Massive simulations and arrays of sensing devices, in combination with increasing computing resources, have generated large, complex, high-dimensional datasets used to study phenomena across numerous fields of study. Visualization plays an important role in exploring such datasets. We provide a comprehensive survey of advances in high-dimensional data visualization that focuses on the past decade. We aim at providing guidance for data practitioners to navigate through a modular view of the recent advances, inspiring the creation of new visualizations along the enriched visualization pipeline, and identifying future opportunities for visualization research.

Resampling raw surface meshes is one of the most fundamental operations used by nearly all digital geometry processing systems. The vast majority of this work has focused on triangular remeshing, yet quadrilateral meshes are preferred for many surface PDE problems, especially fluid dynamics, and are best suited for defining Catmull-Clark subdivision surfaces. We describe a fundamentally new approach to the quadrangulation of manifold polygon meshes using Laplacian eigenfunctions, the natural harmonics of the surface. These surface functions distribute their extrema evenly across a mesh, which connect via gradient flow into a quadrangular base mesh. An iterative relaxation algorithm simultaneously refines this initial complex to produce a globally smooth parameterization of the surface. From this, we can construct a well-shaped quadrilateral mesh with very few extraordinary vertices. The quality of this mesh relies on the initial choice of eigenfunction, for which we describe algorithms and hueristics to efficiently and effectively select the harmonic most appropriate for the intended application.

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Abstract-The Morse-Smale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalar-valued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closure-finite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divide-and-conquer strategy allows for memory-efficient computation of the MS complex and simplification on-the-fly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementation-specific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on off-the-shelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/1024 3 node grid on a laptop computer with 2Gb memory.Index Terms-Topology-based analysis, Morse-Smale complex, large scale data.

An important goal of scientific data analysis is to understand the behavior of a system or process based on a sample of the system. In many instances it is possible to observe both input parameters and system outputs, and characterize the system as a high-dimensional function. Such data sets arise, for instance, in large numerical simulations, as energy landscapes in optimization problems, or in the analysis of image data relating to biological or medical parameters. This paper proposes an approach to analyze and visualizing such data sets. The proposed method combines topological and geometric techniques to provide interactive visualizations of discretely sampled high-dimensional scalar fields. The method relies on a segmentation of the parameter space using an approximate Morse-Smale complex on the cloud of point samples. For each crystal of the Morse-Smale complex, a regression of the system parameters with respect to the output yields a curve in the parameter space. The result is a simplified geometric representation of the Morse-Smale complex in the high dimensional input domain. Finally, the geometric representation is embedded in 2D, using dimension reduction, to provide a visualization platform. The geometric properties of the regression curves enable the visualization of additional information about each crystal such as local and global shape, width, length, and sampling densities. The method is illustrated on several synthetic examples of two dimensional functions. Two use cases, using data sets from the UCI machine learning repository, demonstrate the utility of the proposed approach on real data. Finally, in collaboration with domain experts the proposed method is applied to two scientific challenges. The analysis of parameters of climate simulations and their relationship to predicted global energy flux and the concentrations of chemical species in a combustion simulation and their integration with temperature.