Data-driven realizations of kernel and image representations and their application to fault detection and control system design

This paper deals with the data-driven design of observer-based fault detection and control systems. We first introduce the definitions of the data-driven forms of kernel and image representations. It is followed by the study of their identification. In the context of a fault-tolerant architecture, the design of observer-based fault detection, feed-forward and feedback control systems are addressed based on the data-driven realization of the kernel and image representations. Finally, the main results are demonstrated on the laboratory continuous stirred tank heater (CSTH) system.     

On a new procedure for finding nonclassical symmetries

A new technique for deriving the determining equations of nonclassical symmetries associated with a partial differential equation system is introduced. The problem is reduced to computing the determining equations of the classical symmetries associated with a related equation with coefficients which depend on the nonclassical symmetry operator. As a consequence, all the symbolic manipulation programs designed for the latter task can also be used to find the determining equations of the nonclassical symmetries, without any adaptation of the program. The algorithm was implemented as the MAPLE routine GENDEFNC and uses the MAPLE package DESOLV (authors Carminati and Vu). As an example, we consider the Huxley partial differential equation.     

Parameterization in Finite Precision

Certain classes of algebraic curves and surfaces admit both parametric and implicit representations. Such dual forms are highly useful in geometric modeling since they combine the strengths of the two representations. We consider the problem of computing the rational parameterization of an implicit curve or surface in a finite precision domain. Known algorithms for this problem are based on classical algebraic geometry, and assume exact arithmetic involving algebraic numbers. In this work we investigate the behavior of published parameterization algorithms in a finite precision domain and derive succinct algebraic and geometric error characterizations. We then indicate numerically robust methods for parameterizing curves and surfaces which yield no error in extended finite precision arithmetic and, alternatively, minimize the output error under fixed finite precision calculations. Received January 8, 1997; revised August 27, 1998.     

Exact Algorithms for Linear Programming over Algebraic Extensions

Abstract. We study the computational complexity of linear programs with coefficients that are real algebraic numbers under a Turing machine model of computation. After reviewing a method for exact representation of algebraic numbers under the Turing model, we show that the fundamental tasks of comparison and arithmetic can be performed in polynomial time. Our technique for establishing polynomial-time algorithms for comparison and arithmetic is distinct from the usual resultant-based approaches, and has the advantage that it provides a natural framework for analysis of the complexity of computational tasks, such as Gaussian elimination, that involve a sequence of arithmetic operations. Our main contribution is to show that a variant of the ellipsoid method can be used to solve linear programming in time polynomial in the encoding size of the problem coefficients and the degree of any algebraic extension that contains those coefficients.     

Discrete Critical Values: a General Framework for Silhouettes Computation

Many shapes resulting from important geometric operations in industrial applications such as Minkowski sums or volume swept by a moving object can be seen as the projection of higher dimensional objects. When such a higher dimensional object is a smooth manifold, the boundary of the projected shape can be computed from the critical points of the projection. In this paper, using the notion of polyhedral chains introduced by Whitney, we introduce a new general framework to define an analogous of the set of critical points of piecewise linear maps defined over discrete objects that can be easily computed. We illustrate our results by showing how they can be used to compute Minkowski sums of polyhedra and volumes swept by moving polyhedra.     

Contouring Discrete Indicator Functions

We present a method for calculating the boundary of objects from Discrete Indicator Functions that store 2‐material volume fractions with a high degree of accuracy. Although Marching Cubes and its derivatives are effective methods for calculating contours of functions sampled over discrete grids, these methods perform poorly when contouring non‐smooth functions such as Discrete Indicator Functions. In particular, Marching Cubes will generate surfaces that exhibit aliasing and oscillations around the exact surface. We derive a simple solution to remove these problems by using a new function to calculate the positions of vertices along cell edges that is efficient, easy to implement, and does not require any optimization or iteration. Finally, we provide empirical evidence that the error introduced by our contouring method is significantly less than is introduced by Marching Cubes.     

Solving more general index-2 differential algebraic equations

The paper deals with a class of quasilinear index-2 differential algebraic equations, which covers both linear variable coefficient systems as well as Hessenberg form equations. Supposing low smoothness only, the solvability of initial value problems is stated via classical analytical techniques. For that class of differential algebraic equations, backward differentiation formulas and Runge-Kutta methods as well as projected versions are discussed with respect to feasibility, (in)stability, convergence, and asymptotical behaviour.     

On rotation representations in computational robot kinematics

Various methods of implementing forward and inverse kinematics of six-axes industrial robots are analyzed in this paper from the viewpoint of numerical conditioning and convergence speed both close to a solution and away from it. Computational complexities are derived in terms of the number of arithmetic operations and comparisons are made by observing the actual CPU time consumption. The formulations presented make use of different sets of invariants describing the orientation of the gripper. It is shown that, in inverse kinematics, there is a tradeoff between numerical stability and computational speed.An abridged version of this paper was presented in the 1992 IEEE International Conference on Robotics and Automation, 1992 [1].     

