Epilepsy detection method combining uncorrelated multi-linear discriminant analysis and deep metric learning

Abstract

The invention discloses an epilepsy detection method combining uncorrelated multi-linear discriminant analysis and deep metric learning, and belongs to the technical field of epilepsy detection.The epilepsy detection method comprises the steps that S1, original electroencephalogram signals are collected and preprocessed; s2, forming a third-order electroencephalogram tensor by using the electroencephalogram signals subjected to the improved S-transform time-frequency decomposition processing, and extracting electroencephalogram features through uncorrelated multi-linear discriminant analysis; s3, performing epilepsy disease classification by using a triple network metric learning classifier; s4, a Softmax function is used for outputting probability distribution of the test data, and a detection label is obtained; s5, a moving average filter is used for eliminating burst noisy points and isolated abnormal values, and a classification result and post-processing are output; according to the epilepsy detection method combining uncorrelated multi-linear discriminant analysis and deep metric learning provided by the invention, the workload of doctors can be effectively reduced, the efficiency and accuracy of epilepsy detection are improved, and the epilepsy detection method has a remarkable promotion effect on clinical diagnosis and treatment.

Description

Technical Field

[0001] The present invention relates to the technical field of epilepsy detection, and in particular to an epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning. Background Art

[0002] Epilepsy is a complex chronic brain disease caused by abnormal discharges of neurons, which often manifests as abnormalities in movement, sensation, cognition and psychology, seriously affecting patients' daily life and mental health. With the increasing incidence of epilepsy, timely and accurate diagnosis and treatment are particularly critical. Electroencephalogram (EEG), as the main tool for epilepsy diagnosis, can record abnormal signals in the brain's neural electrical activity. However, the high-dimensional complexity of EEG signals makes manual analysis time-consuming and laborious, and is easily interfered by the doctor's subjective experience, leading to misdiagnosis. Therefore, it is of great clinical significance to develop an automated epilepsy detection system that uses computer algorithms to process EEG data to improve diagnostic accuracy and efficiency.

[0003] Existing automatic epilepsy detection systems mainly rely on feature extraction and classifier accuracy to distinguish between ictal and non-ictal signals. However, traditional unsupervised learning methods, such as MPCA and UMPCA, due to the lack of prior knowledge, easily lead to overfitting of the model to data noise or irrelevant features, thereby affecting the stability and reliability of the detection effect. Therefore, to address this problem, the present invention proposes an automatic epilepsy detection model based on uncorrelated multilinear discriminant analysis (UMLDA) and deep metric learning, aiming to more efficiently extract structured features from EEG data and improve the generalization ability of the model. Summary of the invention

[0004] The purpose of the present invention is to provide an epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning to solve the problems existing in the above-mentioned background technology.

[0005] To achieve the above object, the present invention provides an epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning, comprising the following steps:

[0006] S1, collect raw EEG signals and preprocess them;

[0007] S2, using the EEG signal processed by improved S-transform time-frequency decomposition to form a third-order EEG tensor and extract EEG features through uncorrelated multilinear discriminant analysis (UMLDA). UMLDA is a supervised learning algorithm, and its projection features are reduced-order vector features;

[0008] S3, epilepsy disease classification using triplet network metric learning classifier;

[0009] S4. Use the Softmax function to output the probability distribution of the test data and obtain the detection label;

[0010] S5. Use a moving average filter to eliminate sudden noise points and isolated outliers, which helps to make the data more stable, reduce the impact of sudden noise, and make the data more continuous and smooth, output classification results and post-processing.

[0011] Preferably, the conventional S transform lacks parameters for adjusting the Gaussian window width in frequency and time, and the improved S transform introduces additional parameter settings to more effectively control the characteristics of the window. In the improved S transform, the form of the window function is similar to the Gaussian window, and the standard deviation σ of the improved S transform in step S2 is set to:

[0012]

[0013] Among them, λ represents the parameter used to adjust the standard deviation of the function; f represents the frequency;

[0014] Therefore, the modified S transform is defined as:

[0015]

[0016] Among them, S x (t,f,λ) represents the signal The time-frequency transformation result is represents the current moment of signal analysis, represents the input signal, Indicates the center is The window function depends on f and λ. represents the modulation term of frequency f.

