Pulmonary nodule detection method based on multi-kernel representation learning

Abstract

The invention discloses a pulmonary nodule detection method based on multi-kernel representation learning, and belongs to the field of medical image analysis. The method comprises the following steps: S1, obtaining suspected region image block data of which the form is similar to that of a nodule, and carrying out standardization processing; s2, performing feature extraction through principal component analysis according to the normalized image blocks; s3, according to the extracted feature representation, calculating corresponding kernel matrixes by using multiple kernel functions; s4, according to the obtained multiple kernel matrixes, based on spectral entropy weighted fusion, obtaining a mixed kernel matrix; s5, training a single-class support vector machine model according to the mixed kernel matrix, and establishing a discrimination boundary of a normal sample; and S6, in a test stage, repeatedly executing the steps S1 to S4 on suspected nodule region image blocks acquired from the CT image of the patient, extracting features, constructing a mixed kernel matrix, inputting the trained single-class support vector machine model for judgment, and if the suspected nodule region image blocks are abnormal, outputting a nodule region until judgment of all candidate regions is completed. The problems that an existing single-core support vector machine model is poor in adaptability and sensitive to data are solved, the deep learning method depends on a large number of labeled samples, and robust pulmonary nodule detection is achieved under the condition that pulmonary nodule samples are scarce by means of the unsupervised characteristic of anomaly detection.

Description

Technical Field

[0001] The present invention relates to the technical field of medical image analysis, and belongs to a lung nodule recognition method based on anomaly detection, in particular to a semi-supervised lung nodule detection method based on multi-kernel representation learning. Background Art

[0002] Lung cancer is a malignant adenocarcinoma with high morbidity and mortality rates. Early detection is extremely crucial for improving the treatment rate and reducing the mortality rate, and is an important link in protecting human life and health. Lung nodule detection is an important means to assist in the early diagnosis of lung cancer, and abnormal nodules in lung CT images should be accurately calibrated.

[0003] Currently, the mainstream methods for automatic lung nodule recognition are mostly based on deep learning models, which rely on large-scale uniformly standard labeled image samples for training. However, in real scenarios, the acquisition cost of nodule image samples is high, the annotation is complex, the scale of public databases is generally small, and the sample coverage is limited and the generation standards are inconsistent, resulting in the training data being difficult to comprehensively reflect the morphological changes of lung nodules, affecting the generalization performance and clinical adaptability of the model. In contrast, unsupervised anomaly detection methods do not rely on abnormal samples or manual annotations, and can learn the distribution boundary only through normal sample data, so as to identify potential anomalies. In the field of images, the acquisition of normal samples is much easier than that of abnormal samples. Therefore, unsupervised or semi-detection methods can effectively reduce the dependence on training samples in practical applications, reduce the human and time costs of model development, and are more suitable for clinical scenarios where abnormal samples are scarce in lung nodule screening.

[0004] On the other hand, although some existing support vector machines or kernel methods based on single-kernel functions can construct anomaly detection models in situations where abnormal samples are scarce, their kernel function structures are fixed and the expression ability is limited, making it difficult to simultaneously take into account the distribution characteristics of data in multiple scales or directions. When the selected kernel function does not match the internal distribution structure of the data, the discriminant boundary of the model may shift or overfit, resulting in unstable detection performance and insufficient generalization ability.

[0005] Generally speaking, the existing lung nodule detection methods usually have the following problems: (1) Most methods rely on a large number of labeled nodule image samples. However, in practice, image data is scarce and annotation is difficult, making it difficult to meet the training requirements of supervised models; (2) Traditional single-kernel function-based anomaly detection methods have rigid structures and limited expression abilities, and cannot fully characterize the complex non-linear structures of high-dimensional image data, resulting in unstable detection accuracy. Summary of the Invention

[0006] Aiming at the above deficiencies in the prior art, a lung nodule detection method based on multi-kernel representation learning provided by the present invention solves the problem of being unable to effectively train a detection model under the condition of scarce abnormal samples, improves the separability between nodule-like regions and true nodules in lung images under non-linear structures, and thus realizes semi-supervised detection of lung nodules.

