On the fractal dimension of isosurfaces

Box dimension of the border of Kingdom of Saudi Arabia

Fractal dimension unlike topological dimension is (usually) a non-integer number which measures complexity, roughness, or irregularity of an object with respect to the space in which the set lies. It is used to characterize highly irregular objects in nature containing statistical self-similarity such as mountains, snowflakes, clouds, coastlines, borders etc. In this article, box dimension (a version of fractal dimension) of the border of Kingdom of Saudi Arabia (KSA) is computed using a multicore parallel processing algorithm based on the classical box-counting method. A power law relation is obtained from numerical simulations which relates the length of the border with the scale size and provides a very close estimate of the actual length of the KSA border within the scaling regions and scaling effects on the length of KSA border are considered. The algorithm presented in the article is shown to be highly scalable and efficient and the speedup of the algorithm is computed using Amdahl's and Gustafson's laws. For simulations, a high performance parallel computer is employed using Python codes and QGIS software.

Fractals: An Eclectic Survey, Part-I

Fractals are geometric shapes and patterns that may repeat their geometry at smaller or larger scales. It is well established that fractals can describe shapes and surfaces that cannot be represented by the classical Euclidean geometry. An eclectic survey of fractals is presented in two parts encompassing applications of fractals in a variety of diverse and innovative fields. The goal of the first part is to focus on the glossary of fractals, their mathematical description, aesthetic, artistic, and architectural applications, while the second part is focused on engineering, industry, commercial, and futuristic applications of fractals.

Fractal dimension of coastline of Australia

Coastlines are irregular in nature having (random) fractal geometry and are formed by various natural activities. Fractal dimension is a measure of degree of geometric irregularity present in the coastline. A novel multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia. The reliability of the coastline length of Australia is addressed by recovering the power law from our computational results. For simulations, the algorithm is implemented on a parallel computer for multi-core processing using the QGIS software, R-programming language and Python codes.

Efficient Local Statistical Analysis via Point-Wise Histograms in Tetrahedral Meshes and Curvilinear Grids

Local histograms (i.e., point-wise histograms computed from local regions of mesh vertices) have been used in many data analysis and visualization applications. Previous methods for computing local histograms mainly work for regular or rectilinear grids only. In this paper, we develop theory and novel algorithms for computing local histograms in tetrahedral meshes and curvilinear grids. Our algorithms are theoretically sound and efficient, and work effectively and fast in practice. Our main focus is on scalar fields, but the algorithms also work for vector fields as a by-product with small, easy modifications. Our methods can benefit information theoretic and other distribution-driven analysis. The experiments demonstrate the efficacy of our new techniques, including a utility case study on tetrahedral vector field visualization.

Visual Analysis of Multi-Run Spatio-Temporal Simulations Using Isocontour Similarity for Projected Views

Multi-run simulations are widely used to investigate how simulated processes evolve depending on varying initial conditions. Frequently, such simulations model the change of spatial phenomena over time. Isocontours have proven to be effective for the visual representation and analysis of 2D and 3D spatial scalar fields. We propose a novel visualization approach for multi-run simulation data based on isocontours. By introducing a distance function for isocontours, we generate a distance matrix used for a multidimensional scaling projection. Multiple simulation runs are represented by polylines in the projected view displaying change over time. We propose a fast calculation of isocontour differences based on a quasi-Monte Carlo approach. For interactive visual analysis, we support filtering and selection mechanisms on the multi-run plot and on linked views to physical space visualizations. Our approach can be effectively used for the visual representation of ensembles, for pattern and outlier detection, for the investigation of the influence of simulation parameters, and for a detailed analysis of the features detected. The proposed method is applicable to data of any spatial dimensionality and any spatial representation (gridded or unstructured). We validate our approach by performing a user study on synthetic data and applying it to different types of multi-run spatio-temporal simulation data.