[0017] Preferably, the EEG signal is converted into a two-dimensional time-spectrogram consisting of time domain and frequency domain using a modified S transform (MST), and then the channel is taken as one dimension, and the time domain and frequency domain are taken as the other two dimensions to form a third-order EEG tensor, whose three dimensions are frequency, time and channel.

[0018] Preferably, the uncorrelated multilinear discriminant analysis in step S2 is a supervised learning algorithm, and its projection feature is a reduced-order vector feature, which is defined as:

[0019] A set of M training tensor objects samples For training, each tensor object The value of is in tensor space On, among which, I N is the n-modular dimension of the tensor, represents the Kronecker product; the goal of uncorrelated multilinear discriminant analysis is to find a TVP that contains P EMPs, where EMP is the elementary multilinear projection factor (EMP), specifically Convert the original tensor space Mapped to a dimension The vector subspace of in

[0020]

[0021] So that in each EMP direction are all maximized while satisfying the constraint that the P coordinate vectors are unrelated.

[0022] Preferably, the goal of the uncorrelated linear discriminant analysis is to determine a set of P EMPs It is used to maximize the divergence ratio while generating features with zero correlation. Therefore, the objective function of the Pth EMP is:

[0023]

[0024] in, represents the transpose of the projection vector of the nth sample in the pth class, N p represents the number of samples in the pth class, F p represents the objective function, which is used to maximize the divergence ratio between classes;

[0025] Subject to:

[0026]

[0027] Among them, g p and g q represents the eigenvectors of the pth and qth classes, ||g p || and ||g q || is the eigenvector g p and g q The norm of pq is the Kronecker delta symbol, which is 1 when p = q and 0 otherwise, and is used to indicate the orthogonality of the eigenvectors;

[0028] P EMPs Determine in P steps in sequence, where the p-th step obtains the p-th EMP; for each EMP to be determined, estimate the projection vector for each n-mode separately The parameters of EMP are simplified to the following formula:

[0029]

[0030] in, represents the projection vector of the nth sample of the pth class, is the inter-class scatter matrix, which indicates the dispersion between categories. is the intra-class scatter matrix, which indicates the dispersion within the same class;

[0031] Subject to:

[0032]

[0033] in, is the m sample matrix after projection, y (p) and (q) represents the projection samples of the pth and qth categories, y p (n) and q (n) Represents the projection result of the nth sample of the pth class and the qth class. This constraint is used to ensure that the projection vectors of different categories are orthogonal.

[0034] Preferably, the Fisher Discriminant Criterion (FDC) in the linear discriminant analysis in step S2 is the divergence ratio of the vector samples, and the p-th projection (scalar) feature Where M is the number of training samples, is the pth EMP For the mth sample The projection of and intra-class scatter for:

[0035]

[0036] Where C is the number of categories, N c is the number of samples of category c, c m is the category label of the mth training sample, and Therefore, for the pth scalar sample, the Fisher discriminant criterion is In addition, the pth coordinate vector is represented by g p , whose mth component is

[0037] Preferably, the triplet network in step S3 is a metric learning neural network structure, which is usually used to learn embedding vectors and consists of anchor samples, positive samples and negative samples. During the training process, the triplet network optimizes the model through triplet loss. For each training sample, it randomly selects a positive sample and a negative sample from the data set, calculates the triplet loss between the anchor sample, the positive sample and the negative sample through a convolutional neural network, and learns a good embedding vector space (Embedding), which can make similar samples cluster in space and dissimilar samples dispersed, thereby improving the performance of the model on similarity tasks. The Embedding of the triplet network is composed of It means that it embeds the two-dimensional matrix x into the d-dimensional Euclidean space. In addition, this embedding is restricted to the d-dimensional hypersphere, that is, ||f(x)||2=1. The triplet loss function was proposed in the context of nearest neighbor classification. Here, it is ensured that a specific sample (anchor point) than it is with any sample of a different category (Counterexample) Closer to samples of the same class (positive example), therefore, the goal is:

[0038]

[0039] where α is the margin enforced between positive and negative pairs; For the set of all triplets in the training set, the minimized loss L is:

[0040]

[0041] Where N represents the total number of all triplets in the training set.