[0007] In order to achieve the above invention purpose, the technical solution adopted by the present invention is: a lung nodule recognition method based on multi-kernel representation learning, including the following steps:

[0008] S1. Obtain suspected region image patch data similar in morphology to nodules and perform normalization processing;

[0009] S2. Extract features through principal component analysis according to the normalized image patches;

[0010] S3. Calculate corresponding kernel matrices respectively using a variety of kernel functions according to the extracted feature representations;

[0011] S4. Based on spectral entropy weighted fusion, obtain a mixed kernel matrix according to the obtained multiple kernel matrices;

[0012] S5. Train a one-class support vector machine model according to the mixed kernel matrix to establish a discrimination boundary for normal samples;

[0013] S6. In the test stage, repeat S1 to S4 for the suspected nodule region image patches obtained from the patient's CT image, extract features and construct a mixed kernel matrix, and input it into the trained one-class support vector machine model for judgment. If it is abnormal, output it as a nodule region until all candidate regions are judged.

[0014] The beneficial effects of the present invention are as follows: By constructing a variety of kernel functions to calculate the non-linear similarity between samples, the expression ability of the model for the image structure relationship is improved, and the problem of fixed structure and poor adaptability of the single-kernel method is solved; Through the spectral entropy weighting mechanism, the information quantity of each kernel matrix is evaluated and adaptively fused to generate a more discriminative mixed kernel matrix, which improves the stability and robustness of detection; Finally, by learning the normal sample boundary with a one-class support vector machine, without abnormal samples or manual annotation, semi-supervised anomaly detection for lung CT images is realized, which helps the early screening and auxiliary diagnosis of lung cancer.

[0015] Further, the expression of the normalization processing in step S1 is:

[0016]

[0017] Among them, the image block data is a grayscale image in PNG format that has been adjusted by window width and window level and normalized. f(·) is a normalization function. (i, j) represents the pixel position of the i-th row and the j-th column in the image x, and the pixel value is x(i, j) ∈ [0, 255].

[0018] The beneficial effect of the above further solution is: standardizing the image block data, unifying the dimension, and reducing the data calculation amount.

[0019] Further, the specific steps of S2 are as follows:

[0020] S201. Calculate the mean vector of the sample set, and perform centering processing on each sample vector to obtain a de-centered sample matrix. The calculation formula is:

[0021]

[0022] X c = X - μ

[0023] where x i ∈ R d is the d-dimensional pixel vector corresponding to the i-th image block; X is the sample set composed of the image blocks after normalization processing, X = [x1, x2,..., x n T ; μ is the sample set mean; X c is the de-centered sample matrix; n is the number of samples;

[0024] S202. Calculate the covariance matrix according to the de-centered sample matrix:

[0025]

[0026] where, X c is the de-centered sample matrix; is the transpose of X c ; Y is the covariance matrix; n is the number of samples;

[0027] S203. Perform eigenvalue decomposition on the covariance matrix to obtain eigenvalues and corresponding eigenvectors, and select the eigenvectors corresponding to the top k largest eigenvalues to form a projection matrix. The calculation formula is:

[0028] ∑v i = λ i v i

[0029] P = [v1, v2... v k

[0030] where, λ i is the eigenvalue after eigenvalue decomposition of the covariance matrix; v​​i is the eigenvalue λ i corresponding eigenvector; P ∈ R d×k is the projection matrix composed of eigenvectors corresponding to the first k largest eigenvalues;

[0031] S204. Project the sample data onto the k-dimensional principal component subspace to obtain a feature representation matrix, and its calculation formula is:

[0032] Z = X c P

[0033] where X c is the centered sample matrix; P ∈ R d×k is the projection matrix composed of eigenvectors corresponding to the first k largest eigenvalues; Z ∈ R n×k represents the feature matrix after principal component analysis and is used for the subsequent kernel matrix construction step.

[0034] The beneficial effect of the above further scheme is: performing principal component analysis on the image block data compresses the feature dimension, retains the main structural information, reduces feature redundancy and computational complexity.

[0035] Furthermore, step S3 calculates different kernel matrices according to the feature matrix:

[0036]

[0037] where z i ∈ R k represents the feature representation of the i-th image block in the principal component subspace; K lin (i, j), K poly (i, j), K rbf (i, j) are all kernel matrices and are symmetric matrices calculated for the sample feature vectors z i and z j under the corresponding kernel function; γ ∈ R + is the scaling coefficient; r ∈ R is the bias term; d ∈ N + is the degree of the polynomial; σ ∈ R + is the bandwidth parameter of the Gaussian kernel; exp(.) represents the exponential function.

