A heuristic method has been developed for registering two sets of 3-D curves obtained by using an edge-based stereo system, or two dense 3-D maps obtained by using a correlation-based stereo system. Geometric matching in general is a difficult unsolved problem in computer vision. Fortunately, in many practical applications, some a priori knowledge exists which considerably simplifies the problem. In visual navigation, for example, the motion between successive positions is usually approximately known. From this initial estimate, our algorithm computes observer motion with very good precision, which is required for environment modeling (e.g., building a Digital Elevation Map). Objects are represented by a set of 3-D points, which are considered as the samples of a surface. No constraint is imposed on the form of the objects. The proposed algorithm is based on iteratively matching points in one set to the closest points in the other. A statistical method based on the distance distribution is used to deal with outliers, occlusion, appearance and disappearance, which allows us to do subset-subset matching. A least-squares technique is used to estimate 3-D motion from the point correspondences, which reduces the average distance between points in the two sets. Both synthetic and real data have been used to test the algorithm, and the results show that it is efficient and robust, and yields an accurate motion estimate.

Rosin, P.L., A note on the least squares fitting of ellipses, Pattern Recognition Letters 14 (1993) 799-808. The characteristics of two normalisations for the general conic equation are investigated for use in least squares fitting: either setting F = 1 or A + C = 1. The normalisations vary in three main areas: curvature bias, singularities, transformational invariance. It is shown that setting F = 1 is the more appropriate for ellipse fitting since it is less heavily curvature biased. Setting A + C = 1 produces more eccentric conics, resulting either in over-elongated el-lipses or hyperbolae. Although the F = 1 normalisation is less well suited than the A + C = 1 normalisation with respect to singularities and transformational invariance both these problems are solved by normalising the data, shifting it so that it is centred on the origin before fitting, and then re-expressing the fit in the original frame of reference.

We describe several established error of fit (EOF) functions for use in the least square fitting of ellipses, and introduce a further four new EOFs. Four measures are given for assessing the suitability of such EOFs, quantifying their linearity, curvature bias, asymmetry, and overall goodness. These measures enable a better understanding to be gained of the individual merits of the EOF functions.

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...

Determination of relative three-dimensional (3--D) position, orientation, and relative motion between two reference frames is an important problem in robotic guidance, manipulation, and assembly as well as in other fields such as photogrammetry. A solution to this problem that uses two-dimensional (2--D), intensity images from a single camera is desirable for real-time applications. Where the object geometry is unknown, the estimation of structure is also required. A single camera is advantageous because a standard video camera is low in cost, setup and calibration are simple, physical space requirements are small, reliability is high, and low-cost hardware is available for digitizing and processing the images. A di#culty in performing this measurement is the process of projecting 3--D object features to 2--D images, a nonlinear transformation. Noise is present in the form of perturbations to the assumed process dynamics, imperfections in system modeling, and errors in the feature locations extracted from the 2--D images. This dissertation presents solutions to the remote measurement problem for a dynamic system given a sequence of 2--D intensity images of an object where feature positions of the object are known relative to a base reference frame and where the feature positions are unknown relative to a base reference frame. The 3--D transformation is modeled as a nonlinear stochastic system with the state estimate providing six degree-of-freedom motion and position values. The stochastic model uses the iterated extended Kalman filter as an estimator and as a screw representation of the 3--D transformation based on dual quaternions. Dual quaternions provide a means to represent both rotation and translation in a unified notation. The method has been implemented and tes...

The choice of shape representation and the extraction of such representations from images is one of the great challenges of computer vision. This thesis addresses these issues by examining a number of topics in curve representations. Beginning with an examination of the conic fitting problem, a new linear ellipse fitter is developed. Previous ellipse-specific methods have been computationally expensive, and previous linear methods have fitted general conics, rather than ellipses, to the data. The new algorithm is compared with several others and is shown to be extremely stable and insensitive to noise. The comparison is itself of interest as it focusses on the behaviour of the algorithms under occlusion rather than noise, demonstrating that this is the parameter to which they are most sensitive. A comprehensive evaluation of conic fitting algorithms then follows, concluding that occlusion sensitivity is one of the key characteristics of the conic fitting problem. This survey is in itself of interest as it provides specific recommendations for practitioners in the field. The second part of the thesis deals with the question of deciding how well a model describes a given set of data. Two new techniques are discussed, both of which are independent of the noise level of the data, and which are therefore applicable to a wide range of automated processes. The run-distribution test of Chapter 5 is an effective method of determining a posteriori whether a given model accurately describes a data set. Comparisons with a number of standard tests indicate that the run-distribution test outperforms them unless the true noise level is known. The sum-of-variance metric of Chapter 6, on the other hand, provides a parameter-free method of segmenting a dataset into piecewise smooth segments. The behaviour of the metric is demonstrated

. In this paper, we propose an unbiased minimum variance estimator to estimate the parameters of an ellipse. The objective of the optimization is to compute a minimum variance estimator. The experimental results show the dramatic improvement over existed weighted least sum of squares approach especially when the ellipse is occluded. 1 Introduction The methods of estimation of the parameters of quadratic curve can be classified into two categories, the least squares curve fitting [1, 5, 7, 8], and the Kalman filtering techniques, [3, 6]. The general quadratic curve can be written as follows: Q(X;Y ) = aX 2 + bXY + cY 2 + dX + eY + f = 0; (1) with b 2 ! 4ac corresponding to the ellipses. Suppose that points (x i ; y i ); i = 1; 2; :::; n are the detected elliptical points, then the least sum of squares fitting method finds the ellipse parameters (a; b; c; d; e; f) by minimizing following objective function: n X i=1 " i = n X i=1 (ax 2 i + bx i y i + cy 2 i + dx i + ey...

Polyethylene wear in the acetabular components of hip prostheses is implicated in loosening and failure. Radiographic measurement of wear is used to identify patients at risk and to assess prosthesis designs. This paper focuses on analysis of prostheses with cemented acetabular cups from anteroposterior radiographs. The articular surface of the femoral head and the acetabular rim marker are modelled as spherical and circular respectively, resulting in elliptical image projections. Methods for automatically localising these structures in radiographs are presented using robust ellipse fitting and various error functions. Special attention is paid to the acetabular marker since this often projects as a highly eccentric ellipse. Robust fitting enables successful localisation in the presence of clutter without the need for user interaction. Finally, the use of these ellipses as reference structures for wear estimation is investigated and the effect of eccentricity errors is highlighted. 1

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