1. Introduction

To date, in lens production and optometry, the automatic focimeter based on the Hartmann detection method is still the most established instrument for detecting the lens power. In China's national metrology system, the verification results of the automatic focimeter on the lens are used as the standard for lens focality measurement [1,2]. The automatic focimeter based on the Hartmann detection method is mainly used to measure the rear top focus of the lens. In the measurement, the rear vertex of the lens is contacted with the diaphragm of automatic focimeter. The sagitta between rear vertex and diaphragm changes against with the curvature radius of the lens, resulting in a large measurement error. Moreover, owing to the four-point holding tool, there are eccentricities and inclinations in the lens measurement process, resulting in other small measurement errors [3–7]. There are also various problems and errors in the research of other methods [8−10]. For the low coherence speckle interference detection lens, although the full surface parameters of the lens can be obtained without focusing, the detection method is complex and the operability needs to be improved [8]. In the study of the comprehensive optometer, although its inspection accuracy is high and its repeatability is good, the imaging quality of its luminous aperture needs to be improved [9]. In the study of lens image analysis, the lens focal power is deduced by analyzing the image fuzziness under different lenses, but the lens position must be consistent during image analysis during detection; otherwise, there will be a large error in measurement [10].

To solve the above problems, we propose a method to obtain lens images with different focal degrees by using the Hartmann detection method, extracting features from the images, forming a lens feature dataset, classifying them using a support vector machine (SVM) algorithm in machine learning, and introducing the k-nearest neighbor clustering (KNN) algorithm for comparison. First, by restoring the measurement method of the automatic focimeter, a self-built image acquisition system is used to obtain sufficient rear vertex power images of single focus lenses with different powers. Second, the spot centroid is extracted from the collected spot image through image processing, and the target centroid is connected to obtain the feature distance of the corresponding power lens, to obtain the feature set of different power lenses. Finally, the feature set is trained using a support vector machine (SVM) algorithm and the k-nearest neighbor clustering (KNN) algorithm, and the test results are obtained by the models. The confusion matrix is used to display the number of model test samples and the final test results. All lenses are measured using a NIDEK-LM-1800P automatic focimeter, and the measurement results are compared with each other.

2. Optical imaging principle

Figure 1 (a) shows a schematic of the Hartmann detection method. The collimated beam is deflected through the tested lens and images on the acquisition camera screen through the Hartmann diaphragm. A is a small hole on the Hartmann diaphragm, and B is the spot imaging on the camera screen after the deflected beam passes through hole of the Hartmann diaphragm. The optical path diagram of the beam emitted from the light source to the camera screen through the measured lens is shown in Fig. 1 (b). The spot distribution of the collimated beam under the Hartmann diaphragm through the single focus lens is shown in Fig. 1 (c), where $O,A,B,C,D$ are the five large circular spots on the Hartmann plate, ${\varDelta _1} = \overline {AB} $, ${\Delta _2} = \overline {BC} $, ${\Delta _3} = \overline {CD} $ and ${\Delta _4} = \overline {AD} $, ${\Delta _1} = {\Delta _2} = {\Delta _3} = {\Delta _4}$. When the flat lens is placed or no lens is placed, the display spot on the camera screen does not deflect. When the negative lens is placed, the spots move from $O,A,B,C,D$ to ${A{''}}$, ${B{''}}$, ${C^{''}}$, ${D^{''}}$. The display spot on the camera screen shows an expanding trend. When the positive lens is placed, the spots move from $O,A,B,C,D$ to ${A'}$, ${B'}$, ${C'}$, ${D'}$, and the display spot on the camera screen shows a converging trend.

 

Fig. 1. Schematic diagram of optical imaging acquisition system. (a) the Hartmann method; (b) the optical path; (c) the spot distribution of the collimated beam under the Hartmann diaphragm

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3. Experimental device and feature extraction

3.1 Image acquisition system

The image acquisition system consists of a motor transmission module, motor control module, 220 V to 24 V voltage converter, collimating light source driving module, collimating light source, lens platform, and industrial camera. The 220 V AC voltage is converted into a 24 V DC voltage through the voltage converter to supply power to the entire system. By controlling the motor, the lens placed on the lens platform is moved above the Hartmann diaphragm through the motor, and the image acquisition is automatically completed after alignment with the optical center, to achieve rapid and efficient image acquisition. The lens image acquisition system is shown in Fig. 2.

 

Fig. 2. Lens image acquisition system

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As the optical path shown in Fig. 1 (b), the collimated light through the lens is split into small beams and their intensity are detected by a camera. The system is designed to test the power from -20 m^-1 to 20 m^-1. This indicates a wide range of the distances between the small holes. Therefore, a camera with a high resolution and large photosensitive area is needed. Table 1 lists the core parameters of the acquisition camera.