[0038] The beneficial effect of the above further scheme is: simultaneously using multiple kernel functions to characterize the non-linear structural differences between image blocks from multiple perspectives, solving the problems of fixed structure and poor adaptability of the single-kernel method, and improving the sensitivity and discrimination ability of the model to complex image structures.

[0039] Furthermore, the specific steps of step S4 are:

[0040] S401. Normalize each obtained kernel matrix to get the normalized kernel matrix, and its expression is:

[0041]

[0042] where \(K\) m is the \(m\)-th kernel matrix, \(K\) m (i, j) is the element value of the kernel matrix \(K\) m at (i, j), max(.) is the maximum value function, min(.) is the minimum value function, and \(K'\) m is the normalized kernel matrix;

[0043] S402. For each normalized kernel matrix, calculate its corresponding weight coefficient and perform fusion, and the expression is as follows:

[0044]

[0045] where \(K'\) m is the normalized kernel matrix; is the \(i\)-th eigenvalue of the normalized kernel matrix \(K'\) m ; \(H\) m is the spectral entropy of the \(m\)-th kernel matrix; \(K\) mix is the final mixed kernel matrix.

[0046] The beneficial effect of the above further solution is that by introducing the spectral entropy weighting mechanism, the information amount of the kernel matrices constructed by multiple kernel functions is evaluated and adaptively fused, and the fused mixed kernel matrix is constructed, which effectively improves the comprehensive expression ability of the model for different scales and different structural features, and enhances the stability and robustness of the detection process.

[0047] Furthermore, the expression for training the single-core support vector machine model in step S5 is:

[0048] Objective function:

[0049]

[0050] Constraints:

[0051]

[0052] Decision function:

[0053]

[0054] where \(z\) i ∈R k represents the feature vector of the \( - \)th image block sample in the principal component subspace; \(z\in R\) k represents any sample to be detected; \(K\) mix (zi , z j ) represents the sample z i and z j in the fused kernel space; α i ∈R is the weight coefficient corresponding to the i-th support vector; ν ∈ (0, 1] is the relaxation parameter in the one-class support vector machine; n is the total number of training samples; ρ ∈ R is the model bias term; f(z) ∈ {-1, +1} is the discrimination result for the input sample z, where +1 indicates that the sample is judged to be normal and -1 indicates that the sample is judged to be abnormal; sign(·) is the sign function, which outputs +1 when its input is greater than or equal to 0 and otherwise outputs -1.

[0055] The beneficial effects of the above further solution are as follows: By training a one-class support vector machine model based on the hybrid kernel matrix to construct the distribution boundary of normal samples, the identification of abnormal regions is realized without the participation of abnormal samples, reducing the dependence on manually labeled data and improving the adaptability and practicality of the model in real scenarios. Description of the Drawings

[0056] Figure 1 It is the flowchart of the method of the present invention. Detailed Embodiments

[0057] The following describes the detailed embodiments of the present invention to facilitate those skilled in the art of the present technology to understand the present invention. However, it should be clear that the present invention is not limited to the scope of the detailed embodiments. For those of ordinary skill in the art of the present technology, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions and creations using the concept of the present invention are within the scope of protection.

[0058] Embodiment 1

[0059] In this embodiment, the lung nodule detection method based on multi-kernel representation learning is an image intelligent discrimination model for unsupervised anomaly detection. This method combines multiple kernel functions and spectral entropy weighting strategies, fully utilizes the feature information of nodule-like regions in normal lung CT images, and learns its distribution boundary in the high-dimensional kernel space through a one-class support vector machine, so as to identify potential real nodule regions, which is applicable to actual screening scenarios where nodule samples are scarce or difficult to label.

[0060] Specifically, the method first extracts suspected nodule region image patches from normal lung CT images, performs normalization and unified size processing, and compresses their dimensions using the principal component analysis method to extract the main structural features, forming an image patch feature set. Subsequently, multiple kernel matrices are constructed using linear kernel, polynomial kernel, and Gaussian kernel functions in sequence to model the non-linear similarity between samples from different perspectives. To integrate the information contained in each kernel matrix, a spectral entropy mechanism is further introduced to evaluate the information content of each kernel matrix, and weighted fusion is performed according to the normalized results of spectral entropy to construct the final hybrid kernel matrix, effectively enhancing the adaptability of the model to the diversity of image structures.

