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12
Statistical optimization for geometric fitting: Theoretical accuracy analysis and high order error analysis
 Int. J. Comput. Vis
, 2008
"... A rigorous accuracy analysis is given to various techniques for estimating parameters of geometric models from noisy data for computer vision applications. First, it is pointed out that parameter estimation for vision applications is very different in nature from traditional statistical analysis and ..."
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A rigorous accuracy analysis is given to various techniques for estimating parameters of geometric models from noisy data for computer vision applications. First, it is pointed out that parameter estimation for vision applications is very different in nature from traditional statistical analysis and hence a different mathematical framework is necessary in such a domain. After general theories on estimation and accuracy are given, typical existing techniques are selected, and their accuracy is evaluated up to higher order terms. This leads to a “hyperaccurate ” method that outperforms existing methods. 1.
N.: Error analysis for circle fitting algorithms
 Electronic Journal of Statistics
"... We study the problem of fitting circles (or circular arcs) to data points observed with errors in both variables. A detailed error analysis for all popular circle fitting methods – geometric fit, K˚asa fit, Pratt fit, and Taubin fit – is presented. Our error analysis goes deeper than the traditional ..."
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We study the problem of fitting circles (or circular arcs) to data points observed with errors in both variables. A detailed error analysis for all popular circle fitting methods – geometric fit, K˚asa fit, Pratt fit, and Taubin fit – is presented. Our error analysis goes deeper than the traditional expansion to the leading order. We obtain higher order terms, which show exactly why and by how much circle fits differ from each other. Our analysis allows us to construct a new algebraic (noniterative) circle fitting algorithm that outperforms all the existing methods, including the (previously regarded as unbeatable) geometric fit.
Y.: Renormalization Returns: Hyperrenormalization and Its Applications
 ECCV 2012, Part III. LNCS
, 2012
"... Abstract. 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 renormalizati ..."
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Abstract. 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 hyperrenormalization. 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. 1
Temporal super resolution from a single quasiperiodic image sequence based on phase registration
, 2010
"... Abstract. This paper describes a method for temporal super resolution from a single quasiperiodic image sequence. A socalled reconstructionbased method is applied to construct a one period image sequence with high framerate based on phase registration data in subframe order among multiple perio ..."
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Abstract. This paper describes a method for temporal super resolution from a single quasiperiodic image sequence. A socalled reconstructionbased method is applied to construct a one period image sequence with high framerate based on phase registration data in subframe order among multiple periods of the image sequence. First, the periodic image sequence to be reconstructed is expressed as a manifold in the parametric eigenspace of the phase. Given an input image sequence, phase registration and manifold reconstruction are alternately executed iteratively within an energy minimization framework that considers data fitness and the smoothness of both the manifold and the phase evolution. The energy minimization problem is solved through threestep coarsetofine procedures to avoid local minima. The experiments using both simulated and real data confirm the realization of temporal super resolution from a single image sequence. 1
Optimization Techniques for Geometric Estimation: Beyond Minimization
"... Abstract. We overview techniques for optimal geometric estimation from noisy observations for computer vision applications. We first describe estimation techniques based on minimization of given cost functions: least squares (LS), maximum likelihood (ML), which includes reprojection error minimizati ..."
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Abstract. We overview techniques for optimal geometric estimation from noisy observations for computer vision applications. We first describe estimation techniques based on minimization of given cost functions: least squares (LS), maximum likelihood (ML), which includes reprojection error minimization (Gold Standard) as a special case, and Sampson error minimization. We then formulate estimation techniques not based on minimization of any cost function: iterative reweight, renormalization, and hyperrenormalization. Showing numerical examples, we conclude that hyperrenormalization is robust to noise and currently is the best method. 1
Hyper Least Squares and Its Applications
 Proc. 10th Asian Conf. Comput. Vision.
, 2011
"... We present a new least squares (LS) estimator, called "HyperLS", specifically designed for parameter estimation in computer vision applications. It minimizes the algebraic distance under a special scale normalization, which is derived by rigorous error analysis in such a way that statisti ..."
