Uncertainty-Aware Principal Component Analysis

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets. As a special case, we show how to propagate multivariate normal distributions through PCA in closed form. Furthermore, we discuss extensions and limitations of our approach

Capturing and viewing gigapixel images

We present a system to capture and view "Gigapixel images": very high resolution, high dynamic range, and wide angle imagery consisting of several billion pixels each. A specialized camera mount, in combination with an automated pipeline for alignment, exposure compensation, and stitching, provide the means to acquire Gigapixel images with a standard camera and lens. More importantly, our novel viewer enables exploration of such images at interactive rates over a network, while dynamically and smoothly interpolating the projection between perspective and curved projections, and simultaneously modifying the tone-mapping to ensure an optimal view of the portion of the scene being viewed.publishe

Interactive level-of-detail rendering of large graphs

Fig. 1 . Application of our visualization technique on a hierarchical data set, zooming from overview (left) to a region of interest (right). The density-based node aggregation field (blue color) guides edge aggregation (orange/red color) to reveal visual patterns at different levels of detail. Abstract-We propose a technique that allows straight-line graph drawings to be rendered interactively with adjustable level of detail. The approach consists of a novel combination of edge cumulation with density-based node aggregation and is designed to exploit common graphics hardware for speed. It operates directly on graph data and does not require precomputed hierarchies or meshes. As proof of concept, we present an implementation that scales to graphs with millions of nodes and edges, and discuss several example applications

An algorithm for the proportional division of indivisible items

An allocation of indivisible items among n ≥ 2 players is proportional if and only if each player receives a proportional subset—one that it thinks is worth at least 1/n of the total value of all the items. We show that a proportional allocation exists if and only if there is an allocation in which each player receives one of its minimal bundles, from which the subtraction of any item would make the bundle worth less than 1/n. We give a practicable algorithm, based on players’ rankings of minimal bundles, that finds a proportional allocation if one exists; if not, it gives as many players as possible minimal bundles. The resulting allocation is maximin, but it may be neither envy-free nor Pareto-optimal. However, there always exists a Pareto-optimal maximin allocation which, when n = 2, is also envy-free. We compare our algorithm with two other 2-person algorithms, and we discuss its applicability to real-world disputes among two or more players