[0061] On this basis, the hybrid kernel matrix is input into a one-class support vector machine model for training, and the decision boundary is constructed using only normal samples. This model does not rely on any abnormal samples to participate. By learning the feature distribution of normal image patches, it can determine whether there is an abnormal region in the test image during the inference stage. Finally, in the detection of new samples, the above steps of image patch extraction, feature processing, and kernel fusion are repeated, and the suspected region is input into the trained model for judgment. If it is determined to be abnormal, it is output as a lung nodule candidate region.

[0062] The method described in this embodiment gives full play to the advantages of multi-kernel representation and semi-supervised detection, improves the robustness and detection accuracy of the model in an image environment with complex structures and high heterogeneity, and has strong practical application value.

[0063] The expression for the normalization process in step S1 is:

[0064]

[0065] Among them, the image patch data is a grayscale image in PNG format that has been adjusted by window width and window level and normalized. f(·) is the normalization function. (i, j) represents the pixel position at the i-th row and j-th column in image x, and the pixel value is x(i, j) ∈ [0, 255].

[0066] In this embodiment, the image patch data we obtained is a 64×64 grayscale image. Through linear normalization operation, the pixel values of the image patches are mapped to the real number interval from 0 to 1 to unify the feature scale and eliminate the difference in dimension of the original data.

[0067] Step S2 is specifically as follows:

[0068] S201. Calculate the mean vector μ of the sample set, and perform centering processing on each sample vector to obtain the de-centered sample matrix X c , and its calculation formula is:

[0069]

[0070] X c= X - μ

[0071] where x i ∈ R d is the d - dimensional pixel vector corresponding to the i - th image block; X is the sample set composed of the image blocks after normalization processing, X = [x1, x2,..., x n T ;

[0072] S202. Calculate the covariance matrix Y according to the decentralized sample matrix X c :

[0073]

[0074] where represents the transpose of X c ;

[0075] S203. Perform eigenvalue decomposition on the covariance matrix Y to obtain the eigenvalues λ and the corresponding eigenvectors v, and select the eigenvectors corresponding to the first k largest eigenvalues to form the projection matrix P ∈ R d×k , and its calculation formula is:

[0076] ∑v i = λ i v i

[0077] P = [v1, v2... v k

[0078] S204. Project the sample data onto the k - dimensional principal component subspace to obtain the feature representation matrix Z ∈ R n×k , and its calculation formula is:

[0079] Z = X c P

[0080] where Z ∈ R n×k represents the feature matrix after principal component analysis and is used for the subsequent kernel matrix construction step.

[0081] In this embodiment, the normalized data is subjected to feature extraction by the principal component analysis method. While retaining the main structural information of the image blocks, the feature dimension is reduced and redundant information is reduced, providing a concise and stable feature input for the subsequent construction of the kernel matrix. In addition, the eigenvectors corresponding to the first 128 largest eigenvalues are selected in this embodiment.

[0082] Furthermore, in step S3, different kernel functions are used to calculate the kernel matrix for the feature matrix:

[0083]

[0084] where z i ​​∈R k represents the feature representation of the i-th image patch in the principal component subspace; K lin (i, j), K poly (i, j), K rbf (i, j) are all kernel matrices, which are symmetric matrices calculated for the sample feature vectors z i and z j under the corresponding kernel function; γ ∈ R + is the scaling coefficient; r ∈ R is the bias term; d ∈ N + is the degree of the polynomial; σ ∈ R + is the bandwidth parameter of the Gaussian kernel; exp(.) represents the exponential function.

[0085] In this embodiment, the kernel functions used include the linear kernel function, the polynomial kernel function, and the radial basis kernel function. The linear kernel is used to characterize the linear relationship between features, the polynomial kernel is used to model the high-order polynomial interaction relationship, and the radial basis kernel function is used to model the non-linear change structure in the local space.

[0086] In this embodiment, the order of the polynomial kernel function is taken as 3, and the bias constant is 1; the parameter γ of the RBF kernel function adopts the default setting.

[0087] The specific method of step S4 is as follows:

[0088] S401. Normalize each obtained kernel matrix to obtain the normalized kernel matrix, and its expression is:

[0089]

[0090] where K m is the m-th kernel matrix, K m (i, j) is the element value of the kernel matrix K m at (i, j), max(.) is the maximum value function, min(.) is the minimum value function, and K’ m is the normalized kernel matrix;

[0091] S402. Calculate the corresponding weight coefficients for each normalized kernel matrix and perform fusion, and the expression is as follows:

[0092]

[0093] where, K’ m is the normalized kernel matrix; is the i-th eigenvalue of the normalized kernel matrix K’ m ; H m is the spectral entropy of the m-th kernel matrix; K mix is the final mixed kernel matrix.