Exploring Flow Fields Using Space-Filling Analysis of Streamlines

Large scale scientific simulations frequently use streamline based techniques to visualize flow fields. As the shape of a streamline is often related to some underlying property of the field, it is important to identify streamlines (or their parts) with unique geometric features. In this paper, we introduce a metric, called the box counting ratio, which measures the geometric complexity of streamlines by measuring their space-filling capacity at different scales. We propose a novel interactive visualization framework which utilizes this metric to extract, organize and visualize features of varying density and complexity hidden in large numbers of streamlines. The proposed framework extracts complex regions of varying density from the streamlines, and organizes and presents them on an interactive 2D information space, allowing user selection and visualization of streamlines. We also extend this framework to support exploration using an ensemble of measures including box counting ratio. Our framework allows the user to easily visualize and interact with features otherwise hidden in large vector field data. We strengthen our claims with case studies using combustion and climate simulation data sets.

A Web platform for the interactive visualization and analysis of the 3D fractal dimension of MRI data

This study presents a Web platform (http://3dfd.ujaen.es) for computing and analyzing the 3D fractal dimension (3DFD) from volumetric data in an efficient, visual and interactive way. The Web platform is specially designed for working with magnetic resonance images (MRIs) of the brain. The program estimates the 3DFD by calculating the 3D box-counting of the entire volume of the brain, and also of its 3D skeleton. All of this is done in a graphical, fast and optimized way by using novel technologies like CUDA and WebGL. The usefulness of the Web platform presented is demonstrated by its application in a case study where an analysis and characterization of groups of 3D MR images is performed for three neurodegenerative diseases: Multiple Sclerosis, Intrauterine Growth Restriction and Alzheimer's disease. To the best of our knowledge, this is the first Web platform that allows the users to calculate, visualize, analyze and compare the 3DFD from MRI images in the cloud.

An information-aware framework for exploring multivariate data sets

Information theory provides a theoretical framework for measuring information content for an observed variable, and has attracted much attention from visualization researchers for its ability to quantify saliency and similarity among variables. In this paper, we present a new approach towards building an exploration framework based on information theory to guide the users through the multivariate data exploration process. In our framework, we compute the total entropy of the multivariate data set and identify the contribution of individual variables to the total entropy. The variables are classified into groups based on a novel graph model where a node represents a variable and the links encode the mutual information shared between the variables. The variables inside the groups are analyzed for their representativeness and an information based importance is assigned. We exploit specific information metrics to analyze the relationship between the variables and use the metrics to choose isocontours of selected variables. For a chosen group of points, parallel coordinates plots (PCP) are used to show the states of the variables and provide an interface for the user to select values of interest. Experiments with different data sets reveal the effectiveness of our proposed framework in depicting the interesting regions of the data sets taking into account the interaction among the variables.

Integrating Isosurface Statistics and Histograms

Many data sets are sampled on regular lattices in two, three or more dimensions, and recent work has shown that statistical properties of these data sets must take into account the continuity of the underlying physical phenomena. However, the effects of quantization on the statistics have not yet been accounted for. This paper therefore reconciles the previous papers to the underlying mathematical theory, develops a mathematical model of quantized statistics of continuous functions, and proves convergence of geometric approximations to continuous statistics for regular sampling lattices. In addition, the computational cost of various approaches is considered, and recommendations made about when to use each type of statistic.

Volume analysis using multimodal surface similarity

The combination of volume data acquired by multiple modalities has been recognized as an important but challenging task. Modalities often differ in the structures they can delineate and their joint information can be used to extend the classification space. However, they frequently exhibit differing types of artifacts which makes the process of exploiting the additional information non-trivial. In this paper, we present a framework based on an information-theoretic measure of isosurface similarity between different modalities to overcome these problems. The resulting similarity space provides a concise overview of the differences between the two modalities, and also serves as the basis for an improved selection of features. Multimodal classification is expressed in terms of similarities and dissimilarities between the isosurfaces of individual modalities, instead of data value combinations. We demonstrate that our approach can be used to robustly extract features in applications such as dual energy computed tomography of parts in industrial manufacturing.