Attracting and repelling Lagrangian coherent structures from a single computation

Hyperbolic Lagrangian Coherent Structures (LCSs) are locally most repelling or most attracting material surfaces in a finite-time dynamical system. To identify both types of hyperbolic LCSs at the same time instance, the standard practice has been to compute repelling LCSs from future data and attracting LCSs from past data. This approach tacitly assumes that coherent structures in the flow are fundamentally recurrent, and hence gives inconsistent results for temporally aperiodic systems. Here we resolve this inconsistency by showing how both repelling and attracting LCSs are computable at the same time instance from a single forward or a single backward run. These LCSs are obtained as surfaces normal to the weakest and strongest eigenvectors of the Cauchy-Green strain tensor.Comment: Under consideration for publication in Chaos/AI

Wavelet representation of contour sets

Journal ArticleWe present a new wavelet compression and multiresolution modeling approach for sets of contours (level sets). In contrast to previous wavelet schemes, our algorithm creates a parametrization of a scalar field induced by its contours and compactly stores this parametrization rather than function values sampled on a regular grid. Our representation is based on hierarchical polygon meshes with subdivision connectivity whose vertices are transformed into wavelet coefficients. From this sparse set of coefficients, every set of contours can be efficiently reconstructed at multiple levels of resolution. When applying lossy compression, introducing high quantization errors, our method preserves contour topology, in contrast to compression methods applied to the corresponding field function. We provide numerical results for scalar fields defined on planar domains. Our approach generalizes to volumetric domains, time-varying contours, and level sets of vector fields

The State of the Art in Flow Visualization: Partition-Based Techniques

Flow visualization has been a very active subfield of scientific visualization in recent years. From the resulting large variety of methods this paper discusses partition-based techniques. The aim of these approaches is to partition the flow in areas of common structure. Based on this partitioning, subsequent visualization techniques can be applied. A classification is suggested and advantages/disadvantages of the different techniques are discussed as well

Visualization and Analysis of Flow Fields based on Clifford Convolution

Vector fields from flow visualization often containmillions of data values. It is obvious that a direct inspection of the data by the user is tedious. Therefore, an automated approach for the preselection of features is essential for a complete analysis of nontrivial flow fields. This thesis deals with automated detection, analysis, and visualization of flow features in vector fields based on techniques transfered from image processing. This work is build on rotation invariant template matching with Clifford convolution as developed in the diploma thesis of the author. A detailed analysis of the possibilities of this approach is done, and further techniques and algorithms up to a complete segmentation of vector fields are developed in the process. One of the major contributions thereby is the definition of a Clifford Fourier transform in 2D and 3D, and the proof of a corresponding convolution theorem for the Clifford convolution as well as other major theorems. This Clifford Fourier transform allows a frequency analysis of vector fields and the behavior of vectorvalued filters, as well as an acceleration of the convolution computation as a fast transform exists. The depth and precision of flow field analysis based on template matching and Clifford convolution is studied in detail for a specific application, which are flow fields measured in the wake of a helicopter rotor. Determining the features and their parameters in this data is an important step for a better understanding of the observed flow. Specific techniques dealing with subpixel accuracy and the parameters to be determined are developed on the way. To regard the flow as a superposition of simpler features is a necessity for this application as close vortices influence each other. Convolution is a linear system, so it is suited for this kind of analysis. The suitability of other flow analysis and visualization methods for this task is studied here as well. The knowledge and techniques developed for this work are brought together in the end to compute and visualize feature based segmentations of flow fields. The resulting visualizations display important structures of the flow and highlight the interesting features. Thus, a major step towards robust and automatic detection, analysis and visualization of flow fields is taken

Flow pattern analysis for magnetic resonance velocity imaging

Blood flow in the heart is highly complex. Although blood flow patterns have been investigated by both computational modelling and invasive/non-invasive imaging techniques, their evolution and intrinsic connection with cardiovascular disease has yet to be explored. Magnetic resonance (MR) velocity imaging provides a comprehensive distribution of multi-directional in vivo flow distribution so that detailed quantitative analysis of flow patterns is now possible. However, direct visualisation or quantification of vector fields is of little clinical use, especially for inter-subject or serial comparison of changes in flow patterns due to the progression of the disease or in response to therapeutic measures. In order to achieve a comprehensive and integrated description of flow in health and disease, it is necessary to characterise and model both normal and abnormal flows and their effects. To accommodate the diversity of flow patterns in relation to morphological and functional changes, we have described in this thesis an approach of detecting salient topological features prior to analytical assessment of dynamical indices of the flow patterns. To improve the accuracy of quantitative analysis of the evolution of topological flow features, it is essential to restore the original flow fields so that critical points associated with salient flow features can be more reliably detected. We propose a novel framework for the restoration, abstraction, extraction and tracking of flow features such that their dynamic indices can be accurately tracked and quantified. The restoration method is formulated as a constrained optimisation problem to remove the effects of noise and to improve the consistency of the MR velocity data. A computational scheme is derived from the First Order Lagrangian Method for solving the optimisation problem. After restoration, flow abstraction is applied to partition the entire flow field into clusters, each of which is represented by a local linear expansion of its velocity components. This process not only greatly reduces the amount of data required to encode the velocity distribution but also permits an analytical representation of the flow field from which critical points associated with salient flow features can be accurately extracted. After the critical points are extracted, phase portrait theory can be applied to separate them into attracting/repelling focuses, attracting/repelling nodes, planar vortex, or saddle. In this thesis, we have focused on vortical flow features formed in diastole. To track the movement of the vortices within a cardiac cycle, a tracking algorithm based on relaxation labelling is employed. The constraints and parameters used in the tracking algorithm are designed using the characteristics of the vortices. The proposed framework is validated with both simulated and in vivo data acquired from patients with sequential MR examination following myocardial infarction. The main contribution of the thesis is in the new vector field restoration and flow feature abstraction method proposed. They allow the accurate tracking and quantification of dynamic indices associated with salient features so that inter- and intra-subject comparisons can be more easily made. This provides further insight into the evolution of blood flow patterns and permits the establishment of links between blood flow patterns and localised genesis and progression of cardiovascular disease.Open acces

