Applications of fractals to image data compression

Digital image processing is exploited in many diverse applications but the size of digital images places excessive demands on current storage and transmission technology. Image data compression is required to permit further use of digital image processing. Conventional image compression techniques based on statistical analysis have reached a saturation level so it is necessary to explore more radical methods. This thesis is concerned with novel methods, based on the use of fractals, for achieving significant compression of image data within reasonable processing time without introducing excessive distortion. Images are modelled as fractal data and this model is exploited directly by compression schemes. The validity of this is demonstrated by showing that the fractal complexity measure of fractal dimension is an excellent predictor of image compressibility. A method of fractal waveform coding is developed which has low computational demands and performs better than conventional waveform coding methods such as PCM and DPCM. Fractal techniques based on the use of space-filling curves are developed as a mechanism for hierarchical application of conventional techniques. Two particular applications are highlighted: the re-ordering of data during image scanning and the mapping of multi-dimensional data to one dimension. It is shown that there are many possible space-filling curves which may be used to scan images and that selection of an optimum curve leads to significantly improved data compression. The multi-dimensional mapping property of space-filling curves is used to speed up substantially the lookup process in vector quantisation. Iterated function systems are compared with vector quantisers and the computational complexity or iterated function system encoding is also reduced by using the efficient matching algcnithms identified for vector quantisers.

Services for Satellite Data Processing

Data processing services for Meteosat geostationary satellite are presented. Implemented services correspond to the different levels of remote-sensing data processing, including noise reduction at preprocessing level, cloud mask extraction at low-level and fractal dimension estimation at high-level. Cloud mask obtained as a result of Markovian segmentation of infrared data. To overcome high computation complexity of Markovian segmentation parallel algorithm is developed. Fractal dimension of Meteosat data estimated using fractional Brownian motion models.

Unifractality and multifractality in the Italian stock market

Tests for random walk behaviour in the Italian stock market are presented, based on an investigation of the fractal properties of the log return series for the Mibtel index. The random walk hypothesis is evaluated against alternatives accommodating either unifractality or multifractality. Critical values for the test statistics are generated using Monte Carlo simulations of random Gaussian innovations. Evidence is reported of multifractality, and the departure from random walk behaviour is statistically significant on standard criteria. The observed pattern is attributed primarily to fat tails in the return probability distribution, associated with volatility clustering in returns measured over various time scales. © 2009 Elsevier Inc. All rights reserved.

Basic Principles of Organization of the Medium and Thinking Process of the Human in its Conceptual Presentation

Given in the report conceptual presentation of the main principles of fractal-complexity Ration of the media and thinking processes of the human was formulated on the bases of the cybernetic interpretation of scientific information (basically from neurophysiology and neuropsychology, containing the interpretation giving the best fit to the authors point of view) and plausible hypothesis's, filling the lack of knowledge.

Выявление Фракталоподобных Структур в ДНК-Последовательностях

Разработан и реализован алгоритм выявления фракталоподобных структур в ДНК- последовательностях. Фрактальность трактуется как самоподобие, основанное на свойстве симметрии или комплементарной симметрии. Локальные фракталы интересны своей способностью аккумулировать множественные палиндромно-шпилечные структуры с потенциально возможными регуляторными функциями. Выявлены реальные случаи проявления фрактальности в различных геномах: от вирусов до человека. Рассмотрена возможность использования фракталоподобных структур в качестве маркеров, различающих близкие классы последовательностей.

Suggestion from the Past?

Mathematics Subject Classification: 26A33 (main), 35A22, 78A25, 93A30

Joining the dots:using structured e-portfolio assignments to enhance reflection

Although there has been an increased interest in the use of electronic portfolios in higher education over the last five years, relatively little is known about the potential of such tools to support the development of higher order abilities for students, such as reflection, in a structured way that is suitable for assessment. This paper reports the findings from a small-scale research which sets out to compare the outcomes of reflective assignments in two cohorts of participants in a Postgraduate Certificate in Professional Practice in Higher Education in the UK. Participants in the programme were asked to submit reflective accounts using an e-portfolio system as part of their formal assessment. One cohort completed the assessment using some generic guidelines of how to reflect and construct an e-portfolio page without a given template or structure, whereas another cohort was given a specific template with clear assessment criteria to gauge the assembly of their reflections. The authors, who are also tutors in the programme, analysed the submitted reflections following open coding procedures. The analysis found a tendency for the reflection in the first cohort to be merely descriptive without progressing to speculating objectively about answers to relevant analytical questions about the process involved in the ability under scrutiny. In contrast the assignments of cohort two were found to be more insightful in terms of assimilating random bits of materials, thoughts and self-questions into complete reflective accounts. These findings bring some evidence to support and indeed promote a more structured approach to reflective practice, which can be further enhanced through a carefully created e-portfolio template and associated assessment criteria.

