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.

A bugbear of uncalibrated stereo reconstruction is that cameras which deviate from the pinhole model have to be pre-calibrated in order to correct for nonlinear lens distortion. If they are not, and point correspondence is attempted using the uncorrected images, the matching constraints provided by the fundamental matrix must be set so loose that point matching is significantly hampered. This paper shows how linear estimation of the fundamental matrix from two-view point correspondences may be augmented to include one term of radial lens distortion. This is achieved by (1) changing from the standard radiallens model to another which (as we show) has equivalent power, but which takes a simpler form in homogeneous coordinates, and (2) expressing fundamental matrix estimation as a Quadratic Eigenvalue Problem (QEP), for which efficient algorithms are well known. I derive the new estimator, and compare its performance against bundle-adjusted calibration-grid data. The new estimator is fast enough to be included in a RANSAC-based matching loop, and we show cases of matching being rendered possible by its use. I show how the same lens can be calibrated in a natural scene where the lack of straight lines precludes most previous techniques. The modification when the multi-view relation is a planar homography or trifocal tensor is described. 1.

We present an algorithm to estimate the parameters of a linear model in the presence of heteroscedastic noise, i.e., each data point having a different covariance matrix.

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...

Vision algorithms utilizing camera networks with a common field of view are becoming increasingly feasible and important. Calibration of such camera networks is a challenging and cumbersome task. The current approaches for calibration using planes or a known 3D target may not be feasible as these objects may not be simultaneously visible in all the cameras. In this paper, we present a new algorithm to calibrate cameras using occluding contours of spheres. In general, an occluding contour of a sphere projects to an ellipse in the image. Our algorithm uses the projection of the occluding contours of three spheres and solves for the intrinsic parameters and the locations of the spheres. The problem is formulated in the dual space and the parameters are solved for optimally and efficiently using semi-definite programming. The technique is flexible, accurate and easy to use. In addition, since the contour of a sphere is simultaneously visible in all the cameras, our approach can greatly simplify calibration of multiple cameras with a common field of view. Experimental results from computer simulated data and real world data, both for a single camera and multiple cameras, are presented.

This work presents the first direct method for specifically fitting ellipses in the least squares sense. Previous approaches used either generic conic fitting or relied on iterative methods to recover elliptic solutions. The proposed method is (i) ellipse-specific, (ii) directly solved by a generalised eigen-system, (iii) has a desirable loweccentricity bias, and (iv) is robust to noise. We provide a theoretical demonstration, several examples and the Matlab coding of the algorithm. 1. INTRODUCTION Ellipse fitting is one of the classic problems of pattern recognition and has been subject to considerable attention in the past ten years for its many application. Several techniques for fitting ellipses are based on mapping sets of points to the parameter space (notably the Hough transform). In this paper we are concerned with the more fundamental problem of least squares (LSQ) fitting of ellipses to scattered data. Previous methods achieved ellipse fitting by using generic conic fitters ...

A general method for eliminating the bias of non-linear estimators using bootstrap is presented. Instead of the traditional mean bias we consider the definition of bias based on the median. The method is applied to the problem of fitting ellipse segments to noisy data. No assumption beyond being independent identically distributed (i.i.d.) is made about the error distribution and experiments with both synthetic and real data prove the effectiveness of the technique. Index terms: implicit models, curve fitting, bootstrap, low-level processing. 1 Conic Fitting Image formation is a perspective projection of the 3D visual environment. Features extracted from a 2D image can be useful only if they preserve some of the geometric properties of the 3D object they correspond to. Collinearity and conicity are such properties, and therefore line and conic segments are widely used as geometric primitives in computer vision. Let f(u; `) = 0 be the implicit model of a geometric primitive in the ima...

This paper presents a numerically stable non-iterative algorithm for fitting an ellipse to a set of data points. The approach is based on a least squares minimization and it guarantees an ellipse-specific solution even for scattered or noisy data. The optimal solution is computed directly, no iterations are required. This leads to a simple, stable and robust fitting method which can be easily implemented. The proposed algorithm has no computational ambiguity and it is able to fit more than 100,000 points in a second. Keywords: ellipses, fitting, least squares, eigenvectors INTRODUCTION One of basic tasks in pattern recognition and computer vision is a fitting of geometric primitives to a set of points (see [Duda73] for a summary). The use of primitive models allows reduction and simplification of data and, consequently, faster and simpler processing. A very important primitive is an ellipse, which, being a perspective projection of a circle, is exploited in many applications of ...

Camera calibration is a fundamental problem in computer vision and photogrammetry. We present a new algorithm to calibrate cameras using spheres. In general, an occluding contour of a sphere projects to an ellipse in the image. Our algorithm uses the projection of the occluding contours of three spheres and solves for the intrinsic parameters and the locations of the spheres. The problem is formulated in the dual space and the parameters are solved for optimally and efficiently using semi-definite programming. The technique is flexible, accurate and easy to use. In addition, it can be used to simultaneously calibrate multiple cameras with a common field of view. Experimental results from computer simulated data and real world data, both for a single camera and multiple cameras, are presented.