Chaotic spread spectrum watermark of optimal space-filling curves

Spread spectrum watermarking scheme is becoming an important research subject. In this paper, we present a method based on Peano–Hilbert space-filling curves for enhancing the robustness. Peano–Hilbert curve is a continuous mapping from one-dimensional space onto two-dimensional space. It is useful in many applications including quantum mechanics even, and preserves optimal locality. At the same time, we utilize a specified chaotic dynamic system–ICMIC map, which shows lowpass properties when the controlling parameter is devised. In this case, the watermarking detection resorts to the Neyman–Pearson criterion based on some statistical assumptions. Experimental results show that the proposed scheme can work well under JPEG compression and resist line-removal test.     

Approximation of an open polygonal curve with a minimum number of circular arcs and biarcs

We present an algorithm for approximating a given open polygonal curve with a minimum number of circular arcs. In computer-aided manufacturing environments, the paths of cutting tools are usually described with circular arcs and straight line segments. Greedy algorithms for approximating a polygonal curve with curves of higher order can be found in the literature. Without theoretical bounds it is difficult to say anything about the quality of these algorithms. We present an algorithm which finds a series of circular arcs that approximate the polygonal curve while remaining within a given tolerance region. This series contains the minimum number of arcs of any such series. Our algorithm takes O(n²logn) time for an original polygonal chain with n vertices. Using a similar approach, we design an algorithm with a runtime of O(n²logn), for computing a tangent-continuous approximation with the minimum number of biarcs, for a sequence of points with given tangent directions.     

Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves

Some powerful algorithms for multi-extremal non-convex-constrained optimization problems are based on reducing these multi-dimensional problems to those of one dimension by applying Peano-type space-filling curves mapping a unit interval on the real axis onto a multi-dimensional hypercube. Here is presented and substantiated a new scheme simultaneously employing several joint Peano-type scannings which conducts the property of nearness of points in many dimensions to a property of nearness of pre-images of these points in one dimension significantly better than in the case of a scheme with a single space-filling curve. Sufficient conditions of global convergence for the new scheme are investigated.This paper was presented at the II. IIASA Workshop on Global Optimization, Sopron (Hungary), December 9–14, 1990.     

Space-filling curves for 2-simplicial meshes created with bisections and reflections

Numerical experiments in J. Maubach: Local bisection refinement and optimal order algebraic multilevel preconditioners, PRISM-97 conference Proceedings, 1977, 121–136 indicated that the refinement with the use of local bisections presented in J. Maubach: Local bisection refinement for n-simplicial grids generated by reflections, SIAM J. Sci. Comput. 16 (1995), 210–227 leads to highly locally refined computational 2-meshes which can be very efficiently load-balanced with the use of a space-filling curve. This paper introduces the construction of this curve which can be produced at almost no costs, proofs that all its properties are invariant under local bisection, and comments on the 3-dimensional case.With the use of a space-filling curve (which passes through all triangular elements), load balancing over several processors is trivial: The load can be distributed over N processors by cutting the curve into N almost equilength parts. Each processor then operates on the triangles which are passed by its part of the curve.     

Matching planar maps

The subject of this paper are algorithms for measuring the similarity of patterns of line segments in the plane, a standard problem in, e.g., computer vision, geographic information systems, etc. More precisely, we define feasible distance measures that reflect how close a given pattern H is to some part of a larger pattern G. These distance measures are generalizations of the well-known Fréchet distance for curves. We first give an efficient algorithm for the case that H is a polygonal curve and G is a geometric graph. Then, slightly relaxing the definition of distance measure, we give an algorithm for the general case where both, H and G, are geometric graphs.     

Reductions of Tensor Categories Modulo Primes

We study good (i.e., semisimple) reductions of semisimple rigid tensor categories modulo primes. A prime p is called good for a semisimple rigid tensor category 𝒞 if such a reduction exists (otherwise, it is called bad). It is clear that a good prime must be relatively prime to the Müger squared norm |V|² of any simple object V of 𝒞. We show, using the Ito–Michler theorem in finite group theory, that for group-theoretical fusion categories, the converse is true. While the converse is false for general fusion categories, we obtain results about good and bad primes for many known fusion categories (e.g., for Verlinde categories). We also state some questions and conjectures regarding good and bad primes.     