[0042] Preferably, the collar technique is used while executing step S4 and step S5 to process the decision label of the epileptic seizure and appropriately extend the label of the epileptic seizure stage to compensate for the lack of epileptic seizure components. This means that the method will better capture the start and end of the seizure and ensure that the judgment of the epileptic seizure is more accurate and comprehensive.

[0043] Therefore, the present invention adopts the above-mentioned epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning, which has the following beneficial effects:

[0044] (1) The original EEG signal is decomposed into time and frequency using the modified S-transform (MST), thereby constructing a third-order tensor to more comprehensively capture the time-frequency characteristics of the signal;

[0045] (2) Extract features highly correlated with epileptic seizures from high-dimensional EEG data through uncorrelated multilinear discriminant analysis, and maximize the differences between different categories by imposing uncorrelated constraints to enhance the robustness of the model and its adaptability to complex data;

[0046] (3) Through the triplet deep metric learning network, the distance between samples of the same type can be minimized, while the distance between samples of different types can be maximized, so that the learned features can better distinguish different categories in the embedding space, achieve better classification effects, and improve the accuracy and generalization ability of the model.

[0047] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0048] Figure 1 A flowchart of an epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to the present invention;

[0049] Figure 2 It is a schematic diagram of the triplet network structure of the present invention;

[0050] Figure 3 Schematic diagram of the triplet network training process of the present invention;

[0051] Figure 4 Schematic diagram of a third-order EEG tensor according to an embodiment of the present invention. DETAILED DESCRIPTION

[0052] The following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the invention claimed for protection, but merely represents selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0053] See also Figure 1-3 , an epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning, comprising the following steps:

[0054] S1, collect raw EEG signals and preprocess them;

[0055] S2, using the EEG signal processed by improved S-transform time-frequency decomposition to form a third-order EEG tensor and extract EEG features through uncorrelated multilinear discriminant analysis (UMLDA). UMLDA is a supervised learning algorithm, and its projection features are reduced-order vector features;

[0056] The traditional S transform lacks parameters for adjusting the Gaussian window width in frequency and time. The improved S transform introduces additional parameter settings to more effectively control the characteristics of the window. In the improved S transform, the form of the window function is similar to the Gaussian window. The standard deviation σ of the improved S transform in step S2 is set to:

[0057]

[0058] Among them, λ represents the parameter used to adjust the standard deviation of the function; f represents the frequency;

[0059] Therefore, the modified S transform is defined as:

[0060]

[0061] Among them, S x (t,f,λ) represents the signal The time-frequency transformation result is represents the current moment of signal analysis, represents the input signal, Indicates the center is The window function depends on f and λ. represents the modulation term of frequency f.

[0062] The improved S-transform (MST) shows higher efficiency and performance in EEG signal analysis, providing an effective tool for more accurately capturing time-frequency features. This study uses MST to convert EEG signals into a two-dimensional time-frequency spectrum consisting of time domain and frequency domain. Considering that various noises and artifacts are usually distributed in the high-frequency range of EEG signals, and epileptic seizure activities mainly exist below 35Hz in the spectrum, the frequency band range in MST is selected as 1-35Hz, which not only helps to eliminate high-frequency noise and artifacts in the original EEG records in subsequent analysis, but also can more accurately capture the characteristics of major epileptic seizures.

[0063] After converting the EEG signals of the three lead channels into two-dimensional time-spectrograms, we take the channel as one dimension and the time domain and frequency domain as the other two dimensions to form an EEG tensor with three dimensions: frequency, time and channel, as shown in Figure 4 shown.

