal Transformation (FT) IFS Formulation --- p.39Chapter 4.1.3 --- IFS Genotype and Phenotype Expression --- p.41Chapter 4.2 --- Evolutionary Algorithm --- p.43Chapter 4.2.1 --- Single-point Crossover --- p.45Chapter 4.2.2 --- Arithmetic Gaussian mutation --- p.45Chapter 4.2.3 --- Inferior Elimination --- p.46Chapter 4.3 --- Interactive Fine-tuning using FT IFS --- p.46Chapter 4.4 --- Gaussian Fitness Function --- p.48Chapter 5. --- GAUSSIAN AESTHETIC FITNESS FUNCTION --- p.49Chapter 5.1 --- Fitness Considerations --- p.50Chapter 5.2 --- Fitness Function Formulation --- p.53Chapter 5.3 --- Results and Discussion on Fitness Function --- p.55Chapter 6. --- EXPERIMENT RESULTS and DISCUSSION --- p.59Chapter 6.1 --- Experiment of Evolutionary Generation --- p.59Chapter 6.2 --- Comparison on Different Methods --- p.60Chapter 7. --- 3D FRACTALS RENDERING and APPLICATION --- p.62Chapter 7.1 --- Transforming Property and User Modification --- p.62Chapter 7.2 --- Visualization and Rendering of 3D Fractals --- p.66Chapter 7.3 --- Applications in Design --- p.74Chapter 8. --- CONCLUSIONS and FUTURE WORK --- p.81Chapter 8.1 --- Conclusions --- p.81Chapter 8.2 --- Future Work --- p.81BIBLIOGRAPHY --- p.83Appendix --- p.89Marching Cubes Method --- p.8

Use of wavelet-packet transforms to develop an engineering model for multifractal characterization of mutation dynamics in pathological and nonpathological gene sequences

This study uses dynamical analysis to examine in a quantitative fashion the information coding mechanism in DNA sequences. This exceeds the simple dichotomy of either modeling the mechanism by comparing DNA sequence walks as Fractal Brownian Motion (fbm) processes. The 2-D mappings of the DNA sequences for this research are from Iterated Function System (IFS) (Also known as the Chaos Game Representation (CGR)) mappings of the DNA sequences. This technique converts a 1-D sequence into a 2-D representation that preserves subsequence structure and provides a visual representation. The second step of this analysis involves the application of Wavelet Packet Transforms, a recently developed technique from the field of signal processing. A multi-fractal model is built by using wavelet transforms to estimate the Hurst exponent, H. The Hurst exponent is a non-parametric measurement of the dynamism of a system. This procedure is used to evaluate gene-coding events in the DNA sequence of cystic fibrosis mutations. The H exponent is calculated for various mutation sites in this gene. The results of this study indicate the presence of anti-persistent, random walks and persistent sub-periods in the sequence. This indicates the hypothesis of a multi-fractal model of DNA information encoding warrants further consideration.;This work examines the model\u27s behavior in both pathological (mutations) and non-pathological (healthy) base pair sequences of the cystic fibrosis gene. These mutations both natural and synthetic were introduced by computer manipulation of the original base pair text files. The results show that disease severity and system information dynamics correlate. These results have implications for genetic engineering as well as in mathematical biology. They suggest that there is scope for more multi-fractal models to be developed

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

Cortical complexity as a measure of age-related brain atrophy

The structure of the human brain changes in a variety of ways as we age. While a sizeable literature has examined age-related differences in cortical thickness, and to a lesser degree, gyrification, here we examined differences in cortical complexity, as indexed by fractal dimensionality in a sample of over 400 individuals across the adult lifespan. While prior studies have shown differences in fractal dimensionality between patient populations and age-matched, healthy controls, it is unclear how well this measure would relate to age-related cortical atrophy. Initially computing a single measure for the entire cortical ribbon, i.e., unparcellated gray matter, we found fractal dimensionality to be more sensitive to age-related differences than either cortical thickness or gyrification index. We additionally observed regional differences in age-related atrophy between the three measures, suggesting that they may index distinct differences in cortical structure. We also provide a freely available MATLAB toolbox for calculating fractal dimensionality

Soil Geography and Geostatistics - Concepts and Applications

Geostatistics are a useful tool for understanding and mapping the variation of soil properties across the landscapes. They can be applied at different scales regarding the initial punctual datasets the soil scientist has been provided, and regarding the target resolution of the study. This report is a collection of various studies, all dealing with geostatistical methods, which have been done in Hungary, Russia and Mexico, with the financial support of various research grants. It provides also a chapter about the general concepts of geostatistics and a discussion about limitations of geostatistics with an opening discussion on the usage of pedodiversity index. This report is then particularly recommended to soil scientists who are not so familiar with geostatistics and who need support for applying geostatistics in specific conditions.JRC.H.7-Land management and natural hazard