Showing papers on "Pencil (mathematics) published in 1999"

Computer‐Generated Graphite Pencil Rendering of 3D Polygonal Models

Abstract: Researchers in non-photorealistic rendering have investigated the display of three-dimensional worlds using various display models. In particular, recent work has focused on the modeling of traditional artistic media and styles such as pen-and-ink illustration and watercolor painting. By providing 3D rendering systems that use these alternative display models users can generate traditional illustration renderings of their three-dimensional worlds. In this paper we present our graphite pencil 3D renderer. We have broken the problem of simulating pencil drawing down into four fundamental parts: (1) simulating the drawing materials (graphite pencil and drawing paper, blenders and kneaded eraser), (2) modeling the drawing primitives (individual pencil strokes and mark-making to create tones and textures), (3) simulating the basic rendering techniques used by artists and illustrators familiar with pencil rendering, and (4) modeling the control of the drawing composition. Each part builds upon the others and is essential to developing the framework for higher-level rendering methods and tools. In this paper we present parts 2, 3, and 4 of our research. We present non-photorealistic graphite pencil rendering methods for outlining and shading. We also present the control of drawing steps from preparatory sketches to finished rendering results. We demonstrate the capabilities of our approach with a variety of images generated from 3D models.

Shape reconstruction in projective grid space from large number of images

Abstract: This paper proposes a new scheme for multi-image projective reconstruction based on a projective grid space. The projective grid space is defined by two basis views and the fundamental matrix relating these views. Given fundamental matrices relating other views to each of the two basis views, this projective grid space can be related to any view. In the projective grid space as a general space that is related to all images, a projective shape can be reconstructed from all the images of weakly calibrated cameras. The projective reconstruction is one way to reduce the effort of the calibration because it does not need Euclid metric information, but rather only correspondences of several points between the images. For demonstrating the effectiveness of the proposed projective grid definition, we modify the voxel coloring algorithm for the projective voxel scheme. The quality of the virtual view images re-synthesized from the projective shape demonstrates the effectiveness of our proposed scheme for projective reconstruction from a large number of images.

Using Generalized Cayley Transformations within an Inexact Rational Krylov Sequence Method

Abstract: The rational Krylov sequence (RKS) method is a generalization of Arnoldi's method. It constructs an orthogonal reduction of a matrix pencil into an upper Hessenberg pencil. The RKS method is useful when the matrix pencil may be efficiently factored. This paper considers approximately solving the resulting linear systems with iterative methods. We show that a Cayley transformation leads to a more efficient and robust eigensolver than the usual shift-invert transformation when the linear systems are solved inexactly within the RKS method. A relationship with the recently introduced Jacobi--Davidson method is also established.

Projective Shape Analysis

Abstract: Emulating human vision, computer vision systems aim to recognize object shape from images. The main difficulty in recognizing objects from images is that the shape depends on the viewpoint. This difficulty can be resolved by using projective invariants to describe the shape. For four colinear points the cross-ratio is the simplest statistic that is invariant to projective transformations. Five coplanar sets of points can be described by two independent cross-ratios. Using the six-fold set of symmetries of the cross-ratio, corresponding to six permutations of the points, we introduce an inverse stereographic projection of the linear cross-ratio (c) to a stereographic cross-ratio (ξ). To exploit this symmetry, we study the distribution of cos 3ξ when the four points are randomly distributed under appropriate distributions and find the mapping of the cross-ratio so that the distribution of ξ is uniform. These mappings provide a link between projective invariants and directional statistics so that we...

Multi-input partial eigenvalue assignment for the symmetric quadratic pencil

Abstract: A new algorithm is proposed for the multi-input partial pole placement problem by the state feedback for a quadratic pencil. A set of necessary and sufficient conditions for the existence of a solution is also derived. The important features of the algorithm are that the algorithm requires knowledge of only a small number of the eigenvalues that need to be re-assigned in practice and does not give any spill-over, that is, the eigenvalues that are not required to be changed, remain unchanged. Furthermore, it can take advantage of the exploitable structures of the system matrices such as the sparsity, symmetry and definiteness. The solution may be of particular interest in the stabilization and control of flexible, large space structures where only a small part of the spectrum is to be reassigned and the rest of the spectrum has to remain unchanged.

