A method for locating a payload in a JPEG image based on ensemble learning

Abstract

The application provides a JPEG image load positioning method based on ensemble learning, and specifically comprises the following steps: step 1, collecting N pieces of load-bearing JPEG images based on the same embedding position key and embedded using the same DCT domain steganography algorithm; step 2, performing Huffman decoding on all images to obtain a quantized DCT coefficient matrix; combining the coefficients at the same position of all coefficient blocks in each image to obtain 64 pieces of same-frequency subgraphs; step 3, extracting 8-dimensional residual feature vectors for the same-frequency subgraphs in step 2; step 4, extracting 72-dimensional SPAM feature vectors for the same-frequency subgraphs in step 2; step 5, splicing the 8-dimensional residual feature vectors in step 3 and the 72-dimensional SPAM feature vectors in step 4 into 80-dimensional feature vectors; step 6, using stacking combined with K-fold cross-validation to perform model fusion; and using the 80-dimensional feature vectors in step 5 to train to obtain a final model.

Description

Technical Field

[0001] This invention relates to the field of image steganalysis, and more specifically to a JPEG image payload localization method based on ensemble learning. Background Technology

[0002] In today's era, the rapid development of digital multimedia technology, while bringing convenience, also brings new security threats. The rapid development of mobile big data has led to the dissemination of massive amounts of multimedia information on the network, making information hiding technology an important topic in the field of information security and attracting widespread attention from governments and research institutions around the world. Steganography, as an important branch of information hiding technology, has also received increasing attention. Embedded hidden information does not change the audiovisual effects of the carrier, nor does it change the size or format of the carrier file, thus enabling covert communication that is unknown to others. Unlike traditional encrypted communication, the fact that "communication is in progress" is itself hidden, thus possessing extremely strong concealment. With the rapid development of steganography, a large number of steganography methods have emerged. The abuse of steganography seriously threatens network information security. Therefore, researching anti-steganography technology is an urgent task facing researchers in related fields, among which the most important anti-steganography technology is steganalysis.

[0003] In our research, we typically describe steganalysis in four levels: 1. Determining whether an image contains hidden information; 2. Effectively estimating the size of the embedded information, i.e., the embedding rate; 3. Determining the embedding location of the hidden information (load localization); 4. Reconstructing the embedded hidden information. Currently, most steganalysis methods only focus on determining whether an image contains hidden information, while this invention focuses on load localization for images containing hidden information.

[0004] Current Status: Several advanced algorithms have proposed effective methods for locating the load in densely packed images in the spatial domain. In 2008, after studying and analyzing a large number of steganalysts with the same load location, Ker applied the idea of ​​Weighted Stego (WS) steganalysis to image steganography, proposing a localization method for LSBR: the WS residual method. This method uses a linear filter to estimate the carrier image, thereby calculating the average residual at each pixel embedding location to achieve load localization. This method first introduced the concept of residual, laying the foundation for subsequent steganalysis research. Subsequently, Ker and Lubenko proposed a load localization algorithm based on wavelet filter residuals for spatial LSB matching. This algorithm processes the densely packed image using wavelet transform and controls the low-frequency subband, then uses a wavelet absolute moment (WAM) filter to inversely transform the obtained wavelet residuals into spatial residuals, thereby achieving load localization. Following this, Quach proposed the Maximum aposterior (MAP) carrier estimation method, which improved the accuracy of carrier image estimation and enhanced the detection performance of WS and WAM for payload localization. In 2014, Quach further proposed a method based on Markov models or Markov random field models to model the carrier image, using pairwise constraints to capture the natural two-dimensional statistical information of the carrier image, and employing the Viterbi decoding algorithm and the Minimize Binary Variable Function (QPBO) algorithm for optimal estimation of the carrier image, further improving the estimation accuracy. In the same year, Yan Xiaomeng et al. proposed that the LSBM steganography payload localization problem can be regarded as a binary classification problem. Each pixel position of the encrypted image is used as a sample to be classified. The squared mean difference features of the adjacent pixels in eight directions of each pixel position in the encrypted image set are extracted by borrowing SPAM features. Then, an SVM classifier is used to perform binary classification on each pixel, thereby realizing the payload localization of LSBM steganography.

[0005] Unlike spatial embedding load localization, there are few reliable algorithms for load localization of JPEG images in the DCT domain (frequency domain). This is mainly because the correlation between adjacent DCT coefficients in JPEG images is weak, making it difficult to accurately estimate the carrier image, and traditional carrier estimation-based load localization methods would be very challenging.

