this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula,

We present in this paper a new method for matching points in stereoscopic color images, based on color differential invariants involving only first order derivatives of images. Our method is able to match robustly the images even if they present important transformations like rotation, range of viewpoint and change of intensity between each other. We present here a generalization of a gray level corner detector to the case of color images. This detector is robust and allows us to extract point primitives in stereoscopic images to be matched together, only with first order derivatives. We then describe these points with our set of local color invariants, and we propose a simple and efficient scheme for matching them. The robustness of the matching against local deformations is shown using deformations of single color images, then our stereo matching scheme is evaluated using true stereo color images with viewpoint variations. The results obtained on complex scenes clearly show...

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New algorithms for determining discrete and continuous symmetries of polynomials --- also known as binary forms in classical invariant theory --- are presented. Implementations in Mathematica and Maple are discussed and compared. The results are based on a new, comprehensive theory of moving frames that completely characterizes the equivalence and symmetry properties of submanifolds under general Lie group actions. This work was partially supported by NSF Grant DMS 98--03154. 1 Introduction. The purpose of this paper is to explain the detailed implementation of a new algorithm for determining the symmetries of polynomials (binary forms). The method was first described in the second author's new book [24], and the present paper adds details and refinements. We shall demonstrate that the symmetry group of both real and complex binary forms can be completely determined by solving two simultaneous bivariate polynomial equations, which are based on two fundamental covariants of the for...

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over Q. Remarks are given concerning the extension of these results to forms defined over number fields. This paper has now appeared in the LMS Journal of Computation and Mathematics, Volume 2, pages 62--92, and the full published version, with hyperlinks etc., is available online (to subscribers) at http://www.lms.ac.uk/jcm/2/lms98007/. This version contains the complete text of the published version in a very similar format. 1. Introduction Reduction theory for polynomials has a long history and numerous applications, some of which have grown considerably in importance in recent years with the growth of algorithmic an...

Let G = Zp be a cyclic group of prime order p with a representation G ! GL(V ) over a field K of characteristic p. In 1976, Almkvist and Fossum gave formulas for the decomposition of the symmetric powers of V in the case that V is indecomposable. From these they derived formulas for the Hilbert series of the invariant ring K[V ] G . Following Almkvist and Fossum in broad outline, we start by giving a shorter, self-contained proof of their results. We extend their work to modules which are not necessarily indecomposable. We also obtain formulas which give generating functions encoding the decompositions of all symmetric powers of V into indecomposables. Our results generalize to groups of the type Zp \Theta H with jHj coprime to p. Moreover, we prove that for any finite group G whose order is divisible by p but not by p 2 , the invariant ring K[V ] G is generated by homogeneous invariants of degrees at most dim(V ) \Delta (jGj \Gamma 1). Contents Introduction 1 1 Symmetrization ...

A general method of systematic derivation of affine moment invariants of any weights and orders is introduced. Each invariant is expressed by its generating graph. Techniques for elimination of reducible invariants and dependent invariants are discussed. This approach is illustrated on the set of all affine moment invariants up to the weight ten.

We completely characterize all nonlinear partial differential equations leaving a given finite-dimensional vector space of analytic functions invariant. Existence of an invariant subspace leads to a reduction of the associated dynamical partial differential equations to a system of ordinary differential equations, and provide a nonlinear counterpart to quasi-exactly solvable quantum Hamiltonians. These results rely on a useful extension of the classical Wronskian determinant condition for linear independence of functions. In addition, new approaches to the characterization of the annihilating differential operators for spaces of analytic functions are presented.

this paper we present a new method for point matching in stereoscopic color images. Our approach consists rst in characterizing points of interest using dierential invariants. Then we dene additional rst order invariants using color information, which make sucient the characterization till rst order. In addition, we make our description robust to important image transformations like rotation, range of viewpoint and linear illumination variations. Second, we propose a new incremental technique for point matching using our characterization, whichworks robustly and rapidly whatever the numberofpoints to be matched. Our stereo matching scheme is evaluated using stereo color images, with viewpoint and illumination variations. The very good results obtained clearly show the pertinence of our approach. Our color characterization produces a high rate of good matches, even though only rst order derivatives are used. Results on images holding many points show that our matching process is robust and rapidly implemented even if the points to be matched are numerous. It is a great asset, when matching a high set of points is necessary for example to realize dense depth maps between images

In a previous paper we showed that, for any n ≥ m + 2, most sets of n points in R m are determined (up to rotations, reflections, translations and relabeling of the points) by the distribution of their pairwise distances. But there are some exceptional point configurations which are not reconstructible from the distribution of distances in the above sense. In this paper, we concentrate on the planar case m = 2 and present a reconstructibility test with running time O(n 11). The cases of orientation preserving rigid motions (rotations and translations) and scalings are also discussed.