Doctor of Philosophy

dissertationWith modern computational resources rapidly advancing towards exascale, large-scale simulations useful for understanding natural and man-made phenomena are becoming in- creasingly accessible. As a result, the size and complexity of data representing such phenom- ena are also increasing, making the role of data analysis to propel science even more integral. This dissertation presents research on addressing some of the contemporary challenges in the analysis of vector fields--an important type of scientific data useful for representing a multitude of physical phenomena, such as wind flow and ocean currents. In particular, new theories and computational frameworks to enable consistent feature extraction from vector fields are presented. One of the most fundamental challenges in the analysis of vector fields is that their features are defined with respect to reference frames. Unfortunately, there is no single ""correct"" reference frame for analysis, and an unsuitable frame may cause features of interest to remain undetected, thus creating serious physical consequences. This work develops new reference frames that enable extraction of localized features that other techniques and frames fail to detect. As a result, these reference frames objectify the notion of ""correctness"" of features for certain goals by revealing the phenomena of importance from the underlying data. An important consequence of using these local frames is that the analysis of unsteady (time-varying) vector fields can be reduced to the analysis of sequences of steady (time- independent) vector fields, which can be performed using simpler and scalable techniques that allow better data management by accessing the data on a per-time-step basis. Nevertheless, the state-of-the-art analysis of steady vector fields is not robust, as most techniques are numerical in nature. The residing numerical errors can violate consistency with the underlying theory by breaching important fundamental laws, which may lead to serious physical consequences. This dissertation considers consistency as the most fundamental characteristic of computational analysis that must always be preserved, and presents a new discrete theory that uses combinatorial representations and algorithms to provide consistency guarantees during vector field analysis along with the uncertainty visualization of unavoidable discretization errors. Together, the two main contributions of this dissertation address two important concerns regarding feature extraction from scientific data: correctness and precision. The work presented here also opens new avenues for further research by exploring more-general reference frames and more-sophisticated domain discretizations

Simulação numérica e visualização 3D interativa de objetos sob fluxos irrotacionais em tempo Quase-Real

Resumo: De uma maneira geral, qualquer fluxo irrotacional e incompressível é governado pela equação de Laplace. Esta não possui resolução analítica para problemas reais de engenharia, os quais possuem domínios e condições de contorno complexas, exceto para poucos casos particulares. A Dinâmica dos Fluidos Computacional (DFC) é um método utilizado para resolver numericamente a equação de Laplace, satisfazendo condições iniciais e de contorno. Porém, ao se refinar ou estender um domínio calculado, a quantidade de dados numéricos resultantes aumentará proporcionalmente e a análise destes valores pode se tornar complexa e onerosa. Complementariamente, para a compreensão dos resultados, é importante uma representação visual. A resolução numérica da equação de Laplace está descrita neste trabalho, com um algoritmo de solução inédito para as condições de contorno que atende qualquer forma geométrica em três dimensões. Desenvolveu-se um simulador que possibilita alterações geométricas de objetos 3D, calcula e visualiza interativamente velocidades, linhas de fluxo e força de sustentação para fluxos irrotacionais e incompressíveis em tempo quase-real. O sistema utiliza o método das diferenças finitas para a solução das equações. A interface gráfica foi desenvolvida utilizando, deste modo ineditamente para a DFC, a linguagem C++ e o VTK (Visualization Tool Kit). A quantidade, a origem das linhas de fluxo, a seleção do campo de velocidades, o cálculo da força de sustentação e a visualização estereoscópica são parâmetros que podem ser ajustados e selecionados para a visualização. O algoritmo passou por validações mostrando a capacidade de resolução em três dimensões. Assim, o simulador desenvolvido resolve, ao contrário dos softwares já existentes, o problema do cálculo e visualização interativa imediata ao se fazer modificações em objetos 3D. Este procedimento permitirá que se façam comparações entre formas geométricas imediatamente alteradas para que se possa escolher, entre elas, a que se adequar melhor às necessidades de um projeto