Robust ellipse fitting via alternating direction method of multipliers
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 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: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 .
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).
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