A central problem in object recognition is to determine the transformation that relates the model to the image, given some partial correspondence between the two. This is useful in determining whether an object is present in an image, and if so, determining where the object is. We present a novel method of solving this problem that uses region information. In our approach the model is divided into volumes, and the image is divided into regions. Given a match between subsets of volumes and regions (without any explicit correspondence between different pieces of the regions) the alignment transformation is computed. The method applies to planar objects under similarity, affine, and projective transformations and to projections of 3-D objects undergoing affine and projective transformations. 1 Introduction A fundamental problem in recognition is pose estimation. Given a correspondence between some portions of an object model and some portions of an image, determine the transformation th...

AbstractÐWe have recently proposed an approach to recognition that uses regions to determine the pose of objects while allowing for partial occlusion of the regions. Regions introduce an attractive alternative to existing global and local approaches, since, unlike global features, they can handle occlusion and segmentation errors, and unlike local features they are not as sensitive to sensor errors, and they are easier to match. The region-based approach also uses image information directly, without the construction of intermediate representations, such as algebraic descriptions, which may be difficult to reliably compute. In this paper, we further analyze properties of the method for planar objects undergoing projective transformations. In particular, we prove that three visible regions are sufficient to determine the transformation uniquely and that for a large class of objects, two regions are insufficient for this purpose. However, we show that when several regions are available, the pose of the object can generally be recovered even when some or all regions are significantly occluded. Our analysis is based on investigating the flow patterns of points under projective transformations in the presence of fixed points. Index TermsÐObject recognition, pose estimation with regions. 1

Abstract. J. Tits gave a general recipe for producing an abstract geometry from a semisimple algebraic group. This expository paper describes a uniform method for giving a concrete realization of Tits’s geometry and works through several examples. We also give a criterion for recognizing the automorphism of the geometry induced by an automorphism of the group. The E6 geometry is studied in depth. Contents 1. Tits’s geometry ΓP 3 2. A concrete geometry ΓV, part I 4

Abstract. We describe the first-order variations of the angles of Euclidean, spherical or hyperbolic polygons under infinitesimal deformations such that the lengths of the edges do not change. Using this description, we introduce a vector-valued quadratic invariant b on the space of those isometric deformations which, for convex polygons, has a remarkable positivity property. We give two geometric applications. The first is an isoperimetric statement for hyperbolic polygons: Among the convex hyperbolic polygons with given edge lengths, there is a unique polygon with vertices on a circle, a horocycle, or on one connected component of the space of points at constant distance from a geodesic, and it has maximal area. The second application is a rigidity result for equivariant polyhedral surfaces in the Minkowski space. Résumé. On décrit les déformations infinitésimales des angles d’un polygone euclidien, sphérique ou hyperbolique sous les déformations infinitésimales qui préservent les longueurs des arêtes. Onendéduit la définition d’un invariant quadratique à valeurs vectorielles b sur l’espace de ces déformations isométriques qui, pour les polygones convexes, a une propriétéremarquablede positivité. On donne deux applications géométriques. La première est un énoncé isoperimétrique pour les polygones hyperboliques: Parmi les polygones hyperboliques convexes dont les longueurs des arêtes sont données, il existe un unique élément dont les sommets sont sur un cercle, un horocycle, ou dans une composante connexe de l’ensemble des points à distance constante d’une géodésique, et son aire est maximale. La seconde application est un résultat de rigidité pour les surfaces polyèdrales équivariantes dans l’espace de Minkowski. 1.

Interactive Geometry is a major tool in modern geometry education and various software tools are available. We discuss the requirements of such tools and how they can be fulfilled. We also explain how a geometry tool can benefit from the Internet and present Cinderella's Café, which is an internet-aware geometry tool with a high mathematical background.

Pizlo, Rosenfeld, and Weiss[6] suggest that people perceive two 2-D shapes as the same (i.e., there is shape constancy) when one shape might be a perspective image of the other. We argue that this hypothesis is both too broad and too restrictive. Their view is too broad because, as we show, two such images can appear quite different when there is significant perspective distortion. Their view is too restrictive because factors such as context and the complexity of the shapes compared can play a role in judgements of shape constancy. Moreover, Pizlo et al.'s formulation is asymmetric in requiring that one of the images be considered the model which produces the other. One can consider removing this asymmetry by judging shapes as the same when they could be two perspective views, taken with the same camera, of a third planar shape. We show that this is true if and only if the images are projectively equivalent. However, Pizlo et al. have convincingly rejected projective equivalence as a ...

A human with its highly complex and fine tuned visual system can easily recognize objects. It is however difficult to construct a computer program that can recognize general objects using visual input, e.g. from a CCD-camera. This paper treats recognition of planar non-algebraic curves. It is shown how invariant features, which are robust to occlusion, changes in lighting and viewpoint, can be extracted from a single gray-scale image. The invariant features can then be used in a fast recognition algorithm. Geometrical configurations are selected using edge detection and segmentation. Robust invariant features are then extracted from these configurations by choosing first a canonical or normal reference frame, and then features in this reference frame. These features automatically become invariant. In this work methods are given where a global and stable approach is used both in choosing the normal reference frame and in extracting invariants under affine and projective transformations. These...

. We develop exact algorithms for geometric operations on general circles and circular arcs on the sphere, using integer homogeneous coordinates. The algorithms include testing a point against a circle, computing the intersection of two circles, and ordering three arcs out of the same point. These operations allow robust manipulation of maps on the sphere, providing a reliable framework for GIS, robotics, and other geometric applications. 1 Introduction A spherical map is a partition of the sphere's surface in three elements: vertices (points), edges (circles and arcs of circles), and faces (open regions). Most applications in geographical information systems (GIS) involve maps of this type: they arise from geodetic lines, latitude-longitude grids, stereographic projection of plane polygons, satellite images, etc [4]. Our aim is to develop representations and algorithms for spherical maps which are free from roundoff errors, and therefore robust in the sense of Hoffman and Yap[3...