Elsevier

Signal Processing

Volume 164, November 2019, Pages 30-40
Signal Processing

Robust ellipse fitting via alternating direction method of multipliers

https://doi.org/10.1016/j.sigpro.2019.05.032Get rights and content

Highlights

  • We replace the l2-norm by the lp-norm in the DLSF method to enhance robustness against outliers.

  • We derive the feasible region of the Lagrange multiplier, and analyze the monotonically decreasing property of the resultant problem.

  • We analyze the properties of each subfunction, and devise a simple but efficient method to calculate the corresponding auxiliary variable.

Abstract

The edge point errors, especially outliers, introduced in the edge detection step, will cause severe performance degradation in ellipse fitting. To address this problem, we adopt the ℓp-norm with p < 2 in the direct least square fitting method to achieve outlier resistance, and develop a robust ellipse fitting approach using the alternating direction method of multipliers (ADMM). Especially, to solve the formulated nonconvex and nonlinear problem, we decouple the ellipse parameter vector in the nonlinear ℓp-norm objective function from the nonconvex quadratic constraint via introducing auxiliary variables, and estimate the ellipse parameter vector and auxiliary variables alternately via the derived numerical methods. Simulation and experimental examples are presented to demonstrate the robustness of the proposed approach.

Introduction

The aim of ellipse fitting is to map a set of coplanar points of an image into an ellipse circumference. It plays an important role in the fields of computer vision, automatic manufacture, observational astronomy, structural geology, industry inspection, medical diagnosis, and military and security [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], and thus has received much attention. For example, in the silicon single crystal production industry [18], [19], the obtained diameter estimate via the ellipse fitting technique is employed to stimulate the corresponding control measures accurately in order that the produced cylinder-type silicon single crystal can be used as high-quality source material for semiconductor devices. Additionally, in the spacecraft control system [18], [20], the ellipse fitting technique is commonly utilized to determine the pose of a spacecraft from a single image and then control measures are taken to ensure that the spacecraft can navigate with a prespecified state. In the biomedical field, ellipse fitting has been utilized for automatic cell [21], [22] and epicardial fat segmentation [23] in diagnosis and treatment. It is also employed to assist vision based landing systems in controlling the motions of unmanned aerial vehicles [24].

Generally speaking, ellipse fitting algorithms can be classified into two categories, namely, clustering techniques [10] and least-squares methods [14], [15], [16]. The former include the Hough transform (HT) [10] and its variants, which involve heavy computational load due to the fact that an ellipse is characterized by five parameters and thus fine quantization and peak search in 5-dimensional (5-D) parameter space is required to achieve a satisfactory resolution. On the other hand, the latter can be further divided into geometric and algebraic methods. Geometric methods are based on the orthogonal distances between the data points and the estimated ellipse [11], [15]. For example, in [15], Barwick proposed a geometric approach called the parallel chord method. But it will fail when the parallel chords being perpendicular to a common axis are not available. While algebraic methods are widely used due to their advantages of simplicity and computational efficiency [12], [13], [14], [16]. For example, in [14], Fitzgibbon et al. developed a direct least square fitting (DLSF) method by solving a generalized eigenvalue problem. And the constrained least squares (CLS) method [16] solves a least-squares problem subject to a unit-norm constraint on the ellipse parameters. Besides, based on the probability theory, the Bayesian method [19] constructs a Markov chain and samples the ellipse parameters from the joint posterior distribution, and attains the state of equilibrium after sufficient number of iterations. Liang et al. [25] recovers a low-rank generalized multidimensional scaling matrix, which is a function of ellipse parameters to be determined.

It is worth mentioned that in practical applications, ellipse fitting often follows an edge detection step, and thus the effect of edge point errors, especially outliers, on the fitting performance cannot be ignored. Although some existed ellipse fitting methods work well in normal cases, their performance are sensitive to outliers. Therefore, it is necessary to develop robust ellipse fitting methods to achieve outlier resistance. Recent proposed robust ellipse fitting methods include [17], [18], [28], [29]. The sparsity based method (SBM) [17] selects sparse samples to represent the ellipse coefficient vector in order to alleviate the influence of the outliers on the ellipse fitting. The maximum correntropy criterion (MCC) based method [18] determines the ellipse parameters with the large-weight samples via the half-quadratic and semidefinite relaxation optimization. [28] utilizes the sparsity of outliers and Huber fitting measure to improve the robustness of ellipse fitting [47]. Furthermore, Shao et al. [29] removes the outliers via taking advantages of the phenomenon that outliers generated by ellipse edge point detector are likely to appear as groups.

Recently, Lin et al. proposed an alternating direction method of multipliers (ADMM) for multidimensional ellipsoid-specific fitting where the objective function is separated from the nonconvex positive semidefinite constraint via introduction of an auxiliary vector [30]. Unlike [30], this paper develops a ℓp based ellipse fitting approach [31], [32], [33] with p < 2 to tackle outlier observations. It is worth highlighting three aspects of the proposed algorithm here:

  • 1.

    We replace the ℓ2-norm by the ℓp-norm in the DLSF method to enhance robustness against outliers, which results in a nonconvex and nonlinear optimization problem. In addition, auxiliary variables are introduced to decouple the parameters of the ellipse in the nonlinear ℓp-norm objective function from the nonconvex quadratic constraint. Then we compute the ellipse parameter vector and auxiliary variables alternately via ADMM [31], [32], [33];

  • 2.