Table 1. Industrial camera parameters

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3.2 Diaphragm design

In order to restore the measurement method of automatic focimeter measurement, according to the possibility of processing and manufacturing, to ensure each spot do not overlap in imaging, the Hartmann diaphragm is used as shown in Fig. 3. The small circle diameter is 0.34 mm; the large circle diameter is 0.51 mm; the diameter of the whole Hartmann diaphragm is 8 mm; the spacing between adjacent small holes is 0.57 mm; and the distance between Hartmann diaphragm and camera panel is 18 mm. Figure 3 (a) shows the physical diagram of the diaphragm, and Fig. 3 (b) shows a schematic diagram of the diaphragm.

 

Fig. 3. Hartmann diaphragm

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3.3 Lens selection

In order to consider the change in the curvature radius of the rear surface caused by the change in focal power of the lens, the medium thickness and sagittal height of the selected lens were measured using an altimeter. The medium thickness is expressed as the distance from the vertex of the rear surface of the lens to the vertex of the front surface of the lens, and the sagittal height is the vertical distance from the vertex of the front surface of the lens to the edge of the rear surface of the lens minus the medium thickness of the lens. As shown in Fig. 4(a), the lens is put on the stage of altimeter and the medium thickness is obtained when the measurement probe is aligned with the rear vertex. The sagitta is obtained when turned the lens over as shown in Fig. 4(b). The vertical height equals to the deviation between medium thickness and sagitta.

 

Fig. 4. Measurement of lens medium thickness and sagittal height using an altimeter

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Since the most popular lenses in the lens industry range from -8 m^-1 to +5 m^-1, and the manufacturer’s batch production interval is 0.25 m^-1, so the lenses from -10 m^-1 to +8 m^-1 at an interval of 0.25 m^-1 are selected. With regard to negative lenses, there are myopia patients with - 15 m^-1, so lenses in the interval from -10 m^-1 to - 15 m^-1 with variations of 0.5 m^-1 are considered, and two lenses with different refractive indexes are selected for comparison. The lens diameters from 0 m^-1 to +8 m^-1, 0 m^-1 to -10 m^-1 and - 10 m^-1 to - 15 m^-1 are 68 mm, 72 mm and 70 mm respectively. As shown in Fig. 5, in the case of a positive lens, the sagittal height and medium thickness also increase with an increase in lens focal power. When negative lenses are used, the medium thickness tends to remain unchanged with the increase in focal power, while the sagittal height increases with an increase in lens focal power. The higher the refractive index, the smaller the sagittal height distance. When the lens focal power is -15 m^-1, the sagittal height of the lens with a refractive index of 1.60 is 14 mm, and that of the lens with a refractive index of 1.56 is 16 mm. Figure 5 (a) shows the sagittal height and median thickness of the lens with a refractive index of 1.56, while Fig. 5 (b) shows the sagittal height and median thickness of the lens with a refractive index of 1.60.

 

Fig. 5. Medium thickness and sagittal height of lenses with refractive index of 1.56 and 1.60

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3.4 Lens feature extraction

To ensure the accuracy of the extracted lens features, the collected images are processed. In this study, the distances between the four big circles on the Hartmann diaphragm that enclose the central big circle, are selected as the extraction feature. Because noise, dust, stains and other factors affect the image quality, in the image processing, the spot image is transformed to gray level image first. Then, a median filtering with a 5 × 5 matric sampling window is used to remove the halo around the image spot and the noise. In order to divide the spot and background into two categories, the filtered image is binarized. Because different threshold settings would make the spot connected or lose it, the Otsu method is used to obtain the optimal segmentation threshold [11]. As the distances between the four large circles need to be extracted, the redundant small circles on the image are hidden, and only the five large circle spots are retained; subsequently, the centroids of the five large circle spots are obtained. The gray scale in the spot image can be expressed as the light intensity, and the grayscale weight needs to be added when calculating the centroid. The calculation formula for the centroid coordinates is expressed as,

$d$ is the distance between two centroids:

The obtained spot images with the lens from -15 m^-1 and +8 m^-1 with reflection index 1.56 and 1.60 are shown in Fig. 6. In the process of automatic acquisition, the lens will lead to a series of situations such as eccentricity and inclination. Therefore, the collected image may present a situation in which the four features are not completely equal, that is, ${\Delta _1} \ne {\Delta _2} \ne {\Delta _3} \ne {\Delta _4}$. To reduce the error and avoid extreme changes in the feature distance, the average value of the four feature distances on the lens image is obtained. The calculation formula is as follows:

 

Fig. 6. Spot image processing results of the lens from -15 m^-1 to +8 m^-1 with the reflection index of 1.56 and 1.60

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After all the spot images are processed, the characteristic data sets of lenses with different refractive indexes to acquire. The characteristic datasets were obtained by averaging the four characteristic distances obtained from one image. 2490 lens features with refractive index of 1.56 and 1.60 were obtained. Figure 6 shows the results after two types of refractive index lens processing.

