Mixed Constrained Infinite Horizon Linear Quadratic Optimal Control

For a given initial state, a constrained infinite horizon linear quadratic optimal control problem can be reduced to a finite dimensional problem [12]. To find a conservative estimate of the size of the reduced problem, the existing algorithms require the on‐line solutions of quadratic programs [10] or a linear program [2]. In this paper, we first show based on the Lyapunov theorem that the closed‐loop system with a mixed constrained infinite horizon linear quadratic optimal control is exponentially stable on proper sets. Then the exponentially converging envelop of the closed‐loop trajectory that can be computed off‐line is employed to obtain a finite dimensional quadratic program equivalent to the mixed constrained infinite horizon linear quadratic optimal control problem without any on‐line optimization. The example considered in [2] showed that the proposed algorithm identifies less conservative size estimate of the reduced problem with much less computation.     

Surface Patches from Unorganized Space Curves

Recent 3D sketch tools produce networks of three‐space curves that suggest the contours of shapes. The shapes may be non‐manifold, closed three‐dimensional, open two‐dimensional, or mixed. We describe a system that automatically generates intuitively appealing piecewise‐smooth surfaces from such a curve network, and an intelligent user interface for modifying the automatically chosen surface patches. Both the automatic and the semi‐automatic parts of the system use a linear algebra representation of the set of surface patches to track the topology. On complicated inputs from ILoveSketch [ [BBS08] ], our system allows the user to build the desired surface with just a few mouse‐clicks.     

Improving Performance and Accuracy of Local PCA

Local Principal Component Analysis (LPCA) is one of the popular techniques for dimensionality reduction and data compression of large data sets encountered in computer graphics. The LPCA algorithm is a variant of k‐means clustering where the repetitive classification of high dimensional data points to their nearest cluster leads to long execution times. The focus of this paper is on improving the efficiency and accuracy of LPCA. We propose a novel SortCluster LPCA algorithm that significantly reduces the cost of the point‐cluster classification stage, achieving a speed‐up of up to 20. To improve the approximation accuracy, we investigate different initialization schemes for LPCA and find that the k‐means++ algorithm [ [AV07] ] yields best results, however at a high computation cost. We show that similar ideas that lead to the efficiency of our SortCluster LPCA algorithm can be used to accelerate k‐means++. The resulting initialization algorithm is faster than purely random seeding while producing substantially more accurate data approximation.     

Temporal Visualization of Boundary‐based Geo‐information Using Radial Projection

This work is concerned with a design study by an interdisciplinary team on visualizing a 10‐year record of seasonal and inter‐annual changes in frontal position (advance/retreat) of nearly 200 marine terminating glaciers in Greenland. Whilst the spatiotemporal nature of the raw data presents a challenge to develop a compact and intuitive visual design, the focus on coastal boundaries provides an opportunity for dimensional reduction. In this paper, we report the user‐centered design process carried out by the team, and present several visual encoding schemes that have met the requirements including compactness, intuitiveness, and ability to depict temporal changes and spatial relations. In particular, we designed a family of radial visualization, where radial lines correspond to different coastal locations, and nested rings represent the evolution of the temporal dimension from inner to outer circles. We developed an algorithm for mapping glacier terminus positions from Cartesian coordinates to angular coordinates. Instead of a naive uniform mapping, the algorithm maintains consistent spatial perception of the visually‐sensitive geographical references between their Cartesian and angular coordinates, and distributes other termini positions between primary locations based on coastal distance. This work has provided a useful solution to address the problem of inaccuracy in change evaluation based on pixel‐based visualization [ [BPC*10] ].     

On linear system reduction by stable precompensation and stable state feedback

A unified framework for the problem of reduction of a finite dimensional linear system by stable state feedback and bistable precompensation is presented. The results extend the work of [1] to include in an algebraic setting stability considerations.     

Image Space Rendering of Point Clouds Using the HPR Operator

The hidden point removal (HPR) operator introduced by Katz et al. [KTB07] provides an elegant solution for the problem of estimating the visibility of points in point samplings of surfaces. Since the method requires computing the three‐dimensional convex hull of a set with the same cardinality as the original cloud, the method has been largely viewed as impractical for real‐time rendering of medium to large clouds. In this paper we examine how the HPR operator can be used more efficiently by combining several image space techniques, including an approximate convex hull algorithm, cloud sampling, and GPU programming. Experiments show that this combination permits faster renderings without overly compromising the accuracy.     