Points, lines and diamonds: a two-sorted modal logic for projective planes

Abstract: We introduce a modal language for talking about projective planes. This language is two-sorted, containing formulas to be evaluated at points and at lines, respectively. The language has two diamonds whose intended accessibility relations are the two directions of the incidence relation between points and lines. We provide a sound and complete axiomatization for the formulas that are valid in the class of projective planes. We also show that it is decidable whether a given formula is satisfiable in a projective plane, and we characterize the computational complexity of this satisfaction problem. ∗Institute of Logic, Language and Information, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. yde@wins.uva.nl The research of the author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences

Analysis and computation of projective invariants from multiple views in the geometric algebra framework

Abstract: A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing positions. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes. In this paper we present geometric algebra as a complete framework for the theory and computation of projective invariants formed from points and lines in computer vision. We will look at the formation of 3D projective invariants from multiple images, show how they can be formed from image coordinates and estimated tensors (F, fundamental matrix and T, trilinear tensor) and give results on simulated and real data.

On the isotriviality of families of elliptic surfaces

Abstract: We study non-isotrivial projective families of elliptic surfaces of Kodaira dimension one, over complex projective curves. If the base is an elliptic curve, we show that the family must have a singular fibre, and that over the projective line it must have at least three singular fibres. Similar results, for families of surfaces of general type, have been obtained by Migliorini and Kov\'acs, and they are well-known for projective families of surfaces of Kodaira dimension zero. Revised version: We corrected some minor errors and ambiguities, and we completed the list of references.

Differential invariants for parametrized projective surfaces

Abstract: The differential invariants associated with a transformation group acting on a manifold are the fundamental building blocks for understanding the geometry, equivalence, symmetry and other properties of submanifolds. Moreover, the construction of general invariant differential equations and invariant variational problems requires knowledge of the differential invariants. The basic theory of differential invariants dates back to the work of Lie, [18] and Tresse, [23]. However, a complete classification of differential invariants for many of the fundamental transformation groups of physical and geometrical importance remains undeveloped. In this paper we find complete systems of differential invariants for a particularly interesting example, that of surfaces in real projective space. The classical approach to differential invariants is via the infinitesimal methods pioneered by Lie. The main difficulty in applying Lie's method to complicated examples is that it requires the integration of linear partial differential equations, which can prove to be rather complicated. Cartan, [3, 4], demonstrated how his moving frame method could produce the differential invariants for several groups of geometrical interest, including the geometry of curves in the Euclidean, affine, and projective planes; see also [15]. More recently, the moving frame method was been successfully applied to study the invariants of curves in projective spaces and Grassmannians, [14]. However, extensions to more general examples has proved to be more problematic.

Algebraic and geometric tools to compute projective and permutation invariants

Abstract: Studies the computation of projective invariants in pairs of images from uncalibrated cameras and presents a detailed study of the projective and permutation invariants for configurations of points and/or lines. Two basic computational approaches are given, one algebraic and one geometric. In each case, invariants are computed in projective space or directly from image measurements. Finally, we develop combinations of those projective invariants which are insensitive to permutations of the geometric primitives of each of the basic configurations.

An experimental study of projective structure from motion

Abstract: We describe an essentially algorithm-independent experimental comparison of projective versus Euclidean reconstruction. The Euclidean approach is as accurate as the projective one, even with significant calibration error and for the pure projective structure. Projective optimization has less of a local-minima problem than its Euclidean equivalent. We describe techniques that enhance the convergence of optimization algorithms.

On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Abstract: Let ℘ be a projective space. In this paper we consider sets ℰ of planes of ℘ such that any two planes of ℰ intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:

Complex varieties of general type whose canonical systems are composed with pencils

Abstract: This paper aims to study canonical pencils of higher dimensional projective varieties. It seems that the geometric genus of the general fibre for the derived fibration from the canonical pencil for a 3-fold of general type does not have an upper bound.