A Biorthogonal Wavelet Approach based on Dual Subdivision

In this paper a biorthogonal wavelet approach based on Doo‐Sabin subdivision is presented. In the dual subdivision like Doo‐Sabin scheme, all the old control vertices disappear after one subdivision step, which is a big challenge to the biorthogonal wavelet construction. In our approach, the barycenters of the V‐faces corresponding to the old vertices are selected as the vertices associated with the scaling functions to construct the scaling space. The lifting scheme is used to guarantee the fitting quality of the wavelet transform, and a local orthogonalization is introduced with a discrete inner product operation to improve the computation efficiency. Sharp feature modeling based on extended Doo‐Sabin subdivision rules is also discussed in the framework of our wavelet construction. The presented wavelet construction is proven to be stable and effective by the experimental results.     

Global Structure Optimization of Quadrilateral Meshes

We introduce a fully automatic algorithm which optimizes the high‐level structure of a given quadrilateral mesh to achieve a coarser quadrangular base complex. Such a topological optimization is highly desirable, since state‐of‐the‐art quadrangulation techniques lead to meshes which have an appropriate singularity distribution and an anisotropic element alignment, but usually they are still far away from the high‐level structure which is typical for carefully designed meshes manually created by specialists and used e.g. in animation or simulation. In this paper we show that the quality of the high‐level structure is negatively affected by helical configurations within the quadrilateral mesh. Consequently we present an algorithm which detects helices and is able to remove most of them by applying a novel grid preserving simplification operator (GP‐operator) which is guaranteed to maintain an all‐quadrilateral mesh. Additionally it preserves the given singularity distribution and in particular does not introduce new singularities. For each helix we construct a directed graph in which cycles through the start vertex encode operations to remove the corresponding helix. Therefore a simple graph search algorithm can be performed iteratively to remove as many helices as possible and thus improve the high‐level structure in a greedy fashion. We demonstrate the usefulness of our automatic structure optimization technique by showing several examples with varying complexity.     

Knowledge representations for the interactive selling of financial services

Selling financial services requires deep knowledge about the product domain as well as about potential wishes and needs of customers. In the financial services domain (especially in the retail sector) sales representatives can differ greatly in their expertise and level of sales knowledge. Therefore financial service providers ask for tools effectively supporting sales representatives in the dialog with the customer. In this paper we present technologies which allow a flexible mapping of product, marketing and sales knowledge to the representation of a recommender knowledge-base thus providing an infrastructure for the interactive selling of financial services. Furthermore, we report experiences gained from financial service recommender development projects.

Localized Quadrilateral Coarsening

In this paper we introduce a coarsening algorithm for quadrilateral meshes that generates quality, quad-only connectivity during level-of- coarsening creation. A novel aspect of this work is development and implementation of a localized adaptation of the polychord collapse operator to better control and preserve important surface components. We describe a novel weighting scheme for automatic deletion selection that considers surface attributes, as well as localized queue updates that allow for improved data structures and computational performance opportunities over previous techniques. Additionally, this work supports optional and intuitive user controls for tailored simplification results.     

Semi-regular Quadrilateral-only Remeshing from Simplified Base Domains

Semi-regular meshes describe surface models that exhibit a structural regularity that facilitates many geometric processing algorithms. We introduce a technique to construct semi-regular, quad-only meshes from input surface meshes of arbitrary polygonal type and genus. The algorithm generates a quad-only model through subdivision of the input polygons, then simplifies to a base domain that is homeomorphic to the original mesh. During the simplification, a novel hierarchical mapping method, keyframe mapping , stores specific levels-of-detail to guide the mapping of the original vertices to the base domain. The algorithm implements a scheme for refinement with adaptive resampling of the base domain and backward projects to the original surface. As a byproduct of the remeshing scheme, a surface parameterization is associated with the remesh vertices to facilitate subsequent geometric processing, i.e. texture mapping, subdivision surfaces and spline-based modeling.     

A Part-aware Surface Metric for Shape Analysis

The notion of parts in a shape plays an important role in many geometry problems, including segmentation, correspondence, recognition, editing, and animation. As the fundamental geometric representation of 3D objects in computer graphics is surface-based, solutions of many such problems utilize a surface metric, a distance function defined over pairs of points on the surface, to assist shape analysis and understanding. The main contribution of our work is to bring together these two fundamental concepts: shape parts and surface metric. Specifically, we develop a surface metric that is part-aware. To encode part information at a point on a shape, we model its volumetric context – called the volumetric shape image (VSI) – inside the shape's enclosed volume, to capture relevant visibility information. We then define the part-aware metric by combining an appropriate VSI distance with geodesic distance and normal variation. We show how the volumetric view on part separation addresses certain limitations of the surface view, which relies on concavity measures over a surface as implied by the well-known minima rule. We demonstrate how the new metric can be effectively utilized in various applications including mesh segmentation, shape registration, part-aware sampling and shape retrieval.     

Isosurfaces Over Simplicial Partitions of Multiresolution Grids

We provide a simple method that extracts an isosurface that is manifold and intersection‐free from a function over an arbitrary octree. Our method samples the function dual to minimal edges, faces, and cells, and we show how to position those samples to reconstruct sharp and thin features of the surface. Moreover, we describe an error metric designed to guide octree expansion such that flat regions of the function are tiled with fewer polygons than curved regions to create an adaptive polygonalization of the isosurface. We then show how to improve the quality of the triangulation by moving dual vertices to the isosurface and provide a topological test that guarantees we maintain the topology of the surface. While we describe our algorithm in terms of extracting surfaces from volumetric functions, we also show that our algorithm extends to generating manifold level sets of co‐dimension 1 of functions of arbitrary dimension.