Guaranteed Ellipse Fitting with a Confidence Region and an Uncertainty Measure for Centre, Axes, and Orientation

Szpak, Zygmunt L.; Chojnacki, Wojciech; Hengel, Anton van den

doi:10.1007/s10851-014-0536-x

Cited by 46 publications

(52 citation statements)

References 47 publications

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“…Since minimizing the orthogonal distance between a point and an ellipse has no closed-form solution, these routines resort to iterative techniques that are not guaranteed to converge on an ellipse. A commonly used algebraic method is the simple and robust DEF method developed by Fitzgibbon et al [62] that minimizes the sum of squared algebraic distances between the points and the ellipse, y F ( , ) [59,63] developed an algorithm based on the optimization of the approximate maximum likelihood distance which seeks a balance between the costly geometric methods and stable algebraic techniques. This algorithm-termed the FGEF method-also includes error estimation for the geometrically meaningful ellipse parameters (center coordinates, axes and orientation) which we have extended to include an estimate of the differential phase error, d δϕ .…”

Section: Appendix a Ellipse Fitting Methodsmentioning

confidence: 99%

“…This algorithm-termed the FGEF method-exhibits a smaller bias in the differential phase estimate over a relatively large phase range (centered on 2 π ) compared to the more commonly used 'direct ellipse fit' (DEF) technique [62]. Additionally, [63] includes error estimations for the geometrically meaningful ellipse parameters (center coordinates, axes and orientation). We have extended their work to include an estimate of the statistical uncertainty in the differential phase, d δϕ .…”

Section: Improved Ellipse Fittingmentioning

confidence: 99%

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Correlative methods for dual-species quantum tests of the weak equivalence principle

et al. 2015

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Matter-wave interferometers utilizing different isotopes or chemical elements intrinsically have different sensitivities, and the analysis tools available until now are insufficient for accurately estimating the atomic phase difference under many experimental conditions. In this work, we describe and demonstrate two new methods for extracting the differential phase between dual-species atom interferometers for precise tests of the weak equivalence principle (WEP). The first method is a generalized Bayesian analysis, which uses knowledge of the system noise to estimate the differential phase based on a statistical model. The second method utilizes a mechanical accelerometer to reconstruct single-sensor interference fringes based on measurements of the vibration-induced phase. An improved ellipse-fitting algorithm is also implemented as a third method for comparison. These analysis tools are investigated using both numerical simulations and experimental data from simultaneous 87 Rb and 39 K interferometers, and both new techniques are shown to produce bias-free estimates of the differential phase. We also report observations of phase correlations between atom interferometers composed of different chemical species. This correlation enables us to reject common-mode vibration noise by a factor of 730, and to make preliminary tests of the WEP with a sensitivity of 1.6 10 6 × − per measurement with an interrogation time of T = 10 ms. We study the level of vibration rejection by varying the temporal overlap between interferometers in a symmetric timing sequence. Finally, we discuss the limitations of the new analysis methods for future applications of differential atom interferometry.

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Section: Appendix a Ellipse Fitting Methodsmentioning

confidence: 99%

Section: Improved Ellipse Fittingmentioning

confidence: 99%

Correlative methods for dual-species quantum tests of the weak equivalence principle

et al. 2015

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“…The code fit an ellipse to the points using the approximate maximum‐likelihood (AML) method of Szpak et al (). Crater center (decimal degrees), major ( a ) and minor ( b ) axes (kilometers), tilt angle (0° to 180°, counterclockwise for the major axis, where 0° means the major axis is in an East‐West direction), ellipticity e = a / b , and eccentricity ε = (1– b 2 / a 2 ) 1/2 were recorded along with standard errors for each value.…”

Section: Generating a New Global Lunar Crater Database And Informatimentioning

confidence: 99%

A New Global Database of Lunar Impact Craters >1–2 km: 1. Crater Locations and Sizes, Comparisons With Published Databases, and Global Analysis

2019

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This paper presents a new, global database of lunar impact craters, estimated to be a complete census of all craters with diameters larger than 1–2 km. The database contains over 2 million craters, making it larger in number than any previously published lunar effort by more than a factor of 10. Of those craters, 1.3 million have diameters ≥1 km, approximately 83,000 are ≥5 km, and 6,972 craters are ≥20 km. How the database was constructed along with the reliability of features is described in detail. Comparisons are made with past published databases, demonstrating good agreement for crater size and location. An ellipticity analysis is conducted, illustrating there is no dominant direction for elliptical crater orientation based on location, diameter range, or ellipticity amount, consistent with randomness for craters ≥10 km. A spatial density analysis is described, comparing the spatial density of small versus large craters, and numerous observations about the nonuniformity of the size distributions of craters across the Moon are made. The spatial density is also used in a discussion about kilometer‐scale secondary impact craters and clearly shows that they dominate the crater population in some areas of the lunar surface. This paper presents just a tiny sample of the scientific investigations that could be done with this new crater database.

