Conic sections fitting in disperse data using Differential Evolution

Abstract

Robust conics detection and extraction have received an increased interest due to the potential applications in many critical tasks. In medical imaging, the conics can help to detect optic disk abnormalities in retinographic images, or cranial sections and bone structures in radiography, magnetic resonance, and computed tomography images. Moreover, the physicians can be guided in the prostate cancer diagnosis by assessing the size and shape of the prostate in ultrasound or magnetic resonance images. Some of these structures are composed of discontinuous points, imprecise directions, multiple bifurcations, and variable line–trace width, or may present undesired noise, which makes the curve fitting hard to compute. Image acquisition also plays a fundamental role in adding artifacts, as unexpected outliers, which jeopardizes the applicability and efficiency of the current methods. This article proposes a strategy for determining the general equation of oblique conics in noisy and sparse data sets. As a novelty, this methodology employs four uncorrelated points from a randomly sampled population to estimate the initial space search. Then, Differential Evolution (DE) method iterates until the solution that best fits the oblique conic section is found. Advantages of the method include a DE implementation using a weighted fitness function based on the Mean Squared Error (MSE) and reinforced with the Random Sample Consensus (RANSAC) inliers measurement idea. This objective function allows associating the fitting error to the number of points into the inliers-region. Additionally, the number of parameters to tune is reduced and almost generalized. A distinctive feature is related to the eccentricity, which could conduct to a variety of rich solutions. However, some disadvantages have arisen trying to identify a specific conic. For instance, when some parabolas are considered, the method could evolve optimally to an ellipse. The proposed methodology has been evaluated using synthetic and real images. These images were contaminated with additive white Gaussian noise to prove the method stability. Numerical results showed that the proposed method is a reliable and fast technique for computing any conic from scattered data. Results obtained from two dissimilar applications prove that the proposed method can be used in a variety of sensitive applications. Those applications include the optic disk segmentation and traffic sign classification. The assessment showed an improvement of 2% across the metrics for detecting the optic disk and reached a classification accuracy of 0.9730 for the speed limit traffic signs.