Fatigue driving detection method based on double-source multi-scale feature fusion network

Abstract

The invention discloses a fatigue driving detection method based on a double-source multi-scale feature fusion network, and the method comprises the steps: carrying out the frequency division filtering of a collected electroencephalogram signal, extracting a differential entropy as a channel feature, and extracting a Spearman level correlation coefficient as a brain network feature; according to the method, channel features extracted after filtering are mapped and stacked, the channel features are converted into three-dimensional features, VSCMs of different kernel sizes are input, feature information of different scales is obtained, and compared with a mode of one-dimensional arrangement according to a channel sequence in the traditional technology, actual distribution of electrode channels in a human brain space is well considered, and the method has the advantages of being high in accuracy and high in robustness. The problem that the adjacent electrode channels in the human brain space cannot be normally used in the convolution process originally is solved; meanwhile, point-by-point convolution of different levels is fused by using an up-sampling layer, and the fused features are subjected to a series of convolution and then superposed through a convolution layer to obtain more comprehensive multi-scale features, so that the method is suitable for the condition of a small number of channels, and the fatigue driving detection rate is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of fatigue driving detection, and in particular relates to a fatigue driving detection method based on a dual-source multi-scale feature fusion network. Background Art

[0002] Driving fatigue usually affects the driver's perception, judgment and decision-making ability, reduces physical alertness and prolongs reaction time. Driving fatigue usually increases the probability of traffic accidents and is the main cause of traffic accidents. Driving fatigue accounts for 20% of traffic accidents and 40% of serious traffic accidents. Therefore, a reliable fatigue detection method is very important for driving safety. There are three main types of data used in fatigue detection: vehicle behavior characteristics, video, and physiological signals. Compared with the other two data, physiological signals are less susceptible to uncertain external factors. Among various physiological signals, electroencephalogram (EEG) signals are considered to be the gold standard for detecting brain activity and are widely used to detect driver fatigue. However, due to its nonlinearity and high sensitivity, analyzing and utilizing EEG remains a major challenge.

[0003] In recent studies, the extraction of multi-scale information from different features has emerged. Researchers extracted differential entropy (DE) from each EEG signal channel as the basic feature. Then, the feature matrix was constructed by biharmonic spline interpolation and the feature pyramid network (FPN) was used to improve the performance of electroencephalogram (EEG) emotion recognition. Some researchers constructed four EEG signal feature matrices by using preprocessed signals, differential entropy, symmetric difference and symmetric quotient to fuse different features to obtain more comprehensive feature information. Some researchers used azimuth equidistant projection (AEP) to generate two-dimensional EEG and extracted the average power, variance power and standard deviation power of the three frequency bands a, b and g as the feature data of the EEG feature map. Some methods proposed an ERP-WTC analysis method for EEG signal channel screening based on the original data, and designed the AM-PCNet network to extract the time domain, frequency domain and mixed domain information of the fusion spectrum of EEG signals and surface electromyography signals. Other methods extracted differential entropy (DE), power spectral density (PSD), nonlinear energy (NE) and fractal dimension (FD) to classify EEG signals through deep classification modules. In addition, some people have proposed a method to extract multi-scale spatiotemporal features of EEG signals using stacked multi-scale convolutional neural networks (MS_CNN) with spatially distributed raw EEG signals as input. It can be found that most of the methods proposed currently extract multi-scale information from raw data or several channel features, and do not make full use of EEG information from multiple sources.

[0004] In addition, most of the existing methods for extracting multi-scale information use feature pyramid structures. For example, SleepFC proposed by some researchers uses a convolutional feature pyramid network. Some methods use SleepyPyCo-backbone as the backbone network of the feature pyramid to consider multiple feature sequences at different time and frequency scales. Other methods propose a new EEG signal denoising model DTP-Net, which consists of a densely connected time pyramid (dtp) located at two learnable time-frequency transforms. There are even more methods that directly use feature pyramids to extract multi-scale features. However, this hierarchical increasing structure is designed for the image field and cannot adapt well to the characteristics of EEG data, and is not suitable for EEG data with a small number of channels. The original feature pyramid structure, in which the size of the convolution kernel increases with the number of layers, is designed for the image field and cannot adapt well to the characteristics of EEG data. Summary of the invention

[0005] The purpose of the present invention is to provide a fatigue driving detection method based on a dual-source multi-scale feature fusion network.

