1. Introduction

2. Materials and Methods

3. Results

3.2. Determination of the Weight and Volume of Apples via Digital Methods

Twenty-six models are trained and tested to determine the weight and volume of the apples. The procedure includes the steps shown in Figure 7. Regression models have been developed to predict the data via supervised machine learning.

The training set includes 70% of data for each apple variety, and the remaining 30% of data is used for testing the models. The training and testing sets are compared, and the loss is estimated via cross-validation.

3.2.1. Apple Weight Prediction Results

The features of the images obtained through the analysis that are informative when building the models according to the three algorithms developed in MATLAB have been previously determined. The procedure for selecting informative characteristics to determine the mass for the Aport variety of apple is graphically presented. The results for the two applied algorithms are presented in Figure 8 and Figure 9.

For the remaining varieties, the results are summarized in Table 3.

The results show that feature combinations are more clearly separated when the F test is used. To reduce the time of calculation and training of the model, the first three most informative features are selected. For four of the varieties, these features are h, d, and S. D is informative only for the Ainur apple variety. The models are trained with the three most informative features ranked by the F test algorithm.

The predictions with the best trained model for the Aport apple variety are shown in Figure 10. The models are assessed via the root mean square error (RMSE) and training set. The application also provides information related to the accuracy of the models.

When a model is built, it is necessary to check how well it works. To evaluate the model quality, we can consider the values of various metrics, such as the coefficient of determination, mean absolute error (MAE), and mean squared error.

The MAE indicates how much, on average, the values determined by the model differ from the actual values. The advantage of the MAE is that the results obtained are in the same units as the target variable and are not as affected by the presence of extreme values in the sample.

$$MAE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n\left|y_i-\widehat{y_i}\right|,$$

(3)

where $y_i$ is the actual parameter value and

$\widehat{y_i}$ is the predicted parameter value.

The coefficient of determination (R²) indicates how well the values of the target variable are determined by the model when new data are introduced. R² is a statistical metric that tells us what percentage of the variance of the target variable is predicted by the values of the independent variables.

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}{{\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}},$$

(4)

where y is the mean value of the target variable.

Thus, it is an important measure of how well the regression model fits the data. The R² value is always between zero and one, i.e., 0 ≤ R² ≤ 1. An R² value of 0.9 or above is very good, a value above 0.8 is good, and a value of 0.6 or above may be satisfactory in some applications.

The RMSE is the root mean square of the estimation error and is always positive. This metric is strongly affected by the presence of extreme values in the sample, as squaring further increases their weight.

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2}$$

(5)

Figure 11. Predicted vs. actual plot for the weight training set.

Figure 11. Predicted vs. actual plot for the weight training set.

In an ideal regression model, the predicted responses would equal the true responses, resulting in all points lying on the diagonal line. The vertical distance from the line to each point is the prediction error. A good model has small errors, with predictions scattered close to the line.

The models are also evaluated via a residual plot, which is also used to evaluate the model performance (Figure 12). The residual plot shows the difference between the predicted and true responses. When the models are built, they are tested against the data specified for the test sample. The predicted and actual weight values are visualized in Figure 13.

The summarized data of the developed regression models and the evaluation of the best models in terms of the accuracy of predicting the weights of the apples with training and test samples are presented in Table 4.

The results show that different apple varieties require different models. The linear SVM is applicable only for two apple varieties—Aport and Ainur. The RMSE varies from 7.17 to 12.83 for the five apple varieties. According to the R2 values with the training set, the stepwise linear regression model is the best, followed by rational quadratic Gaussian process regression and linear regression. The linear SVM model is a satisfactory model. According to R2 values with the test set, stepwise linear regression is the best model for the Sinap Almatynski variety, and for the Ainur variety, the linear SVM yields better results than it does with the training sample. The MAEs for both the training and test samples do not exceed 12.29. The training time of the models ranges from 0.60 seconds to 4.27 seconds, with the stepwise linear regression model performing best in both training and testing at 1.88 seconds each.