Projective Differential Geometry

Abstract: In this chapter we give a discussion of projective submanifolds in general, and revisit a few examples (quadrics, in particular) that are important for a number of different reasons. We use thee technique of moving frames a la Elie Cartan. This technique is effective in uncovering various degeneracy phenomena as regards projective subvarieties. Thus the material in the chapter is somewhat separate from the rest of the book; nevertheless serves to provide a context of sort. The main reference for this chapter is the article [Griffiths-Harris2]. See also [Yangl], where the metric geometry of projective submanifolds is discussed.

Dynamic projective geometry

Abstract: The theme of this thesis is dynamic geometry, a new way of exploring classical geometry using interactive computer software. This kind of software allows the user to make geometric constructions on a computer's screen. The constructions might consist of points, lines and conics whose positions have been constrained in various ways. The constraints, which may involve incidences, distances and angles, can be added and removed dynamically. For example, to force a line to always be incident on a point, the user would simply grab the line with the cursor and drop it onto the point. Any object whose position is not completely determined by the constraints can be grabbed and dragged around on the screen. The rest of the objects will then automatically self-adjust in order to keep the constraints satis ed. Dynamic geometry software is primarily used for teaching mathematics, but is useful in any situation where it is important to understand the geometric properties of a dynamic system. Over the last few years, a number of tools for dynamic geometry have been developed. Most of them have focused on elementary Euclidean geometry. In this thesis we present a new software that has been based entirely on projective concepts and thus allows us to illustrate the classical theorems of projective geometry. The software has also extensive support for di erent types of metrics, which makes it possible to explore both Euclidean and non-Euclidean geometry. In fact, the user is given direct access to the absolute elements which de ne the metric. Moreover, the system can handle objects in the complex projective plane, which permits, for example, the circular points in Euclidean geometry to be used in geometric constructions. We discuss how the user interface of a dynamic geometry system should be designed and we identify a number of problems and shortcomings which the user interfaces of all previous systems seem to su er from. Most of these defects are related to the fundamental problem of choosing the right solution of an underdetermined system of constraint equations. We show how this problems can be solved by letting the system automatically add extra constraints if necessary, and by using a richer internal representation based on oriented projective geometry. The thesis is written in English.

Complex varieties of general type whose canonical systems are composed with pencils

Abstract: This paper aims to study canonical pencils of higher dimensional projective varieties. It seems that the geometric genus of the general fibre for the derived fibration from the canonical pencil for a 3-fold of general type does not have an upper bound.

Projective Reconstruction from N Views Having One View in Common

Abstract: Projective reconstruction recovers projective coordinates of 3D scene points from their several projections in 2D images. We introduce a method for the projective reconstruction based on concatenation of trifocal constraints around a reference view. This configuration simplifies computations significantly. The method uses only linear estimates which stay "close" to image data. The method requires correspondences only across triplets of views. However, it is not symmetrical with respect to views. The reference view plays a special role. The method can be viewed as a generalization of Hartley's algorithm, or as a particular application of Triggs' closure relations.

Pencil encased in wood for writing, colouring, drawing and cosmetic purposes

Abstract: The invention relates to a pencil encased in wood for writing, colouring, drawing and cosmetic purposes. The pencil has raised structures, consisting of a synthetic material, which protrude from the surface used to hold the pencil, to form handling surfaces or handling nodules (4).

On Some Numerical Relations of tetragonal Linear Systems

Abstract: Let L be a pencil of degree d on a curve C and let e_1・・・, e_ be scrolar invariants. We already prove that [numerical formula], ...d-2 if [numerical formula] is birationally very ample. In this article, we extend the above result.

Art Class: What To Do When Students Can't Hold a Pencil.

Abstract: (1999). Art Class: What to do When Students can't Hold a Pencil. Art Education: Vol. 52, Teaching Art as if the World Mattered, pp. 18-22.

Measurement Comparability of Paper-and-Pencil and Multimedia Vocational Assessments. ACT Research Report Series 99-1.

Abstract: Table of