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over that lifts to ℤ/p⁷ but not ℤ/p⁸. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem. Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05     

A duality theorem for crossed products of hopf algebras

Abstract

We study the automorphism and collineation groups of plane curves, i.e., in ?², that are not necessarily smooth, and obtain bounds for these curves in terms of, their degree and the number of singularities. We also introduce the notion of bad curves and good curves, and show that all bad curves are rational.     

On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.     

Counting points on curves using a map to P1, II

We introduce a new algorithm to compute the zeta function of a curve over a finite field. This method extends previous work of ours to all curves for which a good lift to characteristic zero is known. We develop all the necessary bounds, analyse the complexity of the algorithm and provide a complete implementation.     

A Framework for Drawing Planar Graphs with Curves and Polylines

We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n) × O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worst-case angular resolution. More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) × O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. All of our algorithms run in O(n) time.     

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.     

Space-filling curves and geodesic laminations. II: Symmetries

In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.     

Space-filling curves and geodesic laminations. II: Symmetries

In 2008, the author introduced a class of space-filling curves associated to fractals that satisfy the a special property. These structures admit geodesic laminations on the disc, which help to understand the geometrical and the dynamical properties of the space-filling curves. In the present article we study the relation between the symmetries of the laminations and the fractals. In particular we prove that the group of symmetries of the lamination is isomorphic to a subgroup of the full group of symmetries of the fractal. We extend the results to a larger class of fractals using the concept of sub-IFS.     

Good Local Behavior of Offsets to Implicit Algebraic Curves

The notion of good local behavior of an offset curve was introduced in [2,4] to denote that the behavior of an offset was locally good from a topological point of view. Also, in these papers the problem of checking whether good local behavior holds for a particular offsetting distance, and of computing intervals of distances with this nice property, was addressed for the case of rational algebraic curves. Thus, here we generalize the results and techniques of these papers to the case when the curve to work with is an implicitly given, possibly singular, non-necessarily rational algebraic curve. Furthermore, a generalization of the (already known) results relating offsets and evolutes of regularly parametrized curves, is presented for the case of possibly singular, implicit algebraic curves.     

Dimensions of the coordinate functions of space-filling curves

The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Pólya and others, are typical examples of self-affine sets, and their Hausdorff dimensions have been the subject of several articles in the mathematical literature. In the first half of this paper, we describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, we present new results on the coordinate functions of Pólya's one-parameter family of space-filling curves. We give a lower bound for the Hausdorff dimension of their graphs which is fairly close to the box-counting dimension. Our techniques are largely probabilistic. The fact that the exact dimension remains elusive seems to indicate the need for further work in the area of self-affine sets.     

Bucket elimination for multiobjective optimization problems

Multiobjective optimization deals with problems involving multiple measures of performance that should be optimized simultaneously. In this paper we extend bucket elimination (BE), a well known dynamic programming generic algorithm, from mono-objective to multiobjective optimization. We show that the resulting algorithm, MO-BE, can be applied to true multi-objective problems as well as mono-objective problems with knapsack (or related) global constraints. We also extend mini-bucket elimination (MBE), the approximation form of BE, to multiobjective optimization. The new algorithm MO-MBE can be used to obtain good quality multi-objective lower bounds or it can be integrated into multi-objective branch and bound in order to increase its pruning efficiency. Its accuracy is empirically evaluated in real scheduling problems, as well as in Max-SAT-ONE and biobjective weighted minimum vertex cover problems.     

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.     

A geometric approach to Euler’s addition formula A direct formulation of the addition law for elliptic curves given in?terms of Weierstra? quartics

Plane quartic curves given by equations of the form y ²=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form ò(1/?{P(x)}) dx\int (1/\sqrt{P(x)})\,\mathrm{d}x with a quartic polynomial P can be derived directly from this addition law.     

Bézier curves and interpolation in Riemannian manifolds

In a connected Riemannian manifold, generalised Bézier curves are C^∞ curves defined by a generalisation, in which line segments are replaced by minimal geodesics, of the classical de Casteljau algorithm. As in Euclidean space, these curves join their first and last control points. We compute the endpoint velocities and (covariant) accelerations of a generalised Bézier curve of arbitrary degree and use the formulae to express the curve's control points in terms of these quantities. These results allow generalised Bézier curves to be pieced together into C² splines, and thereby allow C² interpolation of a sequence of data points. For the case of uniform splines in symmetric spaces, we show that C² continuity is equivalent to a simple relationship, involving the global symmetries at knot points, between the control points of neighbouring curve segments. We also present some examples in hyperbolic 2-space.