The best results for the Sinap Almatynski apples via stepwise linear regression were as follows: RMSEs of 7.17 for training and 4.83 for testing, R2 values of 0.9–0.92 for training and 0.96 for testing, and MAEs of 6.12 for training and 4.14 for testing.

3.2.2. Apple Volume Prediction Results

In the developed models for apple volume prediction, informative parameters are again determined, and the three algorithms integrated into MATLAB are used. For the Aport variety, the results for the two applicable algorithms, i.e., the F test and ReliefF algorithms, are graphically presented in Figure 14 and Figure 15.

The summary information for the five apple varieties and the selection of informative characteristics are presented in Table 5. The results show that the mRMR algorithm is also applicable here. For three of the varieties, only the D parameter is informative. However, the F test algorithm is chosen for ranking the informative parameters.

The predictions with the best trained model for the Aport apple variety are shown in Figure 16. The predicted vs. actual plot for evaluating the performance of the model is shown in Figure 17, the residuals plot is shown in Figure 18, and the predicted and actual weight values for the testing set are shown in Figure 19.

The summarized data of the developed regression models and the evaluation of the best models in terms of the accuracy of predicting the volume of apples with training and test samples are presented in Table 6.

The results show that different apple varieties require different pattern types. The RMSEs range from 9.84 to 22.75 for training and 9.38 to 15.49 for testing for the five apple varieties. According to the R2 with the training data, the linear SVM and linear regression models are the best, followed by the quadratic SVM and robust linear regression. A satisfactory model is the quadratic exponential GPR. According to R² with the test sample, linear regression is the best model for the Nursat variety, and for the Aport variety, linear regression yields better results than it does with the training data. The MAEs for both the training and test samples do not exceed 19.66. The training time of the models ranges from 0.41 seconds to 5.28 seconds, with the robust linear regression model performing best during both training and testing, with a time of 0.41 seconds.

The best results for the Nursat variety via linear regression were as follows: relatively small RMSEs of 9.84 for training and 9.65 for testing, R² values of 0.9–0.91 for training and 0.92 for testing, and relatively small MAEs of 8.12 for training and 7.96 for testing.

5. Conclusions

• With the growth of apple production and the shortage of skilled workers in Kazakhstan, the development and implementation of digital technologies to improve the productivity of apple fruit quality assessment is an urgent issue.
• The literature analysis confirms that technical vision systems, which use color video cameras and flexible algorithms for digital image processing, contribute to the use of information about the object of study and the surrounding background and are widely used for determining the quantitative indicators of various objects to solve the problem of determining fruit quality.
• The developed digital method achieved high accuracy in determining the size d of the fruit, with maximum relative errors of 2.99%, 3.03%, and 4.12% for the e, h, and D parameters, respectively. This methods enables the quantification of the area and the percentage of color, which cannot always be detected with the naked eye, allowing for the automation and optimization of apple quality assessment.
• Experimental studies were conducted, and regression models were developed to determine and predict the mass and volume of apples via a digital method. The best results for the weight prediction of the Sinap Almatynski apples via stepwise linear regression were as follows: RMSEs of 7.17 for training and 4.83 for testing, R2 values of 0.9–0.92 for training and 0.96 for testing, and MAEs of 6.12 for training and 4.14 for testing. The best results for volume prediction of the Nursat variety via linear regression were as follows: relatively small RMSEs of 9.84 for training and 9.65 for testing, R2 values of 0.9–0.91 for training and 0.92 for testing, and relatively small MAEs of 8.12 for training and 7.96 for testing. The accuracy of the procedures could be improved by optimizing the models and increasing the number of datasets.
• The regression equations obtained from the relationship between mass, volume and geometric parameters, together with the numerical method for determining the color area, serve as the basis for the development of a compact instrument for the evaluation of quality indicators and the automatic sorting of apples by commercial variety.

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