When determining the parameters of a parametric planar shape based on a single low-resolution image, common estimation paradigms lead to inaccurate parameter estimates. The reason behind poor estimation results is that standard estimation frameworks fail to model the image formation process at a sufficiently detailed level of analysis. We propose a new method for estimating the parameters of a planar elliptic shape based on a single photon-limited, low-resolution image. Our technique incorporates the effects of several elements—point-spread function, discretization step, quantization step, and photon noise—into a single cohesive and manageable statistical model. While we concentrate on the particular task of estimating the parameters of elliptic shapes, our ideas and methods have a much broader scope and can be used to address the problem of estimating the parameters of an arbitrary parametrically representable planar shape. Comprehensive experimental results on simulated and real imagery demonstrate that our approach yields parameter estimates with unprecedented accuracy. Furthermore, our method supplies a parameter covariance matrix as a measure of uncertainty for the estimated parameters, as well as a planar confidence region as a means for visualizing parameter uncertainty. The mathematical model developed in this paper may prove useful in a variety of disciplines that operate with imagery at the limits of resolution.
The majority of ellipse estimation methods fit a curve to a planar set of points. One distinguishes between point-based ellipse fitting methods by considering the nature of the cost function that the algorithms minimise. Methods which explicitly decrease the distance between the points and the ellipse curve are considered geometric methods. The quintessential geometric method is orthogonal distance regression, which minimises the orthogonal distance from a point to the curve @cite_10 @cite_13 @cite_25 @cite_18 @cite_11 @cite_30 @cite_17 . Algebraic methods, on the other hand, try to ensure that the data points satisfy an ellipse implicit equation as accurately as possible. One differentiates between algebraic methods by considering how they penalise the degree to which a data point fails to satisfy an implicit equation @cite_29 @cite_21 @cite_26 @cite_16 . Algebraic methods, in particular, have been the focus of considerable study, and recent works have concentrated on improving their statistical efficacy and accuracy @cite_12 @cite_14 @cite_1 @cite_22 .
@cite_30 @cite_18 @cite_26 @cite_14 @cite_22 @cite_10 @cite_29 @cite_21 @cite_1 @cite_17 @cite_16 @cite_13 @cite_25 @cite_12 @cite_11
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In this paper, we propose an efficient algorithm, called vector field consensus, for establishing robust point correspondences between two sets of points. Our algorithm starts by creating a set of putative correspondences which can contain a very large number of false correspondences, or outliers, in addition to a limited number of true correspondences (inliers). Next, we solve for correspondence by interpolating a vector field between the two point sets, which involves estimating a consensus of inlier points whose matching follows a nonparametric geometrical constraint. We formulate this a maximum a posteriori (MAP) estimation of a Bayesian model with hidden latent variables indicating whether matches in the putative set are outliers or inliers. We impose nonparametric geometrical constraints on the correspondence, as a prior distribution, using Tikhonov regularizers in a reproducing kernel Hilbert space. MAP estimation is performed by the EM algorithm which by also estimating the variance of the prior model (initialized to a large value) is able to obtain good estimates very quickly (e.g., avoiding many of the local minima inherent in this formulation). We illustrate this method on data sets in 2D and 3D and demonstrate that it is robust to a very large number of outliers (even up to 90 ). We also show that in the special case where there is an underlying parametric geometrical model (e.g., the epipolar line constraint) that we obtain better results than standard alternatives like RANSAC if a large number of outliers are present. This suggests a two-stage strategy, where we use our nonparametric model to reduce the size of the putative set and then apply a parametric variant of our approach to estimate the geometric parameters. Our algorithm is computationally efficient and we provide code for others to use it. In addition, our approach is general and can be applied to other problems, such as learning with a badly corrupted training data set. The problem of accurate parameter estimation of elliptical shapes is addressed. Three methods based on elliptical Fourier descriptors, area moments, and perimeter moments of 2-D elliptical shapes are presented. The mathematics of these methods is discussed and, for area moments, a set of new formulas based on Gauss's theorem is given. To implement a comparative performance study, a method developed by R. Safaee- (1990), the weighted minimum-square-error (MSE) function, is included, and an objective and independent measure of goodness of fit is used. The performance of these methods is compared for two different cases, that of a perfect simulated ellipse and an imperfect imaged ellipse. > We introduce a framework for computing statistically optimal estimates of geometric reconstruction problems. While traditional algorithms often suffer from either local minima or non-optimality--or a combination of both--we pursue the goal of achieving global solutions of the statistically optimal cost-function. Our approach is based on a hierarchy of convex relaxations to solve non-convex optimization problems with polynomials. These convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation. The technique is applied to a number of classical vision problems: triangulation, camera pose, homography estimation and last, but not least, epipolar geometry estimation. Experimental validation on both synthetic and real data is provided. In practice, only a few relaxations are needed for attaining the global optimum. After summarising the conceptual relationship between some old and new techniques for fitting ellipses to data, we conduct a thorough experimental comparison of the discussed methods. The newer techniques promise consistency, unbiasedness, or hyper-accuracy, but our experiments reveal that in practice the performance of these estimators is difficult to distinguish from the performance of established estimators. Since the newer techniques could potentially be extended to other estimation problems, we discuss what implications our experimental findings have for solving more general multiple-view geometry estimation problems.