[0006] The present invention provides a fatigue driving detection method based on a dual-source multi-scale feature fusion network, which comprises the following steps: Step 1: Obtain EEG signals of different subjects to construct a data set; classify fatigue levels into different categories and label the data set with categories; Step 2: Filter the EEG signals in the data set at different frequency bands; calculate the differential entropy and Spearman rank correlation coefficient of the filtered EEG signals respectively, and obtain channel characteristics based on the differential entropy and Spearman rank correlation coefficient and brain-brain network characteristics Add to the dataset; Step 3: Construct a dual-source multi-scale feature fusion network; the dual-source multi-scale feature fusion network includes a brain-brain network feature extraction module, a channel feature extraction module and a fusion module; the inputs of the brain-brain network feature extraction module and the channel feature extraction module are the brain-brain network feature and channel characteristics ; The brain network feature extraction module and the channel feature extraction module both include multiple branches with the same structure; each branch includes a variable kernel size convolution module, a point-by-point convolution, and a convolution layer connected in sequence; the branches in the brain network feature extraction module and the channel feature extraction module are sorted from small to large according to the size of the convolution kernel in the variable kernel size convolution module; the point-by-point convolution is used to reduce the dimension of the input feature map, and the output feature map ;in, ; A is the number of branches; except for the Ath branch, the convolutional layer on each branch receives the feature map And the upsampled feature map As input; the convolutional layer on the last branch A only receives the feature map ; The brain network feature extraction module and the channel feature extraction module fuse multiple feature maps output by different branches through the brain network feature fusion module and the channel feature fusion module respectively; The fusion module is used to fuse the outputs of the brain network feature fusion module and the channel feature extraction module to obtain the prediction results; Step 4: Use the dataset with category labels to train the dual-source multi-scale feature fusion network; Step 5: Collect the subject's EEG signal, perform frequency division filtering and feature extraction on it, and then input it into the trained dual-source multi-scale feature fusion network to detect the subject's fatigue level.

[0007] Preferably, in step 2, the channel characteristics are obtained. and brain network characteristics The method is as follows: Mapping differential entropy one-to-one to 2D matrix, number of rows in the 2D matrix and number of columns Varies with the mapping method and stacked in frequency band order Channel characteristics of size ; Stack the Spearman rank correlation coefficients in frequency band order to get Brain network characteristics of size ;in, is the number of frequency bands; c is the number of channels of the EEG signal.

[0008] Preferably, in the step three, the variable kernel size convolution module includes two layers of convolutional networks connected in sequence, and each layer of the convolutional network includes a convolutional layer, a batch normalization layer and a ReLU activation function connected in sequence.

[0009] Preferably, in the step one, the eyelid closure duration percentage index is used to represent the subject's fatigue level, and the subject's fatigue level is divided into three categories according to the index thresholds of 0.35 and 0.7.

[0010] Preferably, in the step three, the brain network feature fusion module includes a flattening layer, a discard layer and a fully connected layer connected in series; the flattening layer is used to convert multiple feature maps output by the convolutional layer into a vector; the discard layer is used to reduce the risk of overfitting to improve the generalization ability of the model.

[0011] Preferably, in the step three, the channel feature fusion module includes a flattening layer and a fully connected layer connected in series; the flattening layer is used to convert multiple feature maps output by the convolutional layer into a vector.

[0012] Preferably, in the step three, the fusion module includes a fully connected layer and a softmax function connected in sequence; the input of the fully connected layer is a feature vector obtained by concatenating the outputs of the brain network feature fusion module and the channel feature extraction module; and the softmax function is used to predict the probability that the sample belongs to each category.

[0013] Preferably, in step 3, the number of branches in the channel feature extraction module and the brain network feature extraction module is respectively determined by the channel feature. and brain network characteristics The increase in the two-dimensional feature size.

[0014] Preferably, in step three, upsampling is performed using a nearest neighbor interpolation method.