[0064] The uncorrelated multilinear discriminant analysis in step S2 is a supervised learning algorithm, and its projection feature is a reduced-order vector feature, which is defined as:

[0065] A set of M training tensor objects samples For training, each tensor object The value of is in tensor space On, among which, I N is the n-modular dimension of the tensor, represents the Kronecker product; the goal of uncorrelated multilinear discriminant analysis is to find a TVP that contains P EMPs, where EMP is Convert the original tensor space Mapped to a dimension The vector subspace of in

[0066]

[0067] So that in each EMP direction are all maximized while satisfying the constraint that the P coordinate vectors are unrelated.

[0068] The goal of uncorrelated linear discriminant analysis is to determine a set of P EMPs It is used to maximize the divergence ratio while generating features with zero correlation. Therefore, the objective function of the Pth EMP is:

[0069]

[0070] in, represents the transpose of the projection vector of the nth sample in the pth class, N p represents the number of samples in the pth class, F p represents the objective function, which is used to maximize the divergence ratio between classes;

[0071] Subject to:

[0072]

[0073] Among them, g p and g q represents the eigenvectors of the pth and qth classes, ||g p || and ||g q || is the eigenvector g p and g q The norm of pq is the Kronecker delta symbol, which is 1 when p = q and 0 otherwise, and is used to indicate the orthogonality of the eigenvectors;

[0074] P EMPs Determine in P steps in sequence, where the p-th step obtains the p-th EMP; for each EMP to be determined, estimate the projection vector for each n-mode separately The parameters of EMP are simplified to the following formula:

[0075]

[0076] in, represents the projection vector of the nth sample of the pth class, is the inter-class scatter matrix, which indicates the dispersion between categories. is the intra-class scatter matrix, which indicates the dispersion within the same class;

[0077] Subject to:

[0078]

[0079] in, is the m sample matrix after projection, y (p) and (q) represents the projection samples of the pth and qth categories, y p (n) and q (n) Represents the projection result of the nth sample of the pth class and the qth class. This constraint is used to ensure that the projection vectors of different categories are orthogonal.

[0080] The Fisher Discriminant Criterion (FDC) in the linear discriminant analysis in step S2 is the divergence ratio of the vector samples, and the p-th projection (scalar) feature Where M is the number of training samples, is the pth EMP For the mth sample The projection of and intra-class scatter for:

[0081]

[0082] Where C is the number of categories, N c is the number of samples of category c, c m is the category label of the mth training sample, and Therefore, for the pth scalar sample, the Fisher discriminant criterion is In addition, the pth coordinate vector is represented by g p , whose mth component is

[0083] S3, epilepsy disease classification using triplet network metric learning classifier;

[0084] The triplet network is a metric learning neural network structure, usually used to learn embedding vectors. It consists of anchor samples, positive samples, and negative samples. During the training process, the triplet network optimizes the model through triplet loss. For each training sample, it randomly selects a positive sample and a negative sample from the data set, and calculates the triplet loss between the anchor sample, the positive sample, and the negative sample through a convolutional neural network to learn a good embedding vector space (Embedding), which can make similar samples gather in space and dissimilar samples disperse, thereby improving the performance of the model on similarity tasks. The embedding of the triplet network is composed of It means that it embeds the two-dimensional matrix x into the d-dimensional Euclidean space. In addition, this embedding is restricted to the d-dimensional hypersphere, that is, ||f(x)||2=1. The triplet loss function was proposed in the context of nearest neighbor classification. Here, it is ensured that a specific sample (anchor point) than it is with any sample of a different category (Counterexample) Closer to samples of the same class (positive example), therefore, the goal is:

[0085]

[0086] where α is the margin enforced between positive and negative pairs; For the set of all triplets in the training set, the minimized loss L is:

[0087]

[0088] Where N represents the total number of all triplets in the training set.

