Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data across domains. Dimensionality-reduction algorithms involve complex optimizations and the reduced dimensions computed by these algorithms generally lack clear relation to the initial data dimensions. Therefore, interpreting and reasoning about dimensionality reductions can be difficult. In this work, we introduce two interaction techniques, and , for reasoning dynamically about scatter plots of dimensionally reduced data. We also contribute two related visualization techniques, and to facilitate and enrich the effective use of the proposed interactions, which we integrate in a new tool called . To evaluate our techniques, we first analyze their time and accuracy performance across varying sample and dimension sizes. We then conduct a user study in which twelve data scientists use so as to assess the usefulness of the techniques in performing exploratory data analysis tasks. Results suggest that our visual interactions are intuitive and effective for exploring dimensionality reductions and generating hypotheses about the underlying data.
Prior work introduces various visualizations in planar scatter plots of DRs, in order to improve the user experience by communicating projection errors @cite_33 @cite_6 @cite_45 @cite_40 , change in dimensionality projection positions @cite_18 , data properties and clustering results @cite_6 @cite_35 , and contributions of original data dimensions in reduced dimensions @cite_12 . Low-dimensional projections are generally lossy representations of the original data relations: therefore, it is useful to convey both overall and per-point dimensionality-reduction errors to users when desired. Researchers visualized errors in DR scatter plots using Voronoi diagrams @cite_45 @cite_40 and corrected (undistorted) the errors by adjusting the projection layout with respect to the examined point @cite_33 @cite_6 .
@cite_35 @cite_18 @cite_33 @cite_6 @cite_40 @cite_45 @cite_12
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Scatterplots remain a powerful tool to visualize multidimensional data. However, accurately understanding the shape of multidimensional points from 2D projections remains challenging due to overlap. Consequently, there are a lot of variations on the scatterplot as a visual metaphor for this limitation. An important aspect often overlooked in scatterplots is the issue of sensitivity or local trend, which may help in identifying the type of relationship between two variables. However, it is not well known how or what factors influence the perception of trends from 2D scatterplots. To shed light on this aspect, we conducted an experiment where we asked people to directly draw the perceived trends on a 2D scatterplot. We found that augmenting scatterplots with local sensitivity helps to fill the gaps in visual perception while retaining the simplicity and readability of a 2D scatterplot. We call this augmentation the generalized sensitivity scatterplot (GSS). In a GSS, sensitivity coefficients are visually depicted as flow lines, which give a sense of continuity and orientation of the data that provide cues about the way data points are scattered in a higher dimensional space. We introduce a series of glyphs and operations that facilitate the analysis of multidimensional data sets using GSS, and validate with a number of well-known data sets for both regression and classification tasks. We present a framework for robustly estimating registration between a 3D volume image and a 2D projection image and evaluate its precision and robustness in spine interventions for vertebral localization in the presence of anatomical deformation. The framework employs a normalized gradient information similarity metric and multi-start covariance matrix adaptation evolution strategy optimization with local-restarts, which provided improved robustness against deformation and content mismatch. The parallelized implementation allowed orders-of-magnitude acceleration in computation time and improved the robustness of registration via multi-start global optimization. Experiments involved a cadaver specimen and two CT datasets (supine and prone) and 36 C-arm fluoroscopy images acquired with the specimen in four positions (supine, prone, supine with lordosis, prone with kyphosis), three regions (thoracic, abdominal, and lumbar), and three levels of geometric magnification (1.7, 2.0, 2.4). Registration accuracy was evaluated in terms of projection distance error (PDE) between the estimated and true target points in the projection image, including 14 400 random trials (200 trials on the 72 registration scenarios) with initialization error up to ±200 mm and ±10°. The resulting median PDE was better than 0.1 mm in all cases, depending somewhat on the resolution of input CT and fluoroscopy images. The cadaver experiments illustrated the tradeoff between robustness and computation time, yielding a success rate of 99.993 in vertebral labeling (with 'success' defined as PDE <5 mm) using 1,718 664 ± 96 582 function evaluations computed in 54.0 ± 3.5 s on a mid-range GPU (nVidia, GeForce GTX690). Parameters yielding a faster search (e.g., fewer multi-starts) reduced robustness under conditions of large deformation and poor initialization (99.535 success for the same data registered in 13.1 s), but given good initialization (e.g., ±5 mm, assuming a robust initial run) the same registration could be solved with 99.993 success in 6.3 s. The ability to register CT to fluoroscopy in a manner robust to patient deformation could be valuable in applications such as radiation therapy, interventional radiology, and an assistant to target localization (e.g., vertebral labeling) in image-guided spine surgery. We introduce a novel projection-based visualization method for high-dimensional data sets by combining concepts from MDS and the geometry of the hyperbolic spaces. This approach hyperbolic multi-dimensional scaling (H-MDS) is a synthesis of two important concepts for explorative data analysis and visualization: (i) multi-dimensional scaling uses proximity or pair distance data to generate a low-dimensional, spatial presentation of the data; (ii) previous work on the "hyperbolic tree browser" demonstrated the extraordinary advantages for an interactive display of graph-like data in the two-dimensional hyperbolic space (H2).In the new approach, H-MDS maps proximity data directly into the H2. This removes the restriction to "quasihierarchical", graph-based data--a major limitation of (ii). Since a suitable distance function can convert all kinds of data to proximity (or distance-based) data, this type of data can be considered the most general.We review important properties of the hyperbolic space and, in particular, the circular Poincare model of the H2. It enables effective human-computer interaction: by mouse dragging the "focus", the user can navigate in the data without loosing the context. In H2 the "fish-eye" behavior originates not simply by a non-linear view transformation but rather by extraordinary, non-Euclidean properties of the H2. Especially, the exponential growth of length and area of the underlying space makes the H2 a prime target for mapping hierarchical and (now also) high-dimensional data.Several high-dimensional mapping examples including synthetic and real-world data are presented. Since high-dimensional data produce "ring"-shaped displays, we present methods to enhance the display by modulating the dissimilarity contrast. This is demonstrated for an application for unstructured text: i.e., by using multiple film critiques from news:rec.art.movies.reviews and www.imdb.com, each movie is placed within the H2--creating a "space of movies" for interactive exploration. We introduce a set of integrated interaction techniques to interpret and interrogate dimensionality-reduced data. Projection techniques generally aim to make a high-dimensional information space visible in form of a planar layout. However, the meaning of the resulting data projections can be hard to grasp. It is seldom clear why elements are placed far apart or close together and the inevitable approximation errors of any projection technique are not exposed to the viewer. Previous research on dimensionality reduction focuses on the efficient generation of data projections, interactive customisation of the model, and comparison of different projection techniques. There has been only little research on how the visualization resulting from data projection is interacted with. We contribute the concept of probing as an integrated approach to interpreting the meaning and quality of visualizations and propose a set of interactive methods to examine dimensionality-reduced data as well as the projection itself. The methods let viewers see approximation errors, question the positioning of elements, compare them to each other, and visualize the influence of data dimensions on the projection space. We created a web-based system implementing these methods, and report on findings from an evaluation with data analysts using the prototype to examine multidimensional datasets.