[0015] Preferably, in step 4, a cross entropy loss function is used to guide the training, and the expression of the loss function Loss is: Among them, cross_entropy is the cross entropy function; α is the weight parameter; W is the parameter trained in the network; and j are the predicted probability and the true probability respectively; j=1,2,...,N; N is the number of categories of fatigue degree; ||·||2 is the symbol of L2 norm.

[0016] The present invention has the following beneficial effects: 1. The present invention uses an upsampling layer to fuse point-by-point convolutions of different levels, and uses a convolution layer to perform a series of convolutions on the fused features and then superimposes them to obtain more comprehensive multi-scale features. Compared with the traditional fusion method, the present invention more effectively realizes the fusion of feature information of different scales, is suitable for situations where the number of channels is small, and improves the fatigue driving detection rate; at the same time, the present invention extracts channel features and brain network features through DE differential entropy and Spearman rank correlation coefficient, respectively, and uses a brain network feature extraction module and a channel feature extraction module to extract features of the brain network features and channel features, respectively, effectively utilizing feature information of different sources and different scales.

[0017] 2. The present invention maps and stacks the channel features extracted after frequency division filtering, converts them into three-dimensional features, and then inputs VSCMs of different kernel sizes to obtain feature information of different scales. Compared with the one-dimensional arrangement of channels in the traditional technology, the present invention better considers the actual distribution of electrode channels in the human brain space, and solves the problem that electrode channels that are adjacent in the human brain space cannot be used normally during the convolution process. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] Figure 1Schematic diagram of extracting the frequency division feature of EEG signals in the present invention; Figure 2 Schematic diagram of the channel feature mapping method in the present invention.

[0019] Figure 3 Schematic diagram of the structure of the dual-source multi-scale feature fusion network in the present invention. DETAILED DESCRIPTION

[0020] The present invention will be further described below in conjunction with the accompanying drawings.

[0021] like Figure 1 As shown, a fatigue driving detection method based on a dual-source multi-scale feature fusion network includes the following steps: Step 1: Obtain the EEG data set and preprocess the EEG data; the EEG data set includes the SEED-VIG data set and the Cui data set; the SEED-VIG data set includes 23 subjects, the EEG sampling frequency is 1000HZ, which is reduced to 200HZ after preprocessing, and the number of EEG signal channels is 17. The PERCLOS (percentage of duration of eyelid closure) indicator is used to represent the degree of fatigue of the subject, and its value range is between 0 and 1. The larger the value, the more tired the subject is. According to the PERCLOS indicator thresholds of 0.35 and 0.7, the fatigue degree of the subject is divided into three categories, namely awake state (awake<0.35), tired state (0.35≤tired<0.7) and drowsy state (drowsy≥0.7). Every 8 seconds of EEG signal is a sample, and each sample is labeled with the category of fatigue degree.

[0022] The Cui dataset includes 2022 EEG samples from 11 subjects, each of which is 3 seconds long and has a sampling rate downsampled from 500 Hz of the original data to 128 Hz; the number of EEG electrode channels is 30. The reaction time is used to measure whether the subject is in a fatigue state, and the fatigue level of the subject is divided into two categories: fatigue and wakefulness. The Cui dataset is labeled with categories, and each category contains 1011 samples.

[0023] Step 2: Frequency division feature extraction like Figure 1 As shown, the EEG signals in the data set are filtered at different frequency bands to obtain the filtered EEG signals ; Where T is the number of sample points; c is the number of channels of the EEG signal; f is the number of frequency bands. Among them, the filtering on the SEED-VIG dataset is δ (1-3 Hz), θ (4-7 Hz), α (8-13 Hz), β (14-30 Hz) and γ (31-50 Hz), and the filtering on the Cui dataset is δ (1-4Hz), θ (4-8 Hz), α (8-13 Hz), β (13-30 Hz) Four frequency bands. In order to simultaneously utilize and compare the multiple types of features in the EEG data, the DE differential entropy and Spearman rank correlation coefficient are calculated for the filtered EEG data to obtain the two-dimensional features of the channel features (DE differential entropy) and brain network features (Spearman rank correlation coefficient). DE differential entropy and Spearman rank correlation coefficient The calculation formula is:

[0024] in, is the probability density function of channel x.