Distortion‐Guided Structure‐Driven Interactive Exploration of High‐Dimensional Data

Dimension reduction techniques are essential for feature selection and feature extraction of complex high‐dimensional data. These techniques, which construct low‐dimensional representations of data, are typically geometrically motivated, computationally efficient and approximately preserve certain structural properties of the data. However, they are often used as black box solutions in data exploration and their results can be difficult to interpret. To assess the quality of these results, quality measures, such as co‐ranking [ LV09 ], have been proposed to quantify structural distortions that occur between high‐dimensional and low‐dimensional data representations. Such measures could be evaluated and visualized point‐wise to further highlight erroneous regions [ MLGH13 ]. In this work, we provide an interactive visualization framework for exploring high‐dimensional data via its two‐dimensional embeddings obtained from dimension reduction, using a rich set of user interactions. We ask the following question: what new insights do we obtain regarding the structure of the data, with interactive manipulations of its embeddings in the visual space? We augment the two‐dimensional embeddings with structural abstractions obtained from hierarchical clusterings, to help users navigate and manipulate subsets of the data. We use point‐wise distortion measures to highlight interesting regions in the domain, and further to guide our selection of the appropriate level of clusterings that are aligned with the regions of interest. Under the static setting, point‐wise distortions indicate the level of structural uncertainty within the embeddings. Under the dynamic setting, on‐the‐fly updates of point‐wise distortions due to data movement and data deletion reflect structural relations among different parts of the data, which may lead to new and valuable insights.     

Numerische Erfahrungen bei der Lösung des dreidimensionalen Transportproblems

To find an initial basic feasible solution is a special problem on solving the three dimensional transportation problem. It is not possible to do this work in a similar simple way as it can be done in the usual transportation problem. In [3] Haley proposes to solve a new problem that is essentionally the old one with the addition of a set of artificial variables. However, numerical investigations have shown that this procedure enlarges the amount of computing very much. This paper now gives another procedure for finding an initial basic solution for the three dimensional transportation problem. A great number of examples is discussed which demonstrate the efficiency of this procedure.     

Flow‐Induced Inertial Steady Vector Field Topology

Traditionally, vector field visualization is concerned with 2D and 3D flows. Yet, many concepts can be extended to general dynamical systems, including the higher‐dimensional problem of modeling the motion of finite‐sized objects in fluids. In the steady case, the trajectories of these so‐called inertial particles appear as tangent curves of a 4D or 6D vector field. These higher‐dimensional flows are difficult to map to lower‐dimensional spaces, which makes their visualization a challenging problem. We focus on vector field topology, which allows scientists to study asymptotic particle behavior. As recent work on the 2D case has shown, both extraction and classification of isolated critical points depend on the underlying particle model. In this paper, we aim for a model‐independent classification technique, which we apply to two different particle models in not only 2D, but also 3D cases. We show that the classification can be done by performing an eigenanalysis of the spatial derivatives' velocity subspace of the higher‐dimensional 4D or 6D flow. We construct glyphs that depict not only the types of critical points, but also encode the directional information given by the eigenvectors. We show that the eigenvalues and eigenvectors of the inertial phase space have sufficient symmetries and structure so that they can be depicted in 2D or 3D, instead of 4D or 6D.     

Robust controller design for infinite‐dimensional exosystems

In this paper, we consider robust output regulation of distributed parameter systems with infinite‐dimensional exosystems capable of generating polynomially growing signals. We design an observer‐based error feedback controller solving the control problem. The controller is chosen in such a way that it incorporates an internal model of the infinite‐dimensional exosystem. The remaining parameters of the controller are chosen to stabilize the closed‐loop system strongly. We also analyze the classes of signals generated by the exosystem. In particular, we explore the connection between the smoothness properties of the reference and disturbance signals and the strictness of the conditions required for the existence of a controller solving the robust output regulation problem. Copyright © 2012 John Wiley & Sons, Ltd.     

2D-discontinuity detection from scattered data

We describe a numerical approach for the detection of discontinuities of a two dimensional function distorted by noise. This problem arises in many applications as computer vision, geology, signal processing. The method we propose is based on the two-dimensional continuous wavelet transform and follows partially the ideas developed in [2], [6] and [8]. It is well-known that the wavelet transform modulus maxima locate the discontinuity points and the sharp variation points as well. Here we propose a statistical test which, for a suitable scale value, allows us to decide if a wavelet transform modulus maximum corresponds to a function value discontinuity. Then we provide an algorithm to detect the discontinuity curves fromscattered and noisy data.     

Monotone Smoothing Splines using General Linear Systems

In this paper, a method is proposed to solve the problem of monotone smoothing splines using general linear systems. This problem, also called monotone control theoretic splines, has been solved only when the curve generator is modeled by the second‐order integrator, but not for other cases. The difficulty in the problem is that the monotonicity constraint should be satisfied over an interval which has the cardinality of the continuum. To solve this problem, we first formulate the problem as a semi‐infinite quadratic programming problem, and then we adopt a discretization technique to obtain a finite‐dimensional quadratic programming problem. It is shown that the solution of the finite‐dimensional problem always satisfies the infinite‐dimensional monotonicity constraint. It is also proved that the approximated solution converges to the exact solution as the discretization grid‐size tends to zero. An example is presented to show the effectiveness of the proposed method.