Abstract
Scattered data from edge detection usually involve undesired noise which seriously affects the accuracy of ellipse fitting. In order to alleviate this kind of degradation, a method of direct least absolute deviation ellipse fitting by minimizing the ℓ1 algebraic distance is presented. Unlike the conventional ℓ2 estimators which tend to produce a satisfied performance on ideal and Gaussian noise data, while do a poor job for non-Gaussian outliers, the proposed method shows very competitive results for non-Gaussian noise. In addition, an efficient numerical algorithm based on the split Bregman iteration is developed to solve the resulting ℓ1 optimization problem, according to which the computational burden is significantly reduced. Furthermore, two classes of ℓ2 solutions are introduced as the initial guess, and the selection of algorithm parameters is studied in detail; thus, it does not suffer from the convergence issues due to poor initialization which is a common drawback existing in iterative-based approaches. Numerical experiments reveal that the proposed method is superior to its ℓ2 counterpart and outperforms some of the state-of-the-art algorithms for both Gaussian and non-Gaussian artifacts.
Highlights
Ellipse fitting is a fundamental tool for many computer vision tasks such as object detection, recognition, camera calibration [1], and 3D reconstruction [2]
The scale normalization [4], robust data preprocessing for noise removal [5], optimization of the Sampson distance [6, 7], minimizing the algebraic distance in subject to suitable quadratic constraints [8, 9], and modeling the noise as a sum of random amplitude-modulated complex exponentials [10] and iterative orthogonal transformations [11] are launched
E complete procedure of minimization problem equation (5) with the split Bregman iteration is summarized in Algorithm 1, where nmax is the maximum number of iterations; to avoid the procedure falling into an invalid solution, the iteration stop criterion is set as the algebraic distance calculated on the (k + 1)-th iteration, |Dαk+1|1 is greater than k-th |Dαk|1 or the iterations reach to the maximum
Summary
Ellipse fitting is a fundamental tool for many computer vision tasks such as object detection, recognition, camera calibration [1], and 3D reconstruction [2]. The performance has been improved to some extent, the inherit disadvantage of l2 norm-based methods is that the square term will enlarge the influence of outliers and lead to a rough result. To overcome this shortcoming, the l1 [12] and l0 norms [13] are cited to address this issue; theoretically, it turns to be less sensitive to outliers for algorithms based on lp-norm (0 < p < 2), and the selection of iterative initial value is still pending yet. Given the advantage of l1-norm and direct algebraic distance minimizing strategy, a natural way is to replace the l2 item with l1 in Fitzgibbon’s model comes out our l1 model, and we will explore its efficacy
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