    For the ellipse parameter vector determination step, the Lagrange multiplier method is utilized to transform the vector estimation problem as finding a single Lagrange multiplier. In particular, we derive the feasible region of the Lagrange multiplier, analyze the monotonically decreasing property of the resultant problem, and efficiently find the multiplier by applying the bisection method [34];

  • 3.

    For the auxiliary variable determination step, we separate the corresponding nonlinear objective function into multiple subfunctions to determine the auxiliary variables in parallel, each of which only involves one single auxiliary variable. Then, we analyze the convexity or concavity of each subfunction to devise a simple but efficient method to calculate the corresponding auxiliary variable.

The rest of this paper is organized as follows. The ellipse fitting problem is described in Section 2. The proposed approach is developed in Section 3. Simulation and experimental results are presented in Section 4, and conclusions are drawn in Section 5.

Notation: Vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively. The | · |, ‖ · ‖, and indicates that ARn×n is positive semidefinite. The [A, B], (A, B], [A, B), and (A, B) are the closed, right half-closed, left half-closed, and open intervals, respectively. Also, {zi, λi} denotes a set with elements zi and λi, and adiag{[a1, a2, ⋅⋅⋅, an]} represents the anti-diagonal matrix with elements {a1, a2, ⋅⋅⋅, an} on the anti-diagonal positions and zeros otherwise.

Section snippets

Problem formulation

A general elliptic equation with center (h, k), counter-clockwise rotation angle θ, and semi axes (a, b) can be written as:((xh)cosθ+(yk)sinθ)2a2+((xh)sinθ+(yk)cosθ)2b2=1,where (x, y) denotes any point on the ellipse.

The task of ellipse fitting is to estimate the ellipse parameters {a, b, h, k, θ} from the obtained edge points (xi,yi),i=1,,I.

Proposed algorithm

In this section, we first review the DLSF method, then the robust ADMM based ellipse fitting method is devised.

Simulation and experimental results

In this section, simulations and experiments are conducted to assess the proposed method. For comparison purpose, we implement some representative methods, including the CLS method [16], the DLSF method [14], the Lin method [30], the Bayesian method [19], the HT method [10], the SBM method [17], and the MCC method [18]. Among them, the former four methods do not consider the outlier case; whereas the last three methods adopt some robustness schemes to improve the fitting robustness. A PC with

Conclusion

This paper has focused on the robust ellipse fitting problem. By introducing the ℓp-norm into the DLSF method, we construct a new nonconvex and nonlinear optimization formulation to improve the outlier-rejection ability. To solve it using ADMM, we introduce auxiliary variables to decouple the ellipse parameters in the nonlinear objective function from the nonconvex quadratic constraint. To determine the parameters, we derive a proper feasible region for the corresponding Lagrange multiplier and

Declaration of Competing Interest

None.

Acknowledgement

This work was supported in part by National Natural Science Foundation of China (NSFC 61533014, 61471295) and 973 Project (Grant 2014CB360508).

References (53)

  • H. Junrn et al.

    Shape reconstruction by genetic algorithms and artificial neural networks

    Eng. Comput.

    (2003)
  • H.Y. Tseng

    Welding parameters optimization for economic design using neural approximation and genetic algorithm

    Int. J. Adv. Manuf.Technol.

    (2006)
  • R. Halir et al.

    Numerically stable least squares fitting of ellipses

    Proc. Sixth Int’l Conf. in Central Europe on Computer Graphics, Visualization, and Interactive Digital Media

    (1998)
  • V.F. Leavers

    Shape Detection in Computer Vision Using the Hough Transform. New York

    (1992)
  • E.S. Maini

    Enhanced direct least squares fitting of ellipses

    Int. J. Pattern Recognit.Artif. Intell.

    (2012)
  • A. Fitzgibbon et al.

    Direct least square fitting of ellipses

    IEEE Trans. Pattern Anal. Mach.Intell.

    (1999)
  • D.S. Barwick

    Very fast best-fit circular and elliptical boundaries by chord data

    IEEE Trans. Pattern Anal. Mach.Intell.

    (2009)
  • W. Gander et al.

    Least-squares fitting of circles and ellipses

    BIT

    (1994)
  • J. Liang et al.

    Robust ellipse fitting based on sparse combination of data points

    IEEE Trans. Image Process.

    (2013)
  • J. Liang et al.

    Robust ellipse fittng via half-quadratic and semidefinite relaxation optimization

    IEEE Trans. Image Proces.

    (2015)
  • D. Liu et al.

    A Bayesian approach to diameter estimation in the diameter control system of silicon single crystal growth

    IEEE Trans. Instrum.Meas.

    (2011)
  • C. Liu et al.

    Relative pose estimation for cylinder-shaped spacecrafts using single image

    IEEE Trans. Aerosp. Electron.Syst.

    (2014)
  • C. Panagiotakis et al.

    Cell segmentation via region-based ellipse fitting

    2018 25th IEEE International Conference on Image Processing (ICIP)

    (2018)
  • V. Zlokolica et al.

    Semiautomatic epicardial fat segmentation based on fuzzy c-means clustering and geometric ellipse fitting

    J. Healthcare Eng.

    (2017)
  • 2018, Accessed 16 Feb....
  • J. Liang et al.

    Ellipse fitting via low-rank generalized multidimensional scaling matrix recovery

    Multidimension. Syst. Signal Process.

    (2018)
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