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.
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@cite_14: 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.
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@cite_10: 1. Introduction.- 2. Least-Squares Orthogonal Distance Fitting.- 3. Orthogonal Distance Fitting of Implicit Curves and Surfaces.- 4. Orthogonal Distance Fitting of Parametric Curves and Surfaces.- 5. Object Reconstruction from Unordered Point Cloud.- 6. Conclusions.
@cite_29: Abstract 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 ellipses 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 shifting fitting, and then re-expressing the fit in the original frame of reference.
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@cite_1: The technique of "renormalization" for geometric estimation attracted much attention when it was proposed in early 1990s for having higher accuracy than any other then known methods. Later, it was replaced by minimization of the reprojection error. This paper points out that renormalization can be modified so that it outperforms reprojection error minimization. The key fact is that renormalization directly specifies equations to solve, just as the "estimation equation" approach in statistics, rather than minimizing some cost. Exploiting this fact, we modify the problem so that the solution has zero bias up to high order error terms; we call the resulting scheme hyper-renormalization. We apply it to ellipse fitting to demonstrate that it indeed surpasses reprojection error minimization. We conclude that it is the best method available today.
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@cite_12: For fitting an ellipse to a point sequence, ML (maximum likelihood) has been regarded as having the highest accuracy. In this paper, we demonstrate the existence of a "hyperaccurate" method which outperforms ML. This is made possible by error analysis of ML followed by subtraction of high-order bias terms. Since ML nearly achieves the theoretical accuracy bound (the KCR lower bound), the resulting improvement is very small. Nevertheless, our analysis has theoretical significance, illuminating the relationship between ML and the KCR lower bound.
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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 .