Digital image technology based on PCA and SVM for detection and recognition of foreign bodies in lyophilized powder

3.1PCA fast algorithm

The operation time of PCA algorithm is usually wasted on the calculation of the eigenvalues and eigenvectors of the covariance matrix of the sample data set. If the data set matrix 𝑿 has a size n×d, then, when calculating the covariance matrix st of the data set, it is equivalent to calculating a square matrix of d×d. If the dimension d is of a large order of magnitude, the operation time complexity will become very high, and a large amount of time is needed to calculate all the eigenvalues of the covariance matrix. In order to improve the time efficiency of the algorithm, a fast PCA algorithm was introduced in the experiment [12].

Suppose 𝒁n×d matrix is the matrix obtained by subtracting m from each sample of the sample matrix 𝑿, then the distribution matrix 𝑺 is (𝒁T⁢𝒁)d×d. Now let’s introduce a new matrix 𝑹

Therefore, the dimension of the matrix is n×n. In general, the dimension d of the sample image is often larger than the number n of samples, and the size of 𝑺 is often larger than 𝑹, so it is easier to calculate the eigenvalue of 𝑹 and the time complexity is lower. In fact, the eigenvalues corresponding to 𝑺 and 𝑹 are the same, so only the eigenvalues of 𝑹 with a small order of magnitude are required to obtain the eigenvalues of 𝑺.

Set the n dimensional column vector 𝒗 to the eigenvector corresponding to 𝑹:

Multiply both sides of the equation by 𝒁T to get the following equation:

𝒁T⁢𝒗 is the eigenvalue of the matrix 𝑺=(𝒁T⁢𝒁)d×d in equation.

Through the above argument, the eigenvalues of matrices with a larger order of magnitude and their corresponding eigenvectors can be obtained based on the eigenvalues of matrices with a smaller order of magnitude and their corresponding eigenvectors. The fast PCA algorithm is used to reduce the time complexity and optimize the algorithm.

The fast PCA algorithm was used to reduce the dimension of the defective lyophilized powder image and extract the principal component characteristic information of the foreign matter according to the training set image. In the multi-classification (fiber, hair, glass debris) experiment, the number of principal component characteristics was set to 15, and the transformation of principal component characteristics was realized through projection. By processing image information with the above method, the feature vectors of each foreign matter sample (fiber, hair, glass debris) in the sample image are trained to decrease from high dimension to 15 dimension. In the following series of experiments, the foreign matter is identified with a 15-dimensional basis.

The core algorithm of fast PCA is as follows:

4.1Third-Nearest Neighbor classification algorithm

In this paper, the third-Nearest Neighbor classification algorithm was used. The third-Nearest Neighbor classification algorithm has mature theory, simple algorithm and can be used for nonlinear classification. However, this algorithm also has some disadvantages: unbalanced sample effect is not good, large amount of prediction calculation, large memory consumption.

K-Nearest Neighbor (KNN) classification algorithm first selects the different eigenvalues corresponding to the images to be tested and the images in the training set, calculates the distance between them, and classifies them based on the minimum distance. The idea of KNN algorithm: the label and data of training set images are determined, then the distance between the corresponding features of training samples and the features of test samples is calculated, K pieces of data with the smallest interval are selected, and the category to which the K pieces of data belong is counted. The class with the largest number of times is the category to which the test sample belongs. The KNN algorithm process is as follows:

Calculating the absolute value of the difference between pixels is an important part of the third-order neighborhood recognition and classification algorithm. The formula to solve the distance is as follows:

The nearest similar image is obtained from Eq. (5), and its category is determined according to the frequency of occurrence of the category.

The main task of the third-Nearest Neighbor classification algorithm is to identify three training set images. At the same time, the classes of the selected three training set images are set as one, two and three respectively. If one and two do not belong to the same category, and two and three do not belong to the same category, it can be concluded that the test graph belongs to one. If one and two belong to the same category, then the test graph belongs to one, but the test graph is similar to two. If two and three belong to the same category, then the test graph belongs to two, but the test graph is similar to three.

4.2SVM classifier

Vapnik, Boser and Guyon first proposed the SVM [13, 14, 15] algorithm. The classification model of SVM is based on linear separable. For non-linear scenarios, it is often used to project from low-dimensional space to high-dimensional feature space for information processing, in which the non-linear mapping algorithm is involved, so as to transform the non-linear data set into the linear separable data set. Therefore, linear algorithm can be used to analyze the original data set in high dimensional space.

SVM aims at the case of linear separability. For the training sample set (xi,yi), and xi∈RN,yi∈{-1,1},i=1,2,3⁢…⁢n is to find a special hyperplane, which can well separate the data of these different categories from each other and maximize the spacing between these different categories.