[0006] To address this problem, Yang Chunfang et al. proposed a JPEG image carrier estimation method based on co-frequency subgraphs in 2020. This method estimates the carrier by constructing co-frequency subgraphs with stronger correlations, effectively improving the carrier estimation accuracy of JPEG images in the DCT domain. Pan Binmin et al., building upon the co-frequency subgraph method, used an adaptive algorithm to calculate weighting factors and obtain weighted residuals, further improving the accuracy of load location. Inspired by these methods, this invention proposes a JPEG image load localization method based on ensemble learning. Summary of the Invention

[0007] The two DCT domain load localization methods mentioned above are both based on data statistics, distinguishing between loaded and unloaded locations by setting a threshold for the residuals. Due to the limitations of the threshold comparison method, the performance of these two methods is not optimal and still has considerable room for improvement. Furthermore, these two methods require their own corresponding residual calculation methods for different embedding algorithms, lacking a universal residual calculation method when the embedding algorithm is unknown. This invention introduces machine learning, using a more efficient feature extraction method to achieve universal detection for different DCT domain embedding algorithms. Simultaneously, this invention uses the stacking method from ensemble learning for model fusion, and the fused model shows a significant improvement in classification accuracy compared to the threshold comparison method.

[0008] To obtain effective statistical features of JPEG images in the DCT domain, we construct co-frequency submaps by combining the coefficients between quantized DCT coefficient blocks and extract features from these submaps using two different methods. To improve classification accuracy, we fuse several mainstream classifiers using a stacking method; the fused model achieves better classification results than any single classifier. Considering the significant differences in DCT coefficient values ​​across frequency bands, we assign a classifier to each frequency band to prevent interference between samples during classification, resulting in better classification performance than a single classifier.

[0009] A JPEG image payload localization method based on ensemble learning specifically includes the following steps:

[0010] Step 1: Collect N JPEG images embedded using the same embedding location key and the same DCT domain steganography algorithm;

[0011] Step 2: Perform Huffman decoding on all images to obtain the quantized DCT coefficient matrix; combine the coefficients at the same position in all coefficient blocks of each image to obtain a total of 64 sub-images with the same frequency.

[0012] Step 3: Extract 8-dimensional residual features from the densely packed co-frequency subgraph in Step 2;

[0013] Step 4: Extract 72-dimensional SPAM features from the densely packed co-frequency subgraph in Step 2;

[0014] Step 5: Concatenate the 8-dimensional residual feature vector from Step 3 with the 72-dimensional SPAM feature vector from Step 4 to form an 80-dimensional feature vector.

[0015] Step 6: Use four classifiers—Lightweight Gradient Boosting Tree (LGBM), Support Vector Machine (SVM), Gradient Boosting Tree (GBM), and Logistic Regression (LR)—as base learners, and use Logistic Regression (LR) as the meta-learner. Use stacking combined with K-fold cross-validation to fuse the models. Train the model using the 80-dimensional feature vector from Step 5 to obtain the final model.

[0016] Preferably, step 3 includes the following sub-steps:

[0017] Step 3.1: Use wavelet filters to obtain the corresponding estimated carrier, and subtract the original carrier density co-frequency subgraph from the corresponding estimated carrier to obtain the residual;

[0018] Step 3.2: Use two different local variance formulas and two different weight calculation methods to obtain a total of four weights;

[0019] Step 3.3: Multiply the residual from step 3.1 with the four weights from step 3.2 to obtain four weighted residuals, and then use two different normalization methods to obtain the final 8-dimensional residual feature vector.

[0020] Preferably, the two different local variance formulas in step 3.2 include:

[0021] The formula for local variance is as follows:

[0022]

[0023] in Let be the DCT coefficients of the t-th co-frequency sub-image of the n-th densely packed image in the 3×3 neighborhood at position (i,j).

[0024] Estimated coefficients for the carrier at the corresponding location;

[0025] Local variance formula 2 is as follows:

[0026]

[0027] Unlike Formula 1, Formula 2 is calculated by subtracting the same fixed coefficient from the neighborhood DCT coefficient. This coefficient is the estimated coefficient of the carrier corresponding to the neighborhood center position.

[0028] The local variance is obtained using Equations 1 and 2, respectively. and

[0029] Preferably, step 4 includes the following sub-steps:

[0030] Step 4.1: Calculate the adjacency DCT coefficient difference matrix for each position in the eight directions;

[0031] Step 4.2: Model the adjacency DCT coefficient difference matrix as a Markov model, and set the difference threshold to 4: count the number of difference values ​​of -4, -3, ..., 0, ..., 3, 4. Each position can obtain 72 transition probabilities.

[0032] Step 4.3: Count the number of times the DCT coefficient at each location in all N images is non-zero, and use this as the number of valid embeddings at that location;

[0033] Step 4.4: Divide the transition probability in step 4.2 by the corresponding number of effective embeddings in step 4.3 to obtain the 72-dimensional SPAM feature vector.

[0034] Preferably, step 6 includes the following sub-steps:

[0035] Step 6.1: Define the input dataset containing m features x and corresponding labels y as... The base learner is denoted as h b The meta-learner is denoted as h. m The fusion classifier is denoted as H;

[0036] Step 6.2: Use cross-validation on the base learner of the first layer to generate a new dataset as input to the meta-learner of the second layer; randomly divide the original dataset d into K equal parts, each part is called a fold, d = {d1, d2, ..., d...} K};

[0037] Step 6.3: Select one fold that was not selected in the previous loop from the dataset as the test set, and the remaining K-1 folds as the training set. Train the four base learners of the first layer using the K-1 folds.