[0094] In this embodiment, the spectral entropy value H m can measure the structural information content contained in each kernel matrix. It is considered that the larger the spectral entropy value, the greater the contribution of the kernel matrix in characterizing the complex structural relationship between samples. Therefore, the spectral entropy values of all kernel matrices are normalized to calculate the weight corresponding to each kernel matrix as its relative importance index in the fusion process.

[0095] Furthermore, the expression for training the one-class support vector machine in step S5 is:

[0096] Objective function:

[0097]

[0098] Constraints:

[0099]

[0100] Decision function:

[0101]

[0102] where z i ∈R k represents the eigenvector of the i-th image patch sample in the principal component subspace; z ∈ R k represents any sample to be detected; K mix (z i , z j ) represents the similarity between the sample z i and z j in the fused kernel space; α i ∈R is the weight coefficient corresponding to the i-th support vector; ν ∈ (0, 1] is the relaxation parameter in the one-class support vector machine; n is the total number of training samples; ρ ∈ R is the model bias term; f(z) ∈ {-1, +1} is the discrimination result for the input sample z, where +1 indicates that the sample is determined to be normal and -1 indicates that the sample is determined to be abnormal; sign(·) is the sign function, which outputs +1 when its input is greater than or equal to 0 and otherwise outputs -1.

[0103] In this embodiment, the mixed kernel matrix K' m of the training samples is input into the one-class support vector machine model, and the support vector learning method is used to model the distribution of normal image patches. The model aims to find a hyperplane with the largest margin, so that most training samples are inside the boundary, while potential abnormal samples are excluded outside the boundary, thereby realizing the construction of a discrimination boundary without the participation of abnormal samples, making the model more suitable for the lung nodule detection scenario where abnormal samples are scarce.

[0104] In this embodiment, the selected ν is 0.01, which is used to control the tolerance of the model to potential outliers in the training samples and enhance the fitting ability of the normal sample distribution boundary.

[0105] Furthermore, step S6 is used to apply the one-class support vector machine model obtained in the training stage to the detection task of lung CT images to discriminate the suspected nodule regions in the images.

[0106] In this embodiment, first, the new patient's CT image is preprocessed to extract suspected nodule image patches in the same form as in the training stage. Subsequently, the normalization process, principal component analysis feature extraction, kernel matrix construction, and spectral entropy weighted fusion operations consistent with the training stage are sequentially performed to obtain the mixed kernel similarity representation between the test samples and the training samples.

[0107] Next, the one-class support vector machine model constructed in the training stage is used to classify and judge these test image patches. Based on the normal sample distribution boundary obtained through training, the model maps the test samples into the kernel space and makes a discrimination according to their positions relative to the boundary. If the projection of the sample in the kernel space falls outside the boundary, it is determined as an abnormal region, that is, it may be a lung nodule region. Through the above steps, the model can automatically identify potential lesion regions in lung images without relying on any abnormal samples for training, significantly reducing the dependence on manual annotation, improving the automation and adaptability of detection, and having important application value for the early screening and auxiliary diagnosis of lung cancer.

Claims

1. A lung nodule detection method based on multi-core representation learning, characterized in that, It includes the following steps: S1. Obtain the suspected region image patch data with a morphology similar to that of nodules and perform normalization processing; S2. Extract features through principal component analysis based on the normalized image patches; S3. Calculate the corresponding kernel matrices respectively using a variety of kernel functions according to the extracted feature representations; S4. Based on the obtained multiple kernel matrices, perform spectral entropy weighted fusion to obtain a hybrid kernel matrix; S5. Train a one-class support vector machine model according to the hybrid kernel matrix and establish a discrimination boundary for normal samples; S6. In the test stage, repeat S1 to S4 for the suspected nodule region image patches obtained from the patient's CT image, extract features and construct a hybrid kernel matrix, input it into the trained one-class support vector machine model for judgment. If it is abnormal, output it as the nodule region until all candidate regions are judged.

