We present a novel method to visualize multidimensional point clouds. While conventional visualization techniques, like scatterplot matrices or parallel coordinates, have issues with either overplotting of entities or handling many dimensions, we abstract the data using topological methods before presenting it. We assume the input points to be samples of a random variable with a high-dimensional probability distribution which we approximate using kernel density estimates on a suitably reconstructed mesh. From the resulting scalar field we extract the join tree and present it as a topological landscape, a visualization metaphor that utilizes the human capability of understanding natural terrains. In this landscape, dense clusters of points show up as hills. The nesting of hills indicates the nesting of clusters. We augment the landscape with the data points to allow selection and inspection of single points and point sets. We also present optimizations to make our algorithm applicable to large data sets and to allow interactive adaption of our visualization to the kernel window width used in the density estimation.

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We introduce a new method that identifies and tracks features in arbitrary dimensions using the merge tree-a structure for identifying topological features based on thresholding in scalar fields. This method analyzes the evolution of features of the function by tracking changes in the merge tree and relates features by matching subtrees between consecutive time steps. Using the time-varying merge tree, we present a structural visualization of the changing function that illustrates both features and their temporal evolution. We demonstrate the utility of our approach by applying it to temporal cluster analysis of high-dimensional point clouds.

Abstract-Analyzing high-dimensional point clouds is a classical challenge in visual analytics. Traditional techniques, such as projections or axis-based techniques, suffer from projection artifacts, occlusion, and visual complexity. We propose to split data analysis into two parts to address these shortcomings. First, a structural overview phase abstracts data by its density distribution. This phase performs topological analysis to support accurate and non-overlapping presentation of the high-dimensional cluster structure as a topological landscape profile. Utilizing a landscape metaphor, it presents clusters and their nesting as hills whose height, width, and shape reflect cluster coherence, size, and stability, respectively. A second local analysis phase utilizes this global structural knowledge to select individual clusters or point sets for further, localized data analysis. Focusing on structural entities significantly reduces visual clutter in established geometric visualizations and permits a clearer, more thorough data analysis. This analysis complements the global topological perspective and enables the user to study subspaces or geometric properties, such as shape.

Figure 1: Island-like visualization of a document point cloud's topological structure. By sharing similar dimensions, documents accumulate in subspaces of the high dimensional information space. Considering dimensions as words, clusters are assumed to describe topics, i.e., islands, in the final visualization. ABSTRACTDuring the last decades, electronic textual information has become the world's largest and most important information source available. People have added a variety of daily newspapers, books, scientific and governmental publications, blogs and private messages to this wellspring of endless information and knowledge. Since neither the existing nor the new information can be read in its entirety, computers are used to extract and visualize meaningful or interesting topics and documents from this huge information clutter.In this paper, we extend, improve and combine existing individual approaches into an overall framework that supports topological analysis of high dimensional document point clouds given by the well-known tf-idf document-term weighting method. We show that traditional distance-based approaches fail in very high dimensional spaces, and we describe an improved two-stage method for topology-based projections from the original high dimensional information space to both two dimensional (2-D) and three dimensional (3-D) visualizations. To show the accuracy and usability of this framework, we compare it to methods introduced recently and apply it to complex document and patent collections.