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“…We found a perfectly suited algorithm recently described by Szpak et al (2015) that offers a solution for exactly this type of problem. The D and R uncertainties can be described as independent Gaussian noise with zero mean that is inhomogeneous, i.e.…”

Section: Size Of the Shadow From A Best-fitting Ellipsementioning

confidence: 99%

“…Finally, a confidence region in the plane of the ellipse is calculated, illustrating the zone in which the true ellipse is located at a given probability. We chose a planar 68.3% confidence region to be consistent with other literature in the field instead of the 95% region suggested by Szpak et al (2015), e.g. for industrial machine vision applications.…”

Section: Size Of the Shadow From A Best-fitting Ellipsementioning

confidence: 99%

Results from a triple chord stellar occultation and far-infrared photometry of the trans-Neptunian object (229762) 2007 UK₁₂₆

Schindler

Wolf

Bardecker³

et al. 2017

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Context.A stellar occultation by a trans-Neptunian object (TNO) provides an opportunity to probe the size and shape of these distant solar system bodies. In the past seven years, several occultations by TNOs have been observed, but mostly from a single location. Only very few TNOs have been sampled simultaneously from multiple locations. Sufficient data that enable a robust estimation of shadow size through an ellipse fit could only be obtained for two objects. Aims. We present the first observation of an occultation by the TNO 2007 UK 126 on 15 November 2014, measured by three observers, one nearly on and two almost symmetrical to the shadow's centerline. This is the first multi-chord dataset obtained for a so-called detached object, a TNO subgroup with perihelion distances so large that the giant planets have likely not perturbed their orbits. We also revisit Herschel/PACS far-infrared data, applying a new reduction method to improve the accuracy of the measured fluxes. Combining both datasets allows us to comprehensively characterize 2007 UK 126 . Methods. We use error-in-variable regression to solve the non-linear problem of propagating timing errors into uncertainties of the ellipse parameters. Based on the shadow's size and a previously reported rotation period, we expect a shape of a Maclaurin spheroid and derive a geometrically plausible size range. To refine our size estimate of 2007 UK 126 , we model its thermal emission using a thermophysical model code. We conduct a parametric study to predict far-infrared fluxes and compare them to the Herschel/PACS measurements. Results. The favorable geometry of our occultation chords, combined with minimal dead-time imaging, and precise GPS time measurements, allow for an accurate estimation of the shadow size (best-fitting ellipse with axes 645.80 ± 5.68 km × 597.81 ± 12.74 km) and the visual geometric albedo (p V = 15.0 ± 1.6%). By combining our analyses of the occultation and the far-infrared data, we can constrain the effective diameter of 2007 UK 126 to d eff = 599−629 km. We conclude that subsolar surface temperatures are in the order of ≈50−55 K.

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Guaranteed Ellipse Fitting with a Confidence Region and an Uncertainty Measure for Centre, Axes, and Orientation

Cited by 46 publications

References 47 publications

Correlative methods for dual-species quantum tests of the weak equivalence principle

Correlative methods for dual-species quantum tests of the weak equivalence principle

A New Global Database of Lunar Impact Craters >1–2 km: 1. Crater Locations and Sizes, Comparisons With Published Databases, and Global Analysis

Results from a triple chord stellar occultation and far-infrared photometry of the trans-Neptunian object (229762) 2007 UK₁₂₆

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Guaranteed Ellipse Fitting with a Confidence Region and an Uncertainty Measure for Centre, Axes, and Orientation

Cited by 46 publications

References 47 publications

Correlative methods for dual-species quantum tests of the weak equivalence principle

Correlative methods for dual-species quantum tests of the weak equivalence principle

A New Global Database of Lunar Impact Craters >1–2 km: 1. Crater Locations and Sizes, Comparisons With Published Databases, and Global Analysis

Results from a triple chord stellar occultation and far-infrared photometry of the trans-Neptunian object (229762) 2007 UK126

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Results from a triple chord stellar occultation and far-infrared photometry of the trans-Neptunian object (229762) 2007 UK₁₂₆