[0089] S4. Use the Softmax function to output the probability distribution of the test data and obtain the detection label; finally, use the moving average filter to eliminate sudden noise and isolated outliers, which helps to make the data more stable, reduce the impact of sudden noise, and make the data more continuous and smooth. At the same time, the collar technology is used to process the decision labels of epileptic seizures, and the labels of the epileptic seizure stage are appropriately extended to make up for the lack of epileptic seizure components. This means that this method will better capture the start and end of the seizure, and ensure that the judgment of epileptic seizures is more accurate and comprehensive, and finally output the classification results and post-processing.

[0090] In this embodiment, the Freiburg database in Germany is used, which contains EEG data of 21 patients. The data set is collected for patients with a history of intractable epilepsy, and these patients are recorded after stopping taking anti-epileptic drugs for a period of time. The EEG data of this data set are collected using the internationally accepted 10-20 EEG system electrode position method, with a resolution of 16 bits, a sampling frequency of 256Hz, and a multi-channel acquisition method. Four patients were selected from this data set for experimental verification. If the EEG data correctly judged by the algorithm as the epileptic seizure period and the intermittent period without seizures are defined as true positives and true negatives, respectively, and the EEG data mistakenly detected as the epileptic seizure period and the intermittent period without seizures are false positives and false negatives, respectively, then the three evaluation indicators are defined as follows:

[0091] Sensitivity = number of true positive segments / (number of true positive segments + number of false negative segments)

[0092] Specificity = number of true negative segments / (number of true negative segments + number of false negative segments)

[0093] Recognition accuracy = (the number of true positive segments + the number of true negative segments) / the total number of EEG segments used for testing. The test results are shown in Table 1.

[0094] Table 1 Detection results for 21 patients in the Freiburg database

[0095] Patient number Sensitivity (%) Specificity (%) Accuracy (%) 1 100 97.88 97.61 2 100 99.06 99.06 3 100 97.41 97.41 4 100 99.78 99.78 5 100 96.87 96.89 6 100 99.00 99.00 7 100 98.41 98.46 8 100 95.12 95.09 9 95.24 99.28 99.27 10 87.29 95.29 94.91 11 100 99.92 99.92 12 100 98.99 98.97 13 100 99.55 99.56 14 100 98.07 98.08 15 100 99.34 99.34 16 100 99.22 99.22 17 100 99.40 99.40 18 96.15 99.67 99.67 19 100 99.82 99.82 20 100 97.99 98.00 21 96.77 99.19 99.11 Mean 98.83 98.53 98.50

[0096] Experiments have shown that the algorithm has achieved excellent performance on the Freiburg database in Germany, with sensitivity, specificity and accuracy of 98.83%, 98.53% and 98.50% respectively.

[0097] Therefore, the present invention adopts the above-mentioned epilepsy detection method combining unrelated multilinear discriminant analysis and deep metric learning, proposes an automatic detection model for epileptic seizures based on unrelated multilinear discriminant analysis (UMLDA) and deep metric learning, and proposes triple network deep metric learning to improve the flexibility and generalization ability of the model. First, the original EEG is decomposed using the improved S transform (MST) time-frequency decomposition to construct a third-order tensor. Then, part of the data is used as training data, and the structure and features of the input data are more accurately extracted using UMLDA. Finally, using the triple network, by constructing anchor points, positive examples and negative examples, a convolutional neural network is used to learn the embedded representation, and the model is optimized by the triple loss function. Finally, the Softmax function is used for classification and post-processing to obtain the final experimental results.

[0098] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. An epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning, characterized in that: The following steps are involved: S1, collect raw EEG signals and preprocess them; S2, using the EEG signal processed by improved S transform time-frequency decomposition to form a third-order EEG tensor and extract EEG features through uncorrelated multilinear discriminant analysis; S3, epilepsy disease classification using triplet network metric learning classifier; S4. Use the Softmax function to output the probability distribution of the test data and obtain the detection label; S5. Use a moving average filter to eliminate sudden noise points and isolated outliers, and output classification results and post-processing.

