A novel circle fitting algorithm is proposed in this paper. The key points of this paper are given as follows: (i) it formulates the circle fitting problem into the special source localization one in wireless sensor networks (WSN); (ii) the multidimensional scaling (MDS) analysis is applied to the data points, and thus the propagator-like method is proposed to represent the circle center parameters as the functions of the circle radius; (iii) the virtual source localization model can be rerepresented as special nonlinear equations of a unique variable (the circle radius) rather than the original three ones (the circle center and radius), and thus the classical fixed-point iteration algorithm is applied to determine the radius and the circle center parameters. The effectiveness of the proposed circle fitting approach is demonstrated using the simulation and experimental results.
Circle fitting receives considerable attention because it plays an important role in computer vision, observational astronomy, structural geology, industry inspection, medical diagnosis, Iris recognition, military, security, and so forth [
The fitting problem can be viewed as follows: estimate the parameters of a circle from a set of coplanar points. Several classical approaches [
In this paper, we develop a novel circle fitting approach by borrowing the idea from source localization in wireless sensor networks (WSN) [ It formulates the circle fitting problem into special source localization one in WSN, where each data point should be understood as an abstract sensor node in sensor networks, and the circle center represents the localized target. However, the propagation delays are unknown, and thus the existing source localization algorithms in WSN cannot be applied to solve the special source localization problem. The multidimensional scaling (MDS) analysis [ The virtual source localization model can be rerepresented as special nonlinear equations, where the radius is the unique variable rather than the original three ones (the circle center and radius), and thus the classical fixed-point iteration algorithm [
The rest of this paper is organized as follows. The circle fitting problem is described in Section
The equation for a circle centered at
The circle fitting problem (CFP) [
Circle fitting problem description.
In this section, we first reformulate CFP into a virtual source localization problem in wireless sensor networks (WSN) [
Let us review the source localization model in WSN [
To reformulate CFP into the source localization problem in WSN, we rewrite (
The source localization model in (
Virtual source localization in WSN.
By comparing (
In the rest of this section, we will develop a novel algorithm for estimating the “target position”
Let
Under the ideal (without noise) case,
Since the rank of
Similar to the conventional propagator method [
Let
From (
Note that
Plugging (
According to the fixed-point iteration theory [
Once circle radius
In this section, some experiments are conducted to evaluate the performance of the proposed method. For comparison, we simultaneously implement the HT method [
The first experiment is implemented on
Estimated results using different algorithms (Experiment 1).
Parameters | Hough | LS | Proposed |
---|---|---|---|
|
10 | 9.9177 | 9.9204 |
|
8 | 8.0354 | 8.0332 |
|
15 | 14.9800 | 14.9703 |
Data points used in the first experiment.
Estimated circle parameters versus iteration number (Experiment 1).
Fitting results using different algorithms.
In this experiment, the proposed algorithm is applied to the real data. Figure
Estimated results using different algorithms (Experiment 2).
Parameters | Reference | Hough | LS | Proposed |
---|---|---|---|---|
|
45.07 | 45.0700 | 44.9838 | 44.9874 |
|
— | 161 | 160.5635 | 160.5635 |
|
— | 123 | 122.8443 | 122.8443 |
Experimental data used in Experiment 2.
Estimated circle parameters versus iteration number (Experiment 2).
Iris recognition is a biometric identification technique based on images of the irides of an individual’s eyes. Since the Iris area lies between the pupil region (a dark ellipse with the lowest intensity) and limbus region, determining the pupil region is an important preprocessing step of Iris localization. In the third experiment, we implement the proposed algorithm on the Iris image, as shown in Figure
Estimated results using different algorithms (Experiment 3).
Parameters | Hough | LS | Proposed |
---|---|---|---|
|
23 | 23.2277 | 23.2237 |
|
120 | 119.7545 | 119.7542 |
|
156 | 156.4885 | 156.4885 |
Iris Image used in Experiment 3.
Edge points used in Experiment 3.
Fitting result using the proposed algorithm.
Estimated circle parameters versus iteration number (Experiment 3).
Although the HT method is of the highest estimation accuracy, it needs to be pointed out that the HT method requires (i) quantizing the three-dimensional space finely enough; otherwise the peaks in the transform plane will be broadened and (ii) the overwhelming burden of the three-dimensional search in the
In this paper, we propose a novel circle fitting algorithm by borrowing the idea from source localization in wireless sensor networks. Since the virtual propagation delays of all sensor nodes are unknown, the existing source localization algorithms cannot be applied. This paper formulates the virtual source localization model of three unknown parameters
This work was supported in part by the National Natural Science Foundation of China under Grants 61172123 and 60901059, by Excellent Youth Research Star (2012KJXX-35), and educational Department Foundation of Shaanxi Province (12JK0526).