We address the problem of reconstructing 3D space in a projective framework from two or more views, and the problem of artificially generating novel views of the scene from two given views (re-projection). We describe an invariance relation which provides a new description of structure, we call projective depth, which is captured by a single equation relating image point correspondences across two or more views and the homographies of two arbitrary virtual planes. The framework is based on knowledge of correspondence of features across views, is linear, extremely simple, and the computations of structure readily extends to over-determination using multiple views. Experimental results demonstrate a high degree of accuracy in both tasks - reconstruction and re-projection. Keywords---Visual Recognition, 3D Reconstruction from 2D Views, Projective Geometry, Algebraic and Geometric Invariants. I. Introduction The geometric relation between objects (or scenes) in the world and their imag...

We propose an affine framework for perspective views, captured by a single extremely simple equation based on a viewer-centered invariant we call relative affine structure. Via a number of corollaries of our main results we show that our framework unifies previous work -- including Euclidean, projective and affine -- in a natural and simple way, and introduces new, extremely simple, algorithms for the tasks of reconstruction from multiple views, recognition by alignment, and certain image coding applications.

AbstractÐWe present a method of indexing three-dimensional objects from single two-dimensional images. Unlike most other methods to solve this problem, ours does not rely on invariant features. This allows a richer set of shape information to be used in the recognition process. We also suggest the kd-tree as an alternative indexing data structure to the standard hash table. This makes hypothesis recovery more efficient in high-dimensional spaces, which are necessary to achieve specificity in large model databases. Search efficiency is maintained in these regimes by the use of Best-Bin First search, a modified kd-tree search algorithm which locates approximate nearest-neighbors. Neighbors recovered from the index are used to generate probability estimates, local within the feature space, which are then used to rank hypotheses for verification. On average, the ranking process greatly reduces the number of verifications required. Our approach is general in that it can be applied to any real-valued feature vector. In addition, it is straightforward to add to our index information from real images regarding the true probability distributions of the feature groupings used for indexing. In this paper, we provide experiments with both synthetic and real images, as well as details of the practical implementation of our system, which has been applied in the domain of telerobotics. Index TermsÐModel-based object recognition; indexing; kd-tree algorithm; nearest-neighbors algorithm. 1

This paper addresses the problem of recovering relative structure, in the form of an invariant, from two views of a 3D scene. The invariant structure is computed without any prior knowledge of camera geometry, or internal calibration, and with the property that perspective and orthographic projections are treated alike, namely, the system makes no assumption regarding the existence of perspective distortions in the input images. We show that

. The double algebra is a system for computations involving subspaces of a general finite dimensional vector space. If this vector space is taken as projective 3-space, the operations of the double algebra can be interpreted as joins and intersections of points, lines and planes. All computations are coordinate free and invariant over linear transformations. The double algebra is therefore a very effective tool for computation of linear invariants of geometric configurations. In this paper we show how to compute linear invariants of general configurations points and lines observed in two images and polyhedral configurations observed in one image. For these cases we derive directly explicit expression of the invariants without reconstructing individual points and lines. 1 Introduction The basic problem facing automatic visual recognition systems is the variability of the image an object can produce due to changes in viewpoint and intrinsic camera parameters. This is the main motivation ...

One of the strongest cues for retrieval of content information from images is shape. However, due to the wide range of transformations that an object might undergo, this is also the most difficult one to handle. It seems that shape retrieval is one of the major barriers nowadays to image databases being commonly used. We present an approach for shape retrieval from pictorial databases which is based on invariant features of the image. In particular we use a combination of semi-local multivalued invariant signatures and global features. Spatial relations and global properties are used to eliminate nonrelevant images before similarity is computed. The advantages of the proposed approach are its ability to handle images distorted by different viewpoint transformations, its ability to retrieve images even in situations in which part of the shape is missing (i.e., in case of occlusion or sketch-based queries), and its ability to support efficient indexing. We have implemented our approach in a heterogeneous database having a SQL-like user interface augmented with sketch-based queries. The system is built on top of a commercial database system and can be activated from the Web. We present experimental results demonstrating the effectiveness of the proposed approach.

Introduction Efficient and reliable recognition of objects in images requires a representation of the object with low complexity for computational efficiency but sufficiently rich in order to separate it from other objects. The geometry of an object, represented by its edges and contours is often ideal for this purpose. This level of description is in general broken down into the levels of feature points, lines and sometimes higher order curve 1 approximations. Recognition is then based on the relations between these features. The complexity of recognizing 3-D objects from 2-D images is increased by the fact that an object in 3-D will project different images depending on the viewpoint of the observer. The desire to achieve efficient viewpoint independent object recognition has therefore recently led to an increased interest in projective invariance for object representation (Barret 1991, Forsyth 1990, Forsyth 1991, Lamdan 1988, Weiss 1988). Projectively invariant des

Recent papers [ Lamdan et al., 1988, Clemens and Jacobs, 1991 ] have shown that indexing is a promising approach to fast model-based object recognition because it allows most of the possible matches between image point groups and model point groups to be quickly eliminated from consideration. Current indexing systems for the problem of recognizing three-dimensional objects from single two-dimensional images require groups of four points to generate a key into the table of model groups and each model group must be represented over an infinite subspace of a multi-dimensional table [ Clemens and Jacobs, 1991, Jacobs, 1992 ] . We present a system that is capable of indexing using groups of three points by taking advantage of the probabilistic peaking effect [ BenArie, 1990 ] . Each model group need only be represented at one point in the index table. To be able to index using groups of three points, we must allow false negatives for point group matches. If there are n model points present ...

The alignment method [ Huttenlocher and Ullman, 1990 ] is a model-based object recognition technique that determines possible object transformations from three hypothesized matches of model and image points. In the absence of grouping, the alignment method must examine each possible matching of three model points with three image points. Thus, if m is the number of model features and n is the number of image features, this method requires O(m 3 n 3 ) transformations to be computed. Each of these transformations must then be tested to determine whether it is correct, a time consuming step itself. For images and/or models with many features, the running time of the alignment method is not satisfactory, even in the presence of current grouping techniques. This paper presents methods of reducing the number of matches that must be examined. The techniques we describe are: 1) using the probabilistic peaking effect [ Ben-Arie, 1990 ] to eliminate unlikely matches, 2) examining the algorit...

. In this paper, we investigate quasi-invariance on a smooth manifold, and show that there exist quasi-invariant parameterisations which are not exactly invariant but approximately invariant under group transformations and do not require high order derivatives. The affine quasi-invariant parameterisation is investigated in more detail and exploited for defining general affine semi-local invariants from second order derivatives only. The new invariants are implemented and used for matching curve segments under general affine motions and extracting symmetry axes of objects with 3D bilateral symmetry. Keywords: quasi-invariant parameterisations, semi-local invariants, integral invariants, differential invariants, curve matching, bilateral symmetry 1. Introduction The distortions of an image curve caused by the relative motion between the observer and the scene can be described by specific transformation groups (Mundy and Zisserman, 1992). For example, the corresponding pair of contour c...