Simulation and the monte carlo method

In recent years, there is an emerging direction that leverages information theory to solve many challenging problems in scientific data analysis and visualization. In this article, we review the key concepts in information theory, discuss how the principles of information theory can be useful for visualization, and provide specific examples to draw connections between data communication and data visualization in terms of how information can be measured quantitatively. As the amount of digital data available to us increases at an astounding speed, the goal of this article is to introduce the interested readers to this new direction of data analysis research, and to inspire them to identify new applications and seek solutions using information theory.

https://www.mdpi.com/1099-4300/13/1/254/pdf

Information Theory in Scientific Visualization

In this paper, we present an abstract model of visualization and inference processes, and describe an information-theoretic measure for optimizing such processes. In order to obtain such an abstraction, we first examined six classes of workflows in data analysis and visualization, and identified four levels of typical visualization components, namely disseminative, observational, analytical and model-developmental visualization. We noticed a common phenomenon at different levels of visualization, that is, the transformation of data spaces (referred to as alphabets) usually corresponds to the reduction of maximal entropy along a workflow. Based on this observation, we establish an information-theoretic measure of cost-benefit ratio that may be used as a cost function for optimizing a data visualization process. To demonstrate the validity of this measure, we examined a number of successful visualization processes in the literature, and showed that the information-theoretic measure can mathematically explain the advantages of such processes over possible alternatives.

What May Visualization Processes Optimize

Effective 3D streamline placement and visualization play an essential role in many science and engineering disciplines The main challenge for effective streamline visualization lies in seed placement, ie, where to drop seeds and how many seeds should be placed Seeding too many or too few streamlines may not reveal flow features and patterns either because it easily leads to visual clutter in rendering or it conveys little information about the flow field Not only does the number of streamlines placed matter, their spatial relationships also play a key role in understanding the flow field Therefore, effective flow visualization requires the streamlines to be placed in the right place and in the right amount This paper introduces hierarchical streamline bundles, a novel approach to simplifying and visualizing 3D flow fields defined on regular grids By placing seeds and generating streamlines according to flow saliency, we produce a set of streamlines that captures important flow features near critical points without enforcing the dense seeding condition We group spatially neighboring and geometrically similar streamlines to construct a hierarchy from which we extract streamline bundles at different levels of detail Streamline bundles highlight multiscale flow features and patterns through clustered yet not cluttered display This selective visualization strategy effectively reduces visual clutter while accentuating visual foci, and therefore is able to convey the desired insight into the flow data

Hierarchical Streamline Bundles

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.

An Information-Aware Framework for Exploring Multivariate Data Sets

The process of visualization can be seen as a visual communication channel where the input to the channel is the raw data, and the output is the result of a visualization algorithm. From this point of view, we can evaluate the effectiveness of visualization by measuring how much information in the original data is being communicated through the visual communication channel. In this paper, we present an information-theoretic framework for flow visualization with a special focus on streamline generation. In our framework, a vector field is modeled as a distribution of directions from which Shannon's entropy is used to measure the information content in the field. The effectiveness of the streamlines displayed in visualization can be measured by first constructing a new distribution of vectors derived from the existing streamlines, and then comparing this distribution with that of the original data set using the conditional entropy. The conditional entropy between these two distributions indicates how much information in the original data remains hidden after the selected streamlines are displayed. The quality of the visualization can be improved by progressively introducing new streamlines until the conditional entropy converges to a small value. We describe the key components of our framework with detailed analysis, and show that the framework can effectively visualize 2D and 3D flow data.