Cross-institutional development of an online open course for educators:confronting current challenges and imagining future possibilities

The Global Dimensions in Higher Education module is a fully online open course for educators, jointly developed by three UK universities through a process of consultation and inquiry involving colleagues within the global education sector. The purpose of the module is two-fold. Firstly, to engage academics within and beyond the partner institutions in a critical exploration of transnational and global issues within higher education. Secondly, to provide the project partners with an opportunity to understand and address the challenges of jointly developing and delivering an online course that is to be offered both openly as well as integrated within credit-bearing and continued professional development provision for academics in the partner institutions. This short paper describes progress to date in developing the Global Dimensions in Higher Education module, our current activity focused on validating and implementing the module, and lessons learned to be considered for the collaborative development of open online courses.

Tracing the evolution of physics on the backbone of citation networks

Many innovations are inspired by past ideas in a nontrivial way. Tracing these origins and identifying scientific branches is crucial for research inspirations. In this paper, we use citation relations to identify the descendant chart, i.e., the family tree of research papers. Unlike other spanning trees that focus on cost or distance minimization, we make use of the nature of citations and identify the most important parent for each publication, leading to a treelike backbone of the citation network. Measures are introduced to validate the backbone as the descendant chart. We show that citation backbones can well characterize the hierarchical and fractal structure of scientific development, and lead to an accurate classification of fields and subfields. © 2011 American Physical Society.

Fibonacci Series, Golden Proportions, and the Human Biology

Pythagoras, Plato and Euclid’s paved the way for Classical Geometry. The idea of shapes that can be mathematically defined by equations led to the creation of great structures of modern and ancient civilizations, and milestones in mathematics and science. However, classical geometry fails to explain the complexity of non-linear shapes replete in nature such as the curvature of a flower or the wings of a Butterfly. Such non-linearity can be explained by fractal geometry which creates shapes that emulate those found in nature with remarkable accuracy. Such phenomenon begs the question of architectural origin for biological existence within the universe. While the concept of a unifying equation of life has yet to be discovered, the Fibonacci sequence may establish an origin for such a development. The observation of the Fibonacci sequence is existent in almost all aspects of life ranging from the leaves of a fern tree, architecture, and even paintings, makes it highly unlikely to be a stochastic phenomenon. Despite its wide-spread occurrence and existence, the Fibonacci series and the Rule of Golden Proportions has not been widely documented in the human body. This paper serves to review the observed documentation of the Fibonacci sequence in the human body.

Análise de superfícies seletivas em frequência com geometria multifractal

Frequency Selective surfaces are increasingly common structures in telecommunication systems due to their geometric and electromagnetic advantages. As a matter of fact, the frequency selective surfaces with fractal geometry type would allow an even bigger reduction of the electrical length which provided greater flexibility in the design of these structures. In this work, we investigated the use of multifractal geometry in frequency selective surfaces. Three structures with different multifractal geometries have been proposed and analyzed. The first structure allowed the design of multiband structures with greater flexibility in controlling the resonant frequencies and bandwidth. The second structure provided a bandwidth increase even with the rising of the fractal level. The third structure showed response with angle stability, dual polarization and provided room for a bandwidth increase with the rising of the structural multifractality. Furthermore, the proposed structures increased the degree of freedom in the multiband designs because they have multiple resonant frequencies ratios between adjacent bands and are easy to deploy. The validation of the proposed structures was initially verified through simulations in Ansoft Designer software and then the structures were constructed and the experimental results obtained

Estudo da variação da periodicidade e do acoplamento entre superfícies seletivas de frequência com elementos fractais e helicoidais em estruturas de multicamadas