4. Machine learning algorithm

4.1 Principle of k-nearest neighbor clustering algorithm

The K-nearest neighbor clustering algorithm is one of the most common algorithms used in machine learning [12−14]. The algorithm means if a certain sample in the feature space, most of the k nearest samples belong to a certain class, then this sample also belongs to this class and has the same features as the class. The classification decision of this method is to determine the class to which the target classification sample should belong based on the class of the most adjacent one or several samples. The algorithm is described as follows:

The characteristic distance in the spot image of lenses with different focal degrees is different. The larger the focal degree difference, the larger becomes the characteristic distance difference. Therefore, this algorithm is suitable for use when the lens focal degree interval is large. Figure 7 is a schematic diagram of the lens feature K-nearest neighbor clustering algorithm. The small circles of different colors represent the characteristic distance, and lenses with different focal degrees have corresponding category ranges, which are represented by the cross lines.

 

Fig. 7. Feature differentiation of KNN algorithm

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4.2 Principle of support vector machine algorithm

The support vector machine (SVM) was first proposed by Vapnik. It is a supervised learning algorithm used in machine learning. The main idea is to establish a classification hyperplane as a decision surface to maximize the isolation edge between positive and negative examples [15–19]. The algorithm flow is as follows:

Get the optimal solution:${a^*} = {(a_1^*,a_2^*, \cdots ,a_N^*)^T}$

As shown in Fig. 8, w is a normal vector that determines the direction of the hyperplane, x is the training sample, and b is the displacement term, which determines the distance between the hyperplane and the origin. By determining the normal vector w and displacement b, a separation hyperplane can be determined.

 

Fig. 8. Feature differentiation of SVM algorithm

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4.3 Parameter settings

In the use of KNN and SVM, the parameters of the different models need to be adjusted. Approximately 70% of the samples of the data set are used as the training set, and 30% of the samples are used as the test set. The kneigborsclassifier class from the sklearn library is used for the KNN model. The purpose of the KNN algorithm is to measure the Euclidean distance, classify the data within the value of K into the same class, achieve the purpose of clustering, and achieve a correspondence between the features and lens, in order to separate different lens focal degrees. When applying the SVM model, it is necessary to find the hyperplane according to the dataset, draw the boundary, and separate different groups in order to achieve the purpose of differentiation. Because different feature distances do not cross after averaging, the SVM model selects a linear kernel function.

4.4 Evaluation index

For the accuracy of the obtained model, the method divides the dataset into ten parts, taking nine parts as training data and one part as test data in turn, and average the F1 score of the model ten times, in order to achieve the accuracy estimation of the model. The confusion matrix is used to represent the number of test samples of lenses with different focal powers and the interval in which the samples fall. Figure 9 shows a schematic diagram of the confusion matrix.

 

Fig. 9. Confusion matrix diagram

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Precision refers to the proportion of samples correctly predicted to be positive in all samples predicted to be positive, and its calculation formula is

The recall rate is the proportion of the samples correctly predicted to be positive to the samples actual predicted to be positive, and its calculation formula is

F1 (F1 score) depends on the accuracy and recall rate. This is the harmonic average of the two scores, and the calculation formula is as follows:

The F1 score weights both accuracy and recall. When TP increases, false positive decreases and accuracy and recall increase.

5. Result analysis

5.1 KNN testing results

The KNN model test results of lenses with refractive indices of 1.56 and 1.60 are shown in Fig. 10. The ordinate of the model represents the real value of the lens power, and the abscissa represents the recognition result of the lens power. Figure 10 (a) shows the KNN recognition model of a 1.56 refractive index lens, and the recognition results range from - 15 m^-1 to + 7.5 m^-1. The number of selected test samples is displayed in the form of a thermal diagram, and the ordinate corresponds to the real value of the lens. In the figure, test samples of - 15 m^-1 and - 14.5 m^-1 are recognized as - 15 m^-1, the two are not distinguished, and the recognition results of other groups are not ideal. The average F1 score of this model is 13.58%. Figure 10 (b) shows the KNN recognition model of the 1.60 refractive index lens, and the recognition results range from - 13.5 m^-1 to + 7.25 m^-1. This recognition model recognizes the lens focal power of - 15 m^-1, - 14.5 m^-1, - 14 m^-1 and - 13.5 m^-1 as the lens focal power of - 13.5 m^-1, and does not distinguish the four values. The ten-fold average F1 score of this model is 12.59%. In a real situation, it is not easy to distinguish the characteristics of lenses with different focal degrees collected every 0.25 m^-1, and the K value selection of the KNN model is not perfect in clustering; thus, lenses with different focal degrees cannot be clustered together, so the purpose of differentiation cannot be achieved.