An Information-Theoretic Framework for Flow Visualization

We present a file layout algorithm for flow fields to improve runtime I/O efficiency for out-of-core streamline computation. Because of the increasing discrepancy between the speed of processors and storage devices, the cost of I/O becomes a major bottleneck for out-of-core computation. To reduce the I/O cost, loading data with better spatial locality has proved to be effective. It is also known that sequential file access is more efficient. To facilitate efficient streamline computation, we propose to reorganize the data blocks in a file following the data access pattern so that more efficient I/O and effective prefetching can be accomplished. To achieve the goal, we divide the domain into small spatial blocks and order the blocks into a linear layout based on the underlying flow directions. The ordering is done using a weighted directed graph model which can be formulated as a linear graph arrangement problem. Our goal is to arrange the file in a way consistent with the data access pattern during streamline computation. This allows us to prefetch a contiguous segment of data at a time from disk and minimize the memory cache miss rate. We use a recursive partitioning method to approximate the optimal layout. Our experimental results show that the resulting file layout reduces I/O cost and hence enables more efficient out-of-core streamline computation.

A flow-guided file layout for out-of-core streamline computation

While there have been intensive efforts in developing better 3D flow visualization techniques, little attention has
been paid to the design of better user interfaces and more effective data exploration work flow. In this paper, we
propose a novel graph-based user interface called Flow Web to enable more systematic explorations of 3D flow
data.
The Flow Web is a node-link graph that is constructed to highlight the essential flow structures where a
node represents a region in the field and a link connects two nodes if there exist particles traveling between
the regions. The direction of an edge implies the flow path, and the weight of an edge indicates the number of
particles traveling through the connected nodes. Hierarchical flow webs are created by splitting or merging nodes
and edges to allow for easy understanding of the underlying flow structures. To draw the Flow Web, we adopt
force based graph drawing algorithms to minimize edge crossings, and use a hierarchical layout to facilitate the
study of flow patterns step by step. The Flow Web also supports user queries to the properties of nodes and
links. Examples of the queries for node properties include the degrees, complexity, and some associated physical
attributes such as velocity magnitude. Queries for edges include weights, flow path lengths, existence of circles
and so on. It is also possible to combine multiple queries using operators such as and , or, not. The FlowWeb
supports several types of user interactions. For instance, the user can select nodes from the subgraph returned
by a query and inspect the nodes with more details at different levels of detail.
There are multiple advantages of using the graph-based user interface. One is that the user can identify
regions of interest much more easily since, unlike inspecting 3D regions, there is very little occlusion. It is also
much more convenient for the user to query statistical information about the nodes and links at different levels of
detail. With the Flow Web, it becomes easier for the user to log and track the progress of data exploration which
is crucial for exploring large data sets. We demonstrate how to construct and draw the Flow Web effectively,
and how to query the Flow Web to retrieve useful information from the data. Case studies are provided to
demonstrate the exploration process.

Flow Web: a graph based user interface for 3D flow field exploration

We present a file reordering method to improve runtime I/O efficiency for out-of-core streamline computation. Because of the increasing discrepancy between the speed of computation and that of I/O on multi-core machines, the cost of I/O becomes a major bottleneck for out-of-core computation. Among techniques that reduce runtime I/O cost, reordering file layout to increase data locality has become popular in recent years. Better layout optimization relies on the knowledge of the data access pattern, which can be acquired from benchmarking. For streamline computation, we observe that the data access pattern is highly dependent on the flow directions. As a disk I/O request can generally be done more efficiently with shorter seek distances, we propose a novel flow-guided file layout method to improve the I/O performance. With a weighted directed graph to model the data access pattern, the file layout problem can be formulated as a linear graph arrangement problem. The goal is to minimized the sum of the disk seek time based on the linear distances between all pairs of adjacent graph nodes. We use a divide-and-conquer algorithm to approximate the optimal layout. The experimental results show that our flow-guided layout outperforms layouts that use space filling curves and some of the more recent cache-oblivious mesh layout methods.

Lijie Xu

Papers

An Information-Theoretic Framework for Flow Visualization

A flow-guided file layout for out-of-core streamline computation

Flow Web: a graph based user interface for 3D flow field exploration

A flow-guided file layout for out-of-core streamline computation