Visual analysis of multidimensional data requires effective ways to reduce data dimensionality to encode them visually. Multidimensional projections (MDP) figure among the most important visualization techniques in this context, transforming multidimensional data into scatter plots where patterns reflect some notion of similarity in the data. However, MDP come with distortions that make visual patterns not trustworthy. Moreover, the patterns present in scatter plots might not be enough to allow an understanding of multidimensional data, motivating the development of layout enrichment methodologies that operate with MDP. This survey attempts to cover the main aspects of MDP as a visualization and visual analytic tool, providing detailed analysis and taxonomies taht organize MDP techniques according to their main properties and traits. The survey also approaches the different types of distortions that can result from MDP mappings while overviewing existing mechanisms to quantitatively evaluate such distortions. A qualitative analysis of the impact of distortions on the different analytic tasks is also presented, providing guidelines for users to choose a proper MDP for an intended. Finally, layout enrichment schemes to debunk MDP distortions and/or reveal relevant information not directly inferable from the scatter plot are reviewed and discussed in the light of new taxonomies.

Multidimensional projection techniques have experienced many improvements lately, mainly regarding computational times and accuracy. However, existing methods do not yet provide flexible enough mechanisms for visualization-oriented fully interactive applications. This work presents a new multidimensional projection technique designed to be more flexible and versatile than other methods. This novel approach, called Local Affine Multidimensional Projection (LAMP), relies on orthogonal mapping theory to build accurate local transformations that can be dynamically modified according to user knowledge. The accuracy, flexibility and computational efficiency of LAMP is confirmed by a comprehensive set of comparisons. LAMP's versatility is exploited in an application which seeks to correlate data that, in principle, has no connection as well as in visual exploration of textual documents.

The problem of projecting multidimensional data into lower dimensions has been pursued by many researchers due to its potential application to data analysis of various kinds. This paper presents a novel multidimensional projection technique based on least square approximations. The approximations compute the coordinates of a set of projected points based on the coordinates of a reduced number of control points with defined geometry. We name the technique Least Square Projections (LSP). From an initial projection of the control points, LSP defines the positioning of their neighboring points through a numerical solution that aims at preserving a similarity relationship between the points given by a metric in mD. In order to perform the projection, a small number of distance calculations is necessary and no repositioning of the points is required to obtain a final solution with satisfactory precision. The results show the capability of the technique to form groups of points by degree of similarity in 2D. We illustrate that capability through its application to mapping collections of textual documents from varied sources, a strategic yet difficult application. LSP is faster and more accurate than other existing high quality methods, particularly where it was mostly tested, that is, for mapping text sets.

Projection (or dimensionality reduction) techniques have been used as a means to handling the growing dimensionality of data sets as well as providing a way to visualize information coded into point relationships. Their role is essential in data interpretation and simultaneous use of different projections and their visualizations improve data understanding and increase the level of confidence in the result. For that purpose, projections should be fast to allow multiple views of the same data set. In this work we present a novel fast technique for projecting multi-dimensional data sets into bidimensional (2D) spaces that preserves neighborhood relationships. Additionally, a new technique for improving 2D projections from multi-dimensional data is presented, that helps reduce the inherent loss of information yielded by dimensionality reduction. The results are stimulating and are presented in the form of comparative visualizations against known and new 2D projection techniques. Based on the projection improvement approach presented here, a new metric for quality of projection is also given, that matches well the visual perception of quality. We discuss the implication of using improved projections in visual exploration of large data sets and the role of interaction in visualization of projected subspaces.

We present a benchmark for the evaluation and comparison of algorithms which reconstruct a surface from point cloud data. Although a substantial amount of effort has been dedicated to the problem of surface reconstruction, a comprehensive means of evaluating this class of algorithms is noticeably absent. We propose a simple pipeline for measuring surface reconstruction algorithms, consisting of three main phases: surface modeling, sampling, and evaluation. We use implicit surfaces for modeling shapes which are capable of representing details of varying size and sharp features. From these implicit surfaces, we produce point clouds by synthetically generating range scans which resemble realistic scan data produced by an optical triangulation scanner. We validate our synthetic sampling scheme by comparing against scan data produced by a commercial optical laser scanner, where we scan a 3D-printed version of the original surface. Last, we perform evaluation by comparing the output reconstructed surface to a dense uniformly distributed sampling of the implicit surface. We decompose our benchmark into two distinct sets of experiments. The first set of experiments measures reconstruction against point clouds of complex shapes sampled under a wide variety of conditions. Although these experiments are quite useful for comparison, they lack a fine-grain analysis. To complement this, the second set of experiments measures specific properties of surface reconstruction, in terms of sampling characteristics and surface features. Together, these experiments depict a detailed examination of the state of surface reconstruction algorithms.

Fig. 1. Large time-varying multivariate volumetric data projection. The information contained within each simulation cell is represented as a tuple in a high-dimensional Cartesian space. The proposed PLMP technique projects the high-dimensional instances into the visual space (bottom-right) in a streaming way. An inspection window (green rectangles) is delimited on the visual space and instances projected inside the inspection window, which correspond to simulation cells with similar features (mainly characterized by high gradient pressure), are rendered in 3D. The streaming projection enables the analysis/visualization of voxels with similar features (projected close to each other in the visual space) on massive time-varying data.Abstract-Most multidimensional projection techniques rely on distance (dissimilarity) information between data instances to embed high-dimensional data into a visual space. When data are endowed with Cartesian coordinates, an extra computational effort is necessary to compute the needed distances, making multidimensional projection prohibitive in applications dealing with interactivity and massive data. The novel multidimensional projection technique proposed in this work, called Part-Linear Multidimensional Projection (PLMP), has been tailored to handle multivariate data represented in Cartesian high-dimensional spaces, requiring only distance information between pairs of representative samples. This characteristic renders PLMP faster than previous methods when processing large data sets while still being competitive in terms of precision. Moreover, knowing the range of variation for data instances in the high-dimensional space, we can make PLMP a truly streaming data projection technique, a trait absent in previous methods.

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Word clouds have become one of the most widely accepted visual resources for document analysis and visualization, motivating the development of several methods for building layouts of keywords extracted from textual data. Existing methods are effective to demonstrate content, but are not capable of preserving semantic relationships among keywords while still linking the word cloud to the underlying document groups that generated them. Such representation is highly desirable for exploratory analysis of document collections. In this paper we present a novel approach to build document clouds, named ProjCloud that aim at solving both semantical layouts and linking with document sets. ProjCloud generates a semantically consistent layout from a set of documents. Through a multidimensional projection, it is possible to visualize the neighborhood relationship between highly related documents and their corresponding word clouds simultaneously. Additionally, we propose a new algorithm for building word clouds inside polygons, which employs spectral sorting to maintain the semantic relationship among words. The effectiveness and flexibility of our methodology is confirmed when comparisons are made to existing methods. The technique automatically constructs projection based layouts the user may choose to examine in the form of the point clouds or corresponding word clouds, allowing a high degree of control over the exploratory process.