 

Fig. 10. KNN model identification results: (a) 1.56 for KNN, (b) 1.60 for KNN

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5.2 SVM testing results

The test results of the SVM model for lenses with refractive indices of 1.56 and 1.60 are shown in Fig. 11. The ordinate of the model represents the real value of the lens power, and the abscissa represents the predicted value of the lens power. The data of the selected test samples are displayed in the form of a thermal diagram. Figure 11 (a) shows the SVM recognition model of the 1.56 refractive index lens, and Fig. 11 (b) shows the SVM recognition model of the 1.60 refractive index lens. The abscissa of the test sample is the recognition result, and the ordinate represents the true lens power. Corresponding to the real lens power, different colors correspond to different sample numbers. The two SVM classification models accurately identify lenses with different focal powers in their respective refractive index lens models, and their ten-fold average F1 score was 100%. The ten-fold cross F1 scores of the four models are listed in Table 2.

 

Fig. 11. SVM model recognition results: (a) 1.56 for SVM, (b) 1.60 for SVM

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Table 2. Model scores

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Because different focal lenses correspond to different distance characteristics, although the lenses are produced at an interval of 0.25 m^-1 in actual production, the characteristic distances between lenses with different focal degrees are very close, and the core of the SVM algorithm involves separating lenses with different focal degrees by using a hyperplane. Therefore, this model has high accuracy, can completely distinguish lenses with different angles, and there is no error.

5.3 Measurement results of focimeter

In order to verify the difference between the lens power results measured by the existing automatic focimeter and the actual lens power results, lenses with different refractive indices used in the training model are measured using a NIDEK-LM-1800P automatic focimeter, and the measurement results are displayed in the form of a broken line diagram.

The verification results of the NIDEK-LM-1800P lenses with different refractive indices are shown in Fig. 12. The lenses for both refractive indices are in the range of - 15 m^-1 to 8 m^-1. In the market, lenses are usually separated by 0.25 m^-1 at the production interval focal power within - 10 m^-1 to + 8 m^-1 and 0.5 m^-1 at the production interval focal power within - 10.5 m^-1 to - 15 m^-1. In the range of - 10 m^-1 to + 8 m^-1, the measurement errors for the two refractive index lenses were within 0.06 m^-1. Owing to the continuous increase in the lens sagittal height, the radius of curvature of the lens also changes with the change in the lens focal power when the lens diameter does not change. In the range of - 10.5 m^-1 to - 15 m^-1, the lens diameter decreases when the lens sagittal height increases and decreases, the sagittal height of the lens will further increase. The measurement results show that when the lens is in the range of - 10 m^-1 to 8 m^-1, the measurement error of the lens changes little. When the lens is in the range of - 10.5 m^-1 to - 15 m^-1, the measurement error of the two refractive index lenses is within 0.15 m^-1, and the error also increases with an increase in the curvature radius of the rear surface of the lens. Under the same range and interval of the two types of focal lenses, the measurement error results show an increasing trend with the increase in focal power and the radius of curvature of the rear surface of the lens. An SVM test model was used to detect the lens focus. All test samples were correctly identified, and there was no error due to the increase in the curvature radius of the rear surface of the lens.

 

Fig. 12. Measurement results of NIDEK-LM-1800P automatic focimeter

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6. Conclusion

A method for measuring a lens using machine learning is proposed. This method solves the measurement error caused by the change in the curvature radius of the rear surface of the measured lens in the traditional focimeter measurement, which prevents the rear vertex of the lens from coinciding with the Hartmann aperture. All lenses were measured using a NIDEK-LM-1800P automatic focimeter. The measurement results show the errors of lenses with different powers. The KNN and SVM algorithms in machine learning are used to train the lens recognition model. The two groups of models compare the test results in the form of a control experiment. The ten-fold average F1 scores of the 1.56 refractive index and 1.60 refractive index lenses by the KNN model are 13.58% and 12.59%, respectively, which cannot correctly identify the lens power. The 10-fold average F1 score of the 1.56 refractive index and 1.60 refractive index lenses by the SVM model was 100%. Both models can correctly identify the focal power of the corresponding refractive index lens and can completely overcome the systematic measurement error caused by the change in the curvature radius of the back surface of the lens.