[0025] When channel x follows a Gaussian distribution, the probability density function The expression is: Here, σ is the standard deviation.

[0026] If the lengths of two consecutive channels x and y are both n, then the Spearman rank correlation coefficient The calculation formula is: in, ; is the EEG signal of channel x at the sample point at time i; y i is the EEG data of the sample point on channel y at time i; ; is the length of the channel.

[0027] Since the extracted point feature data is arranged in one dimension according to the channel order without considering the actual distribution of the electrode channels in the human brain space, for the convolution network, the electrode channels that are adjacent in the human brain space may be far apart in the one-dimensional arrangement, which makes the electrode channels that are originally adjacent in the human brain space unable to be used normally during the convolution process. Therefore, for the channel features, the actual distribution of the channels in the human brain space is used to map the one-dimensional point features into a two-dimensional matrix, such as Figure 2 As shown; by this method, the length The one-dimensional point features of In a two-dimensional matrix of size, the empty positions are filled with 0, and the number of rows in the two-dimensional matrix and number of columns These two-dimensional matrices are stacked in order of frequency bands to obtain All 3D features constitute T channel features. For brain network features, each Spearman rank correlation coefficient is The values ​​in these adjacency matrices represent the weights of the connections between EEG channels. These adjacency matrices are stacked in the order of frequency bands to obtain All 3D features constitute T brain network features. .

[0028] Step 3: Build a dual-source multi-scale feature fusion network like Figure 3 As shown in the figure, the dual-source multi-scale feature fusion network includes a brain network feature extraction module, a channel feature extraction module and a fusion module. The brain network feature extraction module includes four branches and a brain network feature fusion module that fuses the four branches; the four branches have the same structure, including VSCM modules (variable kernel size convolution modules), point-by-point convolution and convolution layers connected in sequence; the VSCM module is an important part of the dual-source multi-scale feature fusion network, which is used to extract brain network features. Feature extraction is performed; according to the size of the convolution kernel in the VSCM module, the four branches are divided into the first branch, the second branch, the third branch and the fourth branch in descending order, and the sizes of the convolution kernels are 1×1, 2×2, 3×3, and 4×4 respectively; the VSCM module includes two layers of convolutional networks connected in sequence, and each layer of convolutional network includes a convolutional layer, a batch normalization layer and a ReLU activation function connected in sequence; the number of convolutional kernels in the convolutional layers of the first layer of convolutional network and the second layer of convolutional network are 32 and 256 respectively; the convolutional layer is used to extract feature maps of different dimensions; since the convolutional layer is followed by the batch normalization layer, no bias is used here; the batch normalization layer is used to reduce the gradient disappearance and gradient explosion problems and accelerate the convergence speed of the network; the ReLU activation function is used to add nonlinear factors to improve the nonlinear expression ability of the neural network; the feature map output by the first layer of convolutional network on branch a for:

[0029] in, (·) is the convolution operation; (·) is batch standardization operation; (·) is processed by ReLU activation function; is the size of the convolution kernel, k∈{1,2,3,4}.

[0030] Feature map of the second convolutional network output on branch a for: The convolution kernel size of point-by-point convolution is 1×1, and the number of convolution kernels is 64; point-by-point convolution is used to reduce the dimension of the feature map output by the VSCM module to reduce the computational complexity and maintain the network depth; the feature map output by point-by-point convolution for: The convolution layer on each branch a (a≠A) receives the feature map And the feature map after upsampling ; The convolutional layer on the last branch A only receives feature maps ; Input of the convolutional layer for: in, is an upsampling operation.

[0031] The feature maps of two different scales are initially fused together through the convolution layer; the number of convolution kernels in the four convolution layers is 256, and the size of the convolution kernel is 3×3.

[0032] The brain network feature fusion module includes a flattening layer, a discarding layer, and a fully connected layer. The flattening layer is used to convert multiple feature maps output by the convolutional layer into a vector; the discarding layer is used to reduce the risk of overfitting to improve the generalization ability of the model; the output length of the fully connected layer is 100. The feature vector finally output by the brain network feature fusion module is for:

[0033] in, (·) is the flattening layer; (·) is the dropout layer; (·) is a fully connected layer.

