Improving light marker accuracy on camera images

The accuracy of measurement systems that use image processing is very dependent on the method used to determine the target center position on the image. The digital image correlation (DIC) method,¹ which combines image matching and sub-pixel estimation using correlation interpolation, enables registration of pairs of images in many processing applications, such as camera calibration and measurement of structures or fluids. However, DIC is unable to determine the exact center position. Therefore, a light marker is installed on the surface of the object and the central point of the marker is estimated by the centroid calculation method (where the geometric center is determined).² Although this method can be realized with the sub-pixel accuracy of DIC, there is a bias error in the measurement, depending on the threshold that separates the bright spot from the region of interest.³ Our study proposes a dynamic threshold method to reduce the bias and error of the calculated value. In addition, we evaluated the measurement accuracy of the center position of a bright spot and analyzed the error in certain imaging conditions using Monte Carlo simulation.

We sought to improve the well-known technique of threshold selection for image binarization (converting to a black and white image) known as discriminant analysis.^{4, 5} This technique selects an optimal threshold by the discriminant criterion, which maximizes the separability of two classes in a gray-level histogram. The sub-pixel position measurement has a systematic error when used with this threshold selection method, which is caused by a higher threshold compared with the average intensity of the background (see Figure 1). Our study proposes an improved dynamic threshold method from discriminant analysis by using a threshold f_th=m₁+kσ₁, where m₁ and σ₁ are the average and variance, respectively, of the gray level of the background (denoted as class 1 in Figure 1) and k is a coefficient. This equation indicates that the threshold varies depending on the average and the variance of background gray values in the partial image.

Figure 1. Gray-level histogram of a light-marker image. The dotted line indicates the threshold value selected by the discriminant analysis method. Class 1 and class 2 denote background and object, respectively.

We carried out Monte Carlo simulation with a synthetic image, including a bright spot for evaluating measurement accuracy. We modeled the spot using a Gaussian function³ and generated it on the synthetic image with sub-pixel resolution. We calculate the center position of the spot by centroid and 1D Gaussian fitting methods. To find the difference between the ideal and the calculated values, we placed the light marker (the ideal spot) 100,000 times at random positions in the range of −0.5 to 0.5 pixels. We defined the total error as the root-mean-square difference between the true and the calculated positions. The total error depends on the spot radius, peak intensity, background level, and the method used to calculate the centroid (see Figure 2). The distribution of the calculated position shows the systematic error (see Figure 3). When the discriminant analysis includes a high peak level, the distribution of the calculated position includes deflection.

Figure 2. Results of Monte Carlo simulations to evaluate the measurement error caused by alteration of the peak intensity. The red and blue lines indicate the centroid method and Gaussian fitting, respectively. The dotted and solid lines indicate the discriminant analysis method and our proposed approach, respectively.

Figure 3. The deflection of calculated positions by discriminant analysis (a) and the proposed dynamic threshold method (b). The pseudocolor map describes the frequency of the calculated sub-pixel position.

This simulation, using synthetic images, shows that the total and systematic error can be better reduced by our dynamic threshold method than by discriminant analysis. Furthermore, our approach keeps the total error at 0.03 pixels or less, using both the centroid and Gaussian methods. Thus, our approach can be used as a threshold selection method for high-accuracy measurement of light marker positions.

Recently, we have been developing a 3D displacement measurement system using a light marker, a method that requires robust tracking of the target. The results indicate that our proposed method of dynamic threshold selection can be applied to measurement using a light marker even in poor light conditions. In the future, we will evaluate the measurement accuracy of our method in real captured images.