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We present a new least squares (LS) estimator, called "HyperLS", specifically designed for parameter estimation in computer vision applications. It minimizes the algebraic distance under a special scale normalization, which is derived by rigorous error analysis in such a way that statistical bias is removed up to second order noise terms. Numerical experiments suggest that our HyperLS is far superior to the standard LS and comparable in accuracy to maximum likelihood (ML), which is known to produce highly accurate results in image applications but may fail to converge if poorly initialized. Our HyperLS is a perfect candidate for ML initialization. In addition, we discuss how imagebased inference problems have different characteristics form conventional statistical applications, with a view to serving as a bridge between mathematicians and computer engineers.
Fundamental Matrix Computation: Theory and Practice
"... We classify and review existing algorithms for computing the fundamental matrix from point correspondences and propose new effective schemes: 7parameter LevenbergMarquardt (LM) search, extended FNS, and EFNSbased bundle adjustment. Doing experimental comparison, we show that EFNS and the 7parame ..."
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We classify and review existing algorithms for computing the fundamental matrix from point correspondences and propose new effective schemes: 7parameter LevenbergMarquardt (LM) search, extended FNS, and EFNSbased bundle adjustment. Doing experimental comparison, we show that EFNS and the 7parameter LM search exhibit the best performance and that additional bundle adjustment does not increase the accuracy to any noticeable degree. 1.
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT 1 Transceiver inphase/quadratureimbalance, ellipse fitting, and the universal software radio peripheral
"... Abstract—In this paper we introduce a method for IQ imbalance parameter estimation based on ellipse fitting. The performance of the method is analytically derived. In particular, it is shown that the method exhibits a small bias (which can be negligible under some standard practical conditions) and ..."
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Abstract—In this paper we introduce a method for IQ imbalance parameter estimation based on ellipse fitting. The performance of the method is analytically derived. In particular, it is shown that the method exhibits a small bias (which can be negligible under some standard practical conditions) and a variance slightly above the CramérRao bound. The method is then applied to measurements from a contemporary BiCMOS transceiver which is used on one of the most popular daughterboards of the universal software peripheral (USRP). In our measurements the phase skew varies up to five degrees with the baseband frequency, while the amplitude imbalance varies between 0−0.3 dB over carrier frequencies and across hardware units. The time variation however is only 0.004 dB in amplitude and 0.06 degrees in phase. This indicates that the units could either be calibrated online when there is no transmission (in a two antenna MIMO system one antenna could transmit a calibration signal to the other), or they could be calibrated during production, in which a case a table with different carrier and baseband frequencies would be needed. However, there is no need to estimate the parameters on every burst. Index Terms—inphase/quadrature (IQ) imbalance, CramérRao bounds, ellipse fitting, universal software radio peripheral
Hyperaccurate Ellipse Fitting without Iterations
, 2009
"... This paper presents a new method for fitting an ellipse to a point sequence extracted from images. It is widely known that the best fit is obtained by maximum likelihood. However, it requires iterations, which may not converge in the presence of large noise. Our approach is algebraic distance minimi ..."
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This paper presents a new method for fitting an ellipse to a point sequence extracted from images. It is widely known that the best fit is obtained by maximum likelihood. However, it requires iterations, which may not converge in the presence of large noise. Our approach is algebraic distance minimization; no iterations are required. Exploiting the fact that the solution depends on the way the scale is normalized, we analyze the accuracy to high order error terms with the scale normalization weight unspecified and determine it so that the bias is zero up to the second order. We demonstrate by experiments that our method is superior to the Taubin method, also algebraic and known to be highly accurate. 1.
The Algorithm on Displacement Differences Calculation and the Error of Surface Displacement Measuring Device
, 2013
"... Abstract: Angle sensor is the core component in the surface displacement measuring device, whose manufacturing process, fixing position and internal device performance can affect the testing accuracy in many aspects. This paper would present a new displacement calculation based on coordinating trans ..."
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Abstract: Angle sensor is the core component in the surface displacement measuring device, whose manufacturing process, fixing position and internal device performance can affect the testing accuracy in many aspects. This paper would present a new displacement calculation based on coordinating transformation and employ particle swarm method to calibrate the data. Experimental results show that the amended data through this method is quite close to standard test bench data, indicating that the proposed displacement differences calculation and error correction algorithm can effectively eliminate the fixed bias of surface displacement