[0034] Since the channel feature map is small in size after being converted into a two-dimensional matrix, the channel feature extraction module only includes three branches and a channel feature fusion module that fuses the three branches; the branch structure of the channel feature extraction module is the same as that of the brain network feature extraction module; the channel feature fusion module includes a flattening layer and a fully connected layer. The flattening layer is used to convert multiple feature maps output by the convolution layer into a vector; the output length of the fully connected layer is 100. The feature vector finally output by the channel feature extraction module is for:

[0035] The fusion module includes a fully connected layer and a softmax function connected in sequence; the input of the fully connected layer is the feature vector and The output length of the concatenated feature vector and the fully connected layer is the number of categories. The softmax function is used to predict the probability that the sample belongs to each category, and the maximum value of the predicted probability is taken as the prediction result y ̂.

[0036] Step 4: Training and evaluating dual-source multi-scale feature fusion network The SEED-VIG dataset and the Cui dataset are used to train and test the dual-source multi-scale feature fusion network respectively; the cross entropy loss function is used to guide the training, and L2 regularization is used to make the weight parameters smoother to alleviate the problem of overfitting. The expression of the loss function Loss is: in, is the cross entropy function; is the weight parameter; W is the parameter trained in the network; and are the true probability and predicted probability respectively; is the symbol of L2 norm; ; N is the number of categories for fatigue severity classification.

[0037] The initial learning rate is uniformly set to 0.001, the training set batch size is 16, the epoch is set to 50, and the L2 regularization coefficient is set to 0.01 and 0.0001 on the SEED-VIG dataset and the Cui dataset, respectively. The dropout parameter in all random dropout layers in the model is set to 0.5. The network constructed by the present invention and the existing network are tested using the SEED-VIG dataset and the Cui dataset, respectively.

[0038] Each subject in the SEED-VIG dataset has 885 samples. The five-fold cross-validation method is used independently on each subject to divide the dataset into five parts. Each part with 177 samples is used as a test set in turn, and the other samples are used as training sets. Finally, the results of the 23 subjects are summarized as the test results of the dataset; the test results of each network are shown in Table 1.

[0039] Table 1 Comparison with the current state-of-the-art methods on the SEED-VIG dataset (%) Since the number of samples for each subject in the Cui dataset is small, a total of 2022 samples from all subjects are mixed together for a five-fold cross-validation experiment. The dataset is divided into five parts, each with 404 samples as the test set in turn, and the other samples as the training set. Finally, the results of the five rounds of experiments are summarized as the test results of the dataset. The test results of each network are shown in Table 2.

[0040] Table 2 Comparison with the current state-of-the-art methods on the Cui dataset (%) As can be seen from Table 1 and Table 2, the accuracy of the present invention on the SEED-VIG dataset is 96.29%, and the accuracy on the Cui dataset is 92.83%; the present invention has obvious advantages over traditional machine learning methods (XGBoost, SVM, ACF-SSMM), and has increased by 13.45%, 9.19%, and 6.45% on the SEED-VIG dataset, respectively. At the same time, compared with the graph convolutional network methods (AMCNN-DGCN, RGNN, AMD-GCN, PGCNN, DGCNN), the present invention better combines the functional network and feature channel information, which makes the accuracy of the present invention increase by 35.47%, 9.76%, 6.35%, 5.15%, and 1.52% on the SEED-VIG dataset, and by 32.65%, 15.88%, 14.69%, and 4.85% on the Cui dataset, respectively. At the same time, compared with the common convolutional network methods (FBN-CNN, PSD-CNN), the present invention has also improved to a certain extent, and the accuracy on the SEED-VIG dataset has increased by 31.97% and 4.98% respectively. In general, the present invention is effective and exceeds the current state-of-the-art methods in terms of accuracy.

[0041] Step five, after preprocessing the EEG signal to be detected, the preprocessed EEG signal is frequency-filtered and features are extracted; the extracted features are input into the trained dual-source multi-scale feature fusion network model to generate the detection result of the fatigue degree of the subject.