[0038] Step 6.4: Input the untrained test set into the trained base learner to obtain the prediction results, and use the prediction results along with the original data as a new data subset d`. k ={x` i ,y i},in

[0039] Step 6.5: After looping for K rounds, the loop ends. The newly generated subsets of data generated in the loop are merged into a new dataset d`= {d`1,d`2,···,d`K}, and use the new dataset to train the second-layer meta-learner to obtain

[0040] Step 6.6: Retrain the base learner of the first layer based on the original dataset d to obtain...

[0041] Step 6.7: The final prediction result of the fusion model H based on feature x on the dataset d is as follows:

[0042] The substantial advancements of this invention are as follows: 1) This invention solves the problem of universality in existing DCT domain load localization methods when facing different embedding algorithms. Existing methods all use threshold comparison methods to calculate residuals, requiring the use of their respective residual calculation methods when facing different embedding algorithms. They lack a universal residual calculation method when the embedding algorithm is unknown. This invention, however, does not require changing the feature extraction method based on the load image embedding method. In practical applications, it is often difficult to know the embedding method of the load image. Even if the embedding method is unknown, the feature extraction method provided by this invention can still extract effective features. 2) This invention improves upon the problem of existing DCT domain load localization methods requiring a large number of load images with the same embedding method. Because existing methods require calculating the average residual at each location to compare with a threshold to distinguish between load and non-load locations, a large number of statistical samples are needed to ensure the accuracy of the classification results. The feature extraction method and the method of assigning a classifier to each frequency band provided by this invention significantly reduce the requirement for the number of samples. With the same number of samples, the classification accuracy of this invention is significantly better than that of existing methods. Attached Figure Description

[0043] Figure 1 Overall flowchart of the JPEG image load localization method based on ensemble learning;

[0044] Figure 2 Diagram of the same frequency subgraph;

[0045] Figure 3 Residual feature extraction flowchart;

[0046] Figure 4 Residual feature composition structure diagram;

[0047] Figure 5 SPAM feature composition structure diagram;

[0048] Figure 6 Schematic diagram of the fusion model structure;

[0049] Figure 7 A schematic diagram of fusion model training / prediction. Detailed Implementation

[0050] The technical solution of the present invention will be further described in detail below through specific embodiments and in conjunction with the accompanying drawings.

[0051] Example 1

[0052] like Figure 1 As shown, step 0.1 assumes there are N JPEG images S of the same size, embedded using the same embedding location key. Huffman decoding is performed on all images to obtain the quantized DCT coefficient matrix. Let S be the DCT coefficient matrix of the nth image. n .

[0053] Step 0.2 Combine the coefficients at the same position in all coefficient blocks of each image to obtain 64 dense co-frequency sub-images S at 64 different positions. n (t), where t∈{1,···,64}.

[0054] like Figure 2 As shown, the residual feature set and SPAM feature set will be extracted from the 64 densely packed co-frequency sub-images.

[0055] Step 1: Calculate the residual feature set, such as Figure 3 and Figure 4 As shown, the specific steps are as follows:

[0056] Step 1.1 Use filters for each sub-image S n (t) is decomposed into a first-order wavelet to obtain the low-frequency subband cA(t), the horizontal subband cH(t), the vertical subband cV(t), and the diagonal subband cD(t).

[0057] Step 1.2 Calculate the maximum a posteriori probability estimate v for each position (i,j) in the low-frequency subband cA(t), horizontal subband cH(t), vertical subband cV(t), and diagonal subband cD(t) using windows of different sizes. (i,j) The formula is as follows:

[0058]

[0059] in Let W represent the squared mean of all coefficients in the N×N neighborhood centered at the coefficient at (i,j) in the wavelet subband W, where W∈{H,V,D}.

[0060] Step 1.3 Set the low-frequency subband cA(t) to 0, and perform quasiWiener filtering on each coefficient of the horizontal subband cH(t), vertical subband cV(t), and diagonal subband cD(t), as shown in the following formula:

[0061]

[0062] Among them, R w,(i,j) (t) represents the coefficient at (i,j) of the filtered wavelet subband W, w i,j This represents the coefficient at (i,j) of the wavelet subband W before filtering.

[0063] Step 1.4 convert the coefficient R w,(i,j) (t) Inverse wavelet transform, and use the value at each position after transformation as the original carrier coefficient estimate of the t-th frequency sub-image of the corresponding n-th carrier image at position (i,j).

[0064] Step 1.5 introduces a weighting coefficient w related to the variance of the carrier estimation to reduce the negative impact of the estimation in complex texture regions. n,(i,j) (t). Next, the local variance σ is calculated using two methods. n,(i,j) (t) and obtain the corresponding weight coefficients. The local variance formula 1 is as follows:

[0065]

[0066] in Let be the DCT coefficients of the t-th co-frequency sub-image of the n-th densely packed image in the 3×3 neighborhood at position (i,j). These are the estimated coefficients for the carrier at the corresponding positions.

[0067] Local variance formula 2 is as follows:

[0068]

[0069] Unlike Formula 1, Formula 2 is calculated by subtracting the same fixed coefficient from the neighborhood DCT coefficient. This coefficient is the estimated coefficient of the carrier corresponding to the neighborhood center position.

[0070] Step 1.6 Obtain