A robust circle detection algorithm based on top-down least-square fitting analysis☆
Introduction
Circle detection is an important and fundamental problem in the fields of pattern recognition and computer vision [1], [2], which has been studied extensively in the past decades. Circles within an imaged scene can be of great help to infer the geometric properties. Therefore, circles can be used in many applications, such as automatic inspection and assembly [2], pupil and iris detection [3], and circular traffic sign recognition [4]. Additionally, locating circles in an image is also challenging in many industrial fields, for instance, particle tracking [5], biology [6] and so on.
Over decades, many methods have been proposed for circle detection in the literature. The three major concerned issues of circle detection are accuracy, robustness and computational performance. The famous Circle Hough Transform (CHT) [7], [8], [9], which extracts circles in three dimensional parameter space with Canny edge map [10], is robust but inefficient and space consuming. In order to reduce the execution time and save the memory space, several improved CHT-based variants have been developed by using geometrical properties or decomposing the parameter space, including probabilistic HT [11], fuzzy HT [12], iterative randomized HT (IRHT) [13], etc. However, the execution time does not get reduced significantly.
In addition to the CHT-based methods, some randomized algorithms for circle detection have been proposed. A parameter-free randomized circle detection algorithm (RCD) is proposed by Chen and Chung [14]. This method takes four edge points as a random sample: three points construct a possible circle, and the remaining one confirms whether the possible circle is a candidate circle or not. The candidate circle is further determined to be a real circle or not by voting. Some other methods with efficient sampling and refining strategies are exploited to improve the computational performance, such as GRCD-R [15]. The variants of RCD have enhanced the computational performance and accuracy, but still not significantly.
Furthermore, some heuristic optimization methods have been used to detect circles, such as genetic algorithm (GA) [16], discrete differential evolution optimization (DDE) [17]. These algorithms produce good results but spend much computing time.
Recently, some real-time circle detection algorithms have been proposed. Frosio and Borghese [18] propose a real-time circle detector based on maximum likelihood estimation, which can detect circular objects with a predefined radius. Akinlar and Topal [19] also propose a real-time circle detector named EDCircles. It extracts line segments [20] from edge segments produced by an edge detector named EDPF [21], and combines line segments turning in the same direction into circular arcs with bottom-up strategy. The circular arcs are further combined together as the candidate circles or near-circular ellipses. To validate the candidates, the algorithm uses the contrario framework based on Helmholtz Principle [22], [23]. Although EDCircles performs well on many images, in our experiments, we find that it is not robust enough to handle shadows, blurred boundaries or Gaussian noise.
In this paper, we develop a robust and efficient circle detection method based on top-down analysis with a novel edge segment extractor. The three main contributions are: first, we propose a novel Canny edge point chaining strategy to extract edge segments; second, we propose a top-down scheme based on least-square fitting analysis to detect circles; third, we improve the validation process by guaranteeing the independence of the points and make the validation more reasonable. The rest of this paper is organized as follows: Section 2 introduces the background; Section 3 describes the proposed algorithm and introduces the important steps; Section 4 presents the experimental results; Section 5 gives a conclusion of the paper.
Section snippets
Definition of the edge segment
In order to clearly present our solution to circle detection problem, we first fix the related notations and definitions. Let’s call I an image defined over a two dimensional rectangular lattice L, and the value at pixel , with . In our circle detection algorithm, we first extract the edges of the image. Hereinafter, we denote all the extracted edge points as . Based on such notations, we give the definition of edge segment as follows:
Edge segment. We define a set of edge
Proposed algorithm
Based on the criterion presented in the preceding section, we develop a novel circle detection algorithm, which consists of three major stages shown in Fig. 1. In the first stage, we extract a few edge segments from the input image. Then we detect candidate circles within each obtained edge segment based on top-down analysis. We further validate the detected circles in the last stage. Details of each stage are introduced in the following subsections.
Experimental results
To test the performance of the proposed algorithm, we take a series of experiments on various images with different sizes. Several images containing circles with blurred boundaries and some noisy images added different levels of Gaussian noise are also tested to demonstrate the robustness of the proposed method. It is worth noting that, all the internal parameters of the proposed circle detector are fixed for all the experiments.
Conclusions
The proposed circle detection algorithm is robust, efficient and parameter-free, which consists of three major contributions: (1) the novel edge segment detector, which can produce complete and smooth edge segments and is more robust than Edge Drawing algorithm for handling the noise, jumps and branches; (2) the top-down least-square fitting analysis scheme, which can detect more valid candidate circles from each obtained edge segment and is more robust than the conventional bottom-up strategy
Acknowledgments
This work is supported by National Natural Science Foundation of China under Grant 61300061 and Beijing Natural Science Foundation under Grant 4132033. We are deeply indebted to the anonymous reviewers for many insightful remarks and valuable suggestions.
Dong Liu received her B.Eng. degree in 2011 from Wuhan University, China. She is currently a graduate student at the Institute of Computer Science & Technology, Peking University.
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Dong Liu received her B.Eng. degree in 2011 from Wuhan University, China. She is currently a graduate student at the Institute of Computer Science & Technology, Peking University.
Yongtao Wang received his Ph.D. degree of Pattern Recognition and Intelligent System in 2009 from Huazhong University of Science and Technology, China. During 2010, he was a research scientist of the Temasek Laboratories@NTU, Singapore. He is currently an assistant professor of Institute of Computer Science & Technology, Peking University, China. His research interests involve document image analysis and understanding.
Zhi Tang received his Ph.D. degree of Applied Computer Technology in 1995 from Peking University, China. He is now a professor at the Institute of Computer Science & Technology, Peking University. His research interests majorly include document analysis and understanding and digital rights management.
Xiaoqing Lu received the B.S. degree and the M.S. degree in Computer Science from BeiHang University, China, in 1990 and 1993, respectively. After graduating, he joined the Institute of Computer Science and Technology at Peking University. He is presently an associate professor. His research interests include shape analysis, computer vision, pattern recognition, and machine leaning.
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Reviews processed and approved for publication by Editor-in-Chief Dr. Manu Malek.