Claims

1. A fatigue driving detection method based on a dual-source multi-scale feature fusion network, characterized by: The following steps are involved: Step 1: Obtain EEG signals of different subjects to construct a data set; classify fatigue levels into different categories and label the data set with categories; Step 2: Filter the EEG signals in the data set at different frequency bands; calculate the differential entropy and Spearman rank correlation coefficient of the filtered EEG signals respectively, and obtain channel characteristics based on the differential entropy and Spearman rank correlation coefficient and brain network characteristics Add to the dataset; Step 3: Construct a dual-source multi-scale feature fusion network; the dual-source multi-scale feature fusion network includes a brain network feature extraction module, a channel feature extraction module and a fusion module; the inputs of the brain network feature extraction module and the channel feature extraction module are the brain network feature and channel characteristics ; The brain network feature extraction module and the channel feature extraction module both include multiple branches with the same structure; each branch includes a variable kernel size convolution module, a point-by-point convolution, and a convolution layer connected in sequence; the branches in the brain network feature extraction module and the channel feature extraction module are sorted from small to large according to the size of the convolution kernel in the variable kernel size convolution module; the point-by-point convolution is used to reduce the dimension of the input feature map, and the output feature map ;in, ; A is the number of branches; Except for the Ath branch, the convolutional layer on each branch receives the feature map And the upsampled feature map As input; the convolutional layer on the last branch A only receives the feature map ; The brain network feature extraction module and the channel feature extraction module fuse multiple feature maps output by different branches through the brain network feature fusion module and the channel feature fusion module respectively; The fusion module is used to fuse the outputs of the brain network feature fusion module and the channel feature extraction module to obtain the prediction results; Step 4: Use the dataset with category labels to train the dual-source multi-scale feature fusion network; Step 5: Collect the subject's EEG signal, perform frequency division filtering and feature extraction on it, and then input it into the trained dual-source multi-scale feature fusion network to detect the subject's fatigue level.

2. The fatigue driving detection method based on a dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step 2, the channel characteristics are obtained and brain network characteristics The method is as follows: Mapping differential entropy one-to-one to 2D matrix, number of rows in the 2D matrix and number of columns Varies with the mapping method and stacked in frequency band order Channel characteristics of size ;Will The Spearman rank correlation coefficient of size is stacked in frequency band order to obtain Brain network characteristics of size ;in, is the number of frequency bands; c is the number of channels of the EEG signal.

3. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step three, the variable kernel size convolution module includes two layers of convolutional networks connected in sequence, and each layer of the convolutional network includes a convolutional layer, a batch normalization layer and a ReLU activation function connected in sequence.

4. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step 1, the eyelid closure duration percentage index is used to represent the subject's fatigue level, and the subject's fatigue level is divided into three categories according to the index thresholds of 0.35 and 0.

7.

5. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step three, the brain network feature fusion module includes a flattening layer, a discarding layer and a fully connected layer connected in series; the flattening layer is used to convert multiple feature maps output by the convolutional layer into a vector; the discarding layer is used to reduce the risk of overfitting to improve the generalization ability of the model.

6. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step three, the channel feature fusion module includes a flattening layer and a fully connected layer connected in series; the flattening layer is used to convert multiple feature maps output by the convolutional layer into a vector.

7. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step three, the fusion module includes a fully connected layer and a softmax function connected in sequence; the input of the fully connected layer is a feature vector obtained by concatenating the outputs of the brain network feature fusion module and the channel feature extraction module; and the softmax function is used to predict the probability that the sample belongs to each category.

8. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step 3, the number of branches in the channel feature extraction module and the brain network feature extraction module is respectively determined by the channel feature and brain network characteristics The increase in the two-dimensional feature size.

9. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step three, upsampling is performed using the nearest neighbor interpolation method.

10. The fatigue driving detection method based on dual-source multi-scale feature fusion network according to claim 1 is characterized in that: In the step 4, the cross entropy loss function is used to guide the training, and the expression of the loss function Loss is: in, is the cross entropy function; is the weight parameter; W is the parameter trained in the network; and are the predicted probability and the true probability respectively; ; N is the number of categories of fatigue level; is the symbol of L2 norm.

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