2. The lung nodule detection method based on multi-core representation learning according to claim 1, wherein The normalization processing expression in step S1 is: Among them, the image patch data is a grayscale image in PNG format after window width and window level adjustment and normalization. f(·) is a normalization function. (i, j) represents the pixel position of the i-th row and the j-th column in image x, and the pixel value is x(i, j) ∈ [0, 255].

3. The lung nodule detection method based on multi-core representation learning according to claim 1, wherein The specific steps of step S2 are: S201. Calculate the mean vector of the sample set and perform centering processing on each sample vector to obtain a de-centered sample matrix, and its calculation formula is: X c = X - μ where x i ∈R d is the d-dimensional pixel vector corresponding to the i-th image patch; X is the sample set composed of the image patches after normalization, X = [x1, x2,..., x n T ; μ is the sample set mean; X c is the de-centered sample matrix; n is the number of samples;​ S202. Calculate the covariance matrix according to the de-centered sample matrix: Among them, X c is a decentralized sample matrix; is the transpose of X c ; Y is the covariance matrix; n is the number of samples; S203. Perform eigenvalue decomposition on the covariance matrix to obtain eigenvalues and corresponding eigenvectors, and select the eigenvectors corresponding to the top k largest eigenvalues to form a projection matrix, and its calculation formula is: ∑v i = λ i v i P = [v1, v2... v k ​ Among them, λ i is the eigenvalue after the eigenvalue decomposition of the covariance matrix; v i is the eigenvector corresponding to the eigenvalue λ i ; P ∈ R d×k is the projection matrix composed of the eigenvectors corresponding to the top k largest eigenvalues; S204. Project the sample data into the k-dimensional principal component subspace to obtain a feature representation matrix, and its calculation formula is: Z = X c P Among them, X c is a decentralized sample matrix; P ∈ R d×k is a projection matrix composed of eigenvectors corresponding to the top k largest eigenvalues; Z ∈ R n×k represents the feature matrix after principal component analysis and is used in the subsequent kernel matrix construction step.

4. The lung nodule detection method based on multi-core representation learning according to claim 1, wherein The expressions for calculating different kernel matrices in step S3 are: where z i ∈R k represents the feature representation of the i-th image patch in the principal component subspace; K lin (i, j), K poly (i, j), K rbf (i, j) are all kernel matrices, which are symmetric matrices calculated for the sample feature vectors z i and z j under the corresponding kernel function; γ ∈ R + is the scaling coefficient; r ∈ R is the bias term; d ∈ N + is the degree of the polynomial; σ ∈ R + is the bandwidth parameter of the Gaussian kernel; exp(.) represents the exponential function.

5. The lung nodule detection method based on multi-core representation learning according to claim 1, characterized in that, The specific steps of step S4 are: S401. Normalize each obtained kernel matrix to obtain a normalized kernel matrix, and its expression is: where K m is the m-th kernel matrix, K m (i, j) is the element value of the kernel matrix K m at (i, j), max(.) is the maximum value function, min(.) is the minimum value function, and K’ m is the normalized kernel matrix; S402. For each normalized kernel matrix, calculate its corresponding weight coefficient and perform fusion, and the expression is as follows: Among them, K' m is the normalized kernel matrix; is the i-th eigenvalue of the normalized kernel matrix K' m ; H m is the spectral entropy of the m-th kernel matrix; K mix is the final mixed kernel matrix.

6. The lung nodule detection method based on multi-core representation learning according to claim 1, characterized in that The expression for training a one-class support vector machine in step S5 is: Objective function: Constraint condition: Decision function: where z i ∈R k represents the eigenvector of the \(i\)-th image patch sample in the principal component subspace; \(z\in R\) k represents any sample to be detected; \(K\) mix (z i , z j ) represents the similarity between the sample \(z\) i and \(z\) j in the fused kernel space; \(\alpha\) i \(\in R\) is the weight coefficient corresponding to the \(i\)-th support vector; \(\nu\in(0, 1]\) is the relaxation parameter in the one-class support vector machine; \(n\) is the total number of training samples; \(\rho\in R\) is the model bias term; \(f(z)\in\{-1, +1\}\) is the discrimination result for the input sample \(z\), where \(+1\) indicates that the sample is determined to be normal and \(-1\) indicates that the sample is determined to be abnormal; \(sign(\cdot)\) is the sign function, which outputs \(+1\) when its input is greater than or equal to \(0\) and otherwise outputs \(-1\).

Images