In this paper we analyze in some detail the geometry of a pair of cameras, i.e. a stereo rig. Contrarily to what has been done in the past and is still done currently, for example in stereo or motion analysis, we do not assume that the intrinsic parameters of the cameras are known (coordinates of the principal points, pixels aspect ratio and focal lengths). This is important for two reasons. First, it is more realistic in applications where these parameters may vary according to the task (active vision). Second, the general case considered here, captures all the relevant information that is necessary for establishing correspondences between two pairs of images. This information is fundamentally projective and is hidden in a confusing manner in the commonly used formalism of the Essential matrix introduced by Longuet-Higgins [40]. This paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 \Theta 3 ma...

Abstract. This paper has two goals. The first is to develop a variety of robust methods for the computation of the Fundamental Matrix, the calibration-free representation of camera motion. The methods are drawn from the principal categories of robust estimators, viz. case deletion diagnostics, M-estimators and random sampling, and the paper develops the theory required to apply them to non-linear orthogonal regression problems. Although a considerable amount of interest has focussed on the application of robust estimation in computer vision, the relative merits of the many individual methods are unknown, leaving the potential practitioner to guess at their value. The second goal is therefore to compare and judge the methods. Comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery. In contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth. A variety of error measures are examined. The experiments allow a statistically satisfying and quasi-optimal method to be synthesized, which is shown to be stable with up to 50 percent outlier contamination, and may still be used if there are more than 50 percent outliers. Performance bounds are established for the method, and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined. The results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers.

This work presents a new efficient method for fitting ellipses to scattered data. Previous algorithms either fitted general conics or were computationally expensive. By minimizing the algebraic distance subject to the constraint 4ac - b² = 1 the new method incorporates the ellipticity constraint into the normalization factor. The proposed method combines several advantages: (i) It is ellipse-specific so that even bad data will always return an ellipse; (ii) It can be solved naturally by a generalized eigensystem and (iii) it is extremely robust, efficient and easy to implement.

: Almost all problems in computer vision are related in one form or another to the problem of estimating parameters from noisy data. In this tutorial, we present what is probably the most commonly used techniques for parameter estimation. These include linear least-squares (pseudo-inverse and eigen analysis); orthogonal least-squares; gradient-weighted least-squares; bias-corrected renormalization; Kalman øltering; and robust techniques (clustering, regression diagnostics, M-estimators, least median of squares). Particular attention has been devoted to discussions about the choice of appropriate minimization criteria and the robustness of the dioeerent techniques. Their application to conic øtting is described. Key-words: Parameter estimation, Least-squares, Bias correction, Kalman øltering, Robust regression (R#sum# : tsvp) Unite de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France) Telephone : (33) 93 65 77 77 -- Telecopie : (33) 9...

The fundamental matrix is a basic tool in the analysis of scenes taken with two uncalibrated cameras, and the 8-point algoritm is a frequent#e cit#3 met#9 d for comput#10 t he fundament al ma t# ix from a set of 8 or more point mat ches. It hast he advant age of simplicit y of implement at ion. The prevailing view is, however,t#(9 it isext#3791( suscept#-43 t o noise and hence virtually useless for most purposes. This paper challengest#en view, by showing t#ng by precedingt he algorit hm wit h a very simple normalizat ion(t ranslat ion and scaling) oft he coordinat es oft he mat ched point#( result# are obt# ined comparable wit# t he best it## at ive algorit#209 This improved performance is just#690 byt#1082 and verified byext#259( e experiment s on real images.

In the course of developing a system for fitting smooth curves to camera input we have developed several direct (i.e. noniterative) methods for fitting a shape (line, circle, conic, cubic, plane, sphere, quadric, etc.) to a set of points, namely exact fit, simple fit, spherical fit, and blend fit. These methods are all dimension-independent, being just as suitable for 3D surfaces as for the 2D curves they were originally developed for. Exact fit...

The Fundamental matrix is a key concept when working with uncalibrated images and multiple viewpoints. It contains all the available geometric information and enables to recover the epipolar geometry from uncalibrated perspective views. This paper addresses the important problem of its robust determination given a number of image point correspondences. We first define precisely this matrix, and show clearly how it is related to the epipolar geometry and to the Essential matrix introduced earlier by Longuet-Higgins. In particular, we show that this matrix, defined up to a scale factor, must be of rank two. Different parametrizations for this matrix are then proposed to take into account these important constraints and linear and non-linear criteria for its estimation are also considered. We then clearly show that the linear criterion is unable to express the rank and normalization constraints. Using the linear criterion leads definitely to the worst result in the determination of the Fu...

AbstractÐWe consider the problem of estimating parameters of a model described by an equation of special form. Specific models arise in the analysis of a wide class of computer vision problems, including conic fitting and estimation of the fundamental matrix. We assume that noisy data are accompanied by (known) covariance matrices characterizing the uncertainty of the measurements. A cost function is first obtained by considering a maximum-likelihood formulation and applying certain necessary approximations that render the problem tractable. A novel, Newton-like iterative scheme is then generated for determining a minimizer of the cost function. Unlike alternative approaches such as Sampson's method or the renormalization technique, the new scheme has as its theoretical limit the minimizer of the cost function. Furthermore, the scheme is simply expressed, efficient, and unsurpassed as a general technique in our testing. An important feature of the method is that it can serve as a basis for conducting theoretical comparison of various estimation approaches.

In this paper we evaluate several methods of fitting data to conic sections. Conic fitting is a commonly required task in machine vision, but many algorithms perform badly on incomplete or noisy data. We evaluate several algorithms under various noise and degeneracy conditions, identify the key parameters which affect sensitivity, and present the results of comparative experiments which emphasize the algorithms' behaviours under common examples of degenerate data. In addition, complexity analyses in terms of flop counts are provided in order to further inform the choice of algorithm for a specific application. 1 Introduction Vision applications often require the extraction of conic sections from image data. Common examples are the calculation of geometric invariants [2] and estimation of the centers and radii of circles for industrial inspection. Many textbooks [5, 10] provide discussions and algorithms for least-squares approximation of conics, but these often include only the simple...

We describe a model based recognition system, called LEWIS, for the identification of planar objects based on a projectively invariant representation of shape. The advantages of this shape description include simple model acquisition (direct from images), no need for camera calibration or object pose computation, and the use of index functions. We describe the feature construction and recognition algorithms in detail and provide an analysis of the combinatorial advantages of using index functions. Index functions are used to select models from a model base and are constructed from projective invariants based on algebraic curves and a canonical projective coordinate frame. Examples are given of object recognition from images of real scenes, with extensive object libraries. Successful recognition is demonstrated despite partial occlusion by unmodelled objects, and realistic lighting conditions. 1 Introduction 1.1 Overview In the context of this paper, recognition is defined as the prob...