Suppose the equation of this particular hyperplane is w⋅x+v=0, if w⋅x+b>0 then this class is set to the “1” class, if w⋅x+b<0 then this class is set to the “-1” class. In fact, the maximization of the two types of interval is equivalent to the minimization:

Constraint conditions: yi⁢(w⋅x+b)⩾1,∀i∈{1,2,3⁢…⁢n}.

Lagrange multiplier is added here, so that Eq. (5) can be written as Wolf dual form to obtain the following equation:

Solve for α and use α to determine the plane parameters w and b. According to the above, the optimal classification function of SVM can be determined as follows:

For the case of linear indivisibility, the original image sample needs to be projected into a high-dimensional space first. And the kernel function is chosen to be linearly separable in high dimensional space. Thus, the objective function can be converted into the following equation:

In this equation, ε represents the relaxation variable, and C represents the penalty factor, which must meet the following conditions:

The optimal classification function is:

where K⁢(xi⋅x) is the kernel, such as Linear kernel function, Polynomial kernel function, Radial basis kernel function and Radial basis kernel function.

In the lyophilized powder foreign matter image samples, the common three types of visible foreign matter are glass debris foreign matter, fiber foreign bodies and hair foreign matter. There are 10 pictures of foreign matter in each category of lyophilized powder for injection. The first 5 pictures of foreign matter in each category are set as the training set data, and 15 training set data are obtained. Each graph is represented by a row matrix, so 15 graphs form a 15 by 250 by 150 matrix. The remaining images are set as test data Eq. (5). When we read the training sample images, we have identified the labels real_labels and picture_matrix for all training images.

The basis of the foreign matter image in the low-dimensional space is the feature image of the training set, so all images, after dimensionality reduction, can be expressed linearly by the feature image of the training set, which is the basis of image recognition.

The implementation steps of SVM algorithm are as follows:

5.1Experimental environment

Matlab R2014b and Professor Lin Zhiren’s LIBSVM toolbox [16] were used to realize the experiment. Industrial pictures of defective lyophilized powder for injection were used as data sets. The data contains fiber, hair, glass debris and other foreign bodies. The pictures of foreign bodies collected by the industry are shown in Fig. 1. Due to the different shooting angles and light, the form and size of visible foreign bodies in the same category are also different. In the experimental data set, each kind of producing injection has 10 foreign matter images, and put the pictures of the each kind of foreign matter before 5 set as the training set, the rest picture set to test set, after producing a foreign matter image preprocessing, image feature extraction and producing a foreign producing foreign matter image classification and recognition of the experiment, to achieve more than two classes and the class of visible foreign matter classification and recognition. In the process of lyophilized powder foreign matter image preprocessing, first of all, each image was converted to gray, because the foreign matter image collected by the industry is colored. Then, median filtering, Wiener filtering and average filtering were carried out to denoise the image.

5.2Results and analysis

Two links, image preprocessing and image classification and recognition, were compared. The first group of comparative experiments compared the effects of various image preprocessing methods when the classifier was multi-classification (fiber, hair, glass debris) and the image classification and recognition algorithm were the same. The feature extraction algorithm of lyophilized powder foreign matter image is PCA, and the classification and recognition algorithm of foreign matter image is third-order neighborhood classification and recognition algorithm and SVM, respectively. Experimental results are shown in Tables 2 and 3.

Figure 1.

Foreign matters: a. Glass debris, b. Hair, c. Fiber.

As can be seen in Tables 2 and 3, in view of the industrial collection of small sample producing injection foreign matter images, the classifier for classification of classifier and image recognition algorithm under the condition of same, producing defective product image preprocessing effect is best with 3 × 3 median filtering and Wiener template, however 9 × 9 of the template effect of median filtering is the worst. Through the first group of comparative experiments, in the following series of experiments, the image preprocessing method of the defective lyophilized powder selects the median filtering of the 3 × 3 template. The second group of comparison experiments, in the case that the classifier is a multi-classifier and the image preprocessing method is the same, compare the effects of various classification recognition algorithms. The experimental results are shown in Tables 4 and 5.

As can be seen in Tables 4 and 5, in view of the industrial collection of small sample producing injection foreign matter image, image preprocessing method in classifier for multiple classifier and the same situation, based on PCA and SVM algorithm of recognition rate than based on PCA and the third-Nearest Neighbor classification algorithm of high recognition rate. Through the above multiple comparative experiments, the median filter of 3 × 3 template was selected in the image preprocessing method of lyophilized powder defective products, and the classification and recognition effect was the best based on principal component analysis and support vector machine algorithm.