2. The epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to claim 1, characterized in that: The standard deviation σ of the improved S-transform in step S2 is set to: Among them, λ represents the parameter used to adjust the standard deviation of the function; f represents the frequency; Therefore, the modified S transform is defined as: Among them, S x (t,f,λ) represents the signal The time-frequency transformation result is represents the current moment of signal analysis, represents the input signal, Indicates the center is The window function depends on f and λ. represents the modulation term of frequency f.

3. The epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to claim 1, characterized in that: The EEG signal is converted into a two-dimensional time-spectrogram consisting of time domain and frequency domain using the improved S transform. Then the channel is taken as one dimension and the time domain and frequency domain are taken as the other two dimensions to form a third-order EEG tensor, whose three dimensions are frequency, time and channel.

4. The epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to claim 1, characterized in that: The uncorrelated multilinear discriminant analysis in step S2 is a supervised learning algorithm, and its projection feature is The reduced-order vector features are defined as: A set of M training tensor objects samples For training, each tensor object The value of is in tensor space On, among which, I N is the n-modular dimension of the tensor, represents the Kronecker product; the goal of uncorrelated multilinear discriminant analysis is to find a TVP that contains P EMPs, where EMP is the unit multilinear projection factor, specifically Convert the original tensor space Mapped to a dimension The vector subspace of in So that in each EMP direction F p y are all maximized while satisfying the constraint that the P coordinate vectors are unrelated.

5. The epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to claim 1, characterized in that: The goal of uncorrelated linear discriminant analysis is to determine a set of P It is used to maximize the divergence ratio while generating features with zero correlation. Therefore, the objective function of the Pth EMP is: in, represents the transpose of the projection vector of the nth sample in the pth class, N p represents the number of samples in the pth class, F p represents the objective function, which is used to maximize the divergence ratio between classes; Subject to: Among them, g p and g q represents the eigenvectors of the pth and qth classes, ||g p || and ||g q || is the eigenvector g p and g q The norm of pq is the Kronecker delta symbol, which is 1 when p = q and 0 otherwise, and is used to indicate the orthogonality of the eigenvectors; P Determine in P steps in sequence, where the p-th step obtains the p-th EMP; for each EMP to be determined, estimate the projection vector for each n-mode separately The parameters of EMP are simplified to the following formula: in, represents the projection vector of the nth sample of the pth class, is the inter-class scatter matrix, which indicates the dispersion between categories. is the intra-class scatter matrix, which indicates the dispersion within the same class; Subject to: in, is the m sample matrix after projection, y (p) and (q) represents the projection samples of the pth and qth categories, y p (n) and q (n) Represents the projection result of the nth sample of the pth class and the qth class. This constraint is used to ensure that the projection vectors of different categories are orthogonal.

6. The epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to claim 1, characterized in that: The Fisher discriminant criterion in the linear discriminant analysis in step S2 is the divergence ratio of the vector samples, and the p-th projection feature Where M is the number of training samples, It is the pth For the mth sample The projection of and intra-class scatter for: Where C is the number of categories, N c is the number of samples of category c, c m is the category label of the mth training sample, and Therefore, for the pth scalar sample, the Fisher discriminant criterion is In addition, the pth coordinate vector is represented by g p , whose mth component is 7. The epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to claim 1, characterized in that: The triplet network in step S3 consists of anchor samples, positive samples and negative samples. The triplet loss between anchor samples, positive samples and negative samples is calculated through a convolutional neural network to learn an embedding vector space, which is composed of It means that the two-dimensional matrix x is embedded into the d-dimensional Euclidean space to ensure that a specific sample than it is different from any sample of different categories Close to samples of the same class Therefore, the objectives are: where α is the margin enforced between positive and negative pairs; For the set of all triplets in the training set, the minimized loss L is: Where N represents the total number of triplets in the training set.

8. The epilepsy detection method combining uncorrelated multilinear discriminant analysis and deep metric learning according to claim 1, characterized in that: While executing step S4 and step S5, the collar technology is used to process the decision labels of epileptic seizures and extend the labels of the epileptic seizure stage to compensate for the lack of epileptic seizure components.

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