This work aims to investigate the behavior of fractal and helical elements structures in planar microstrip. In particular, the frequency selective surfaces (FSSs) had changed its conventional elements to fractal and helical formats. The dielectric substrate used was fiberglass (FR-4) and has a thickness of 1.5 mm, a relative permittivity 4.4 and tangent loss equal to 0.02. For FSSs, was adopting the Dürer’s fractal geometry and helical geometry. To make the measurements, we used two antennas horns in direct line of sight, connected by coaxial cable to the vector network analyzer. Some prototypes were select for built and measured. From preliminary results, it was aimed to find practical applications for structures from the cascading between them. For FSSs with Dürer’s fractal elements was observed behavior provided by the multiband fractal geometry, while the bandwidth has become narrow as the level of iteration fractal increased, making it a more selective frequency with a higher quality factor. A parametric analysis allowed the analysis of the variation of the air layer between them. The cascading between fractal elements structure were considered, presented a tri-band behavior for certain values of the layer of air between them, and find applications in the licensed 2.5GHz band (2.3-2.7) and 3.5GHz band (3.3-3.8). For FSSs with helical elements, six structures were considered, namely H0, H1, H2, H3, H4 and H5. The electromagnetic behavior of them was analyzed separately and cascaded. From preliminary results obtained from the separate analysis of structures, including the cascade, the higher the bandwidth, in that the thickness of the air layer increases. In order to find practical applications for helical structures cascaded, the helical elements structure has been cascaded find applications in the X-band (8.0-12.0) and unlicensed band (5.25-5.85). For numerical and experimental characterization of the structures discussed was used, respectively, the commercial software Ansoft Designer and a vector network analyzer, Agilent N5230A model.

Écoulements granulaires par avalanches : indices de fluidité, fractales et multifractales

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.

Radiotherapy Treatment Assessment using DCE-MRI

Abstract

The goal of modern radiotherapy is to precisely deliver a prescribed radiation dose to delineated target volumes that contain a significant amount of tumor cells while sparing the surrounding healthy tissues/organs. Precise delineation of treatment and avoidance volumes is the key for the precision radiation therapy. In recent years, considerable clinical and research efforts have been devoted to integrate MRI into radiotherapy workflow motivated by the superior soft tissue contrast and functional imaging possibility. Dynamic contrast-enhanced MRI (DCE-MRI) is a noninvasive technique that measures properties of tissue microvasculature. Its sensitivity to radiation-induced vascular pharmacokinetic (PK) changes has been preliminary demonstrated. In spite of its great potential, two major challenges have limited DCE-MRI’s clinical application in radiotherapy assessment: the technical limitations of accurate DCE-MRI imaging implementation and the need of novel DCE-MRI data analysis methods for richer functional heterogeneity information.

This study aims at improving current DCE-MRI techniques and developing new DCE-MRI analysis methods for particular radiotherapy assessment. Thus, the study is naturally divided into two parts. The first part focuses on DCE-MRI temporal resolution as one of the key DCE-MRI technical factors, and some improvements regarding DCE-MRI temporal resolution are proposed; the second part explores the potential value of image heterogeneity analysis and multiple PK model combination for therapeutic response assessment, and several novel DCE-MRI data analysis methods are developed.

I. Improvement of DCE-MRI temporal resolution. First, the feasibility of improving DCE-MRI temporal resolution via image undersampling was studied. Specifically, a novel MR image iterative reconstruction algorithm was studied for DCE-MRI reconstruction. This algorithm was built on the recently developed compress sensing (CS) theory. By utilizing a limited k-space acquisition with shorter imaging time, images can be reconstructed in an iterative fashion under the regularization of a newly proposed total generalized variation (TGV) penalty term. In the retrospective study of brain radiosurgery patient DCE-MRI scans under IRB-approval, the clinically obtained image data was selected as reference data, and the simulated accelerated k-space acquisition was generated via undersampling the reference image full k-space with designed sampling grids. Two undersampling strategies were proposed: 1) a radial multi-ray grid with a special angular distribution was adopted to sample each slice of the full k-space; 2) a Cartesian random sampling grid series with spatiotemporal constraints from adjacent frames was adopted to sample the dynamic k-space series at a slice location. Two sets of PK parameters’ maps were generated from the undersampled data and from the fully-sampled data, respectively. Multiple quantitative measurements and statistical studies were performed to evaluate the accuracy of PK maps generated from the undersampled data in reference to the PK maps generated from the fully-sampled data. Results showed that at a simulated acceleration factor of four, PK maps could be faithfully calculated from the DCE images that were reconstructed using undersampled data, and no statistically significant differences were found between the regional PK mean values from undersampled and fully-sampled data sets. DCE-MRI acceleration using the investigated image reconstruction method has been suggested as feasible and promising.

Second, for high temporal resolution DCE-MRI, a new PK model fitting method was developed to solve PK parameters for better calculation accuracy and efficiency. This method is based on a derivative-based deformation of the commonly used Tofts PK model, which is presented as an integrative expression. This method also includes an advanced Kolmogorov-Zurbenko (KZ) filter to remove the potential noise effect in data and solve the PK parameter as a linear problem in matrix format. In the computer simulation study, PK parameters representing typical intracranial values were selected as references to simulated DCE-MRI data for different temporal resolution and different data noise level. Results showed that at both high temporal resolutions (<1s) and clinically feasible temporal resolution (~5s), this new method was able to calculate PK parameters more accurate than the current calculation methods at clinically relevant noise levels; at high temporal resolutions, the calculation efficiency of this new method was superior to current methods in an order of 102. In a retrospective of clinical brain DCE-MRI scans, the PK maps derived from the proposed method were comparable with the results from current methods. Based on these results, it can be concluded that this new method can be used for accurate and efficient PK model fitting for high temporal resolution DCE-MRI.

II. Development of DCE-MRI analysis methods for therapeutic response assessment. This part aims at methodology developments in two approaches. The first one is to develop model-free analysis method for DCE-MRI functional heterogeneity evaluation. This approach is inspired by the rationale that radiotherapy-induced functional change could be heterogeneous across the treatment area. The first effort was spent on a translational investigation of classic fractal dimension theory for DCE-MRI therapeutic response assessment. In a small-animal anti-angiogenesis drug therapy experiment, the randomly assigned treatment/control groups received multiple fraction treatments with one pre-treatment and multiple post-treatment high spatiotemporal DCE-MRI scans. In the post-treatment scan two weeks after the start, the investigated Rényi dimensions of the classic PK rate constant map demonstrated significant differences between the treatment and the control groups; when Rényi dimensions were adopted for treatment/control group classification, the achieved accuracy was higher than the accuracy from using conventional PK parameter statistics. Following this pilot work, two novel texture analysis methods were proposed. First, a new technique called Gray Level Local Power Matrix (GLLPM) was developed. It intends to solve the lack of temporal information and poor calculation efficiency of the commonly used Gray Level Co-Occurrence Matrix (GLCOM) techniques. In the same small animal experiment, the dynamic curves of Haralick texture features derived from the GLLPM had an overall better performance than the corresponding curves derived from current GLCOM techniques in treatment/control separation and classification. The second developed method is dynamic Fractal Signature Dissimilarity (FSD) analysis. Inspired by the classic fractal dimension theory, this method measures the dynamics of tumor heterogeneity during the contrast agent uptake in a quantitative fashion on DCE images. In the small animal experiment mentioned before, the selected parameters from dynamic FSD analysis showed significant differences between treatment/control groups as early as after 1 treatment fraction; in contrast, metrics from conventional PK analysis showed significant differences only after 3 treatment fractions. When using dynamic FSD parameters, the treatment/control group classification after 1st treatment fraction was improved than using conventional PK statistics. These results suggest the promising application of this novel method for capturing early therapeutic response.

The second approach of developing novel DCE-MRI methods is to combine PK information from multiple PK models. Currently, the classic Tofts model or its alternative version has been widely adopted for DCE-MRI analysis as a gold-standard approach for therapeutic response assessment. Previously, a shutter-speed (SS) model was proposed to incorporate transcytolemmal water exchange effect into contrast agent concentration quantification. In spite of richer biological assumption, its application in therapeutic response assessment is limited. It might be intriguing to combine the information from the SS model and from the classic Tofts model to explore potential new biological information for treatment assessment. The feasibility of this idea was investigated in the same small animal experiment. The SS model was compared against the Tofts model for therapeutic response assessment using PK parameter regional mean value comparison. Based on the modeled transcytolemmal water exchange rate, a biological subvolume was proposed and was automatically identified using histogram analysis. Within the biological subvolume, the PK rate constant derived from the SS model were proved to be superior to the one from Tofts model in treatment/control separation and classification. Furthermore, novel biomarkers were designed to integrate PK rate constants from these two models. When being evaluated in the biological subvolume, this biomarker was able to reflect significant treatment/control difference in both post-treatment evaluation. These results confirm the potential value of SS model as well as its combination with Tofts model for therapeutic response assessment.

In summary, this study addressed two problems of DCE-MRI application in radiotherapy assessment. In the first part, a method of accelerating DCE-MRI acquisition for better temporal resolution was investigated, and a novel PK model fitting algorithm was proposed for high temporal resolution DCE-MRI. In the second part, two model-free texture analysis methods and a multiple-model analysis method were developed for DCE-MRI therapeutic response assessment. The presented works could benefit the future DCE-MRI routine clinical application in radiotherapy assessment.

Écoulements granulaires par avalanches : indices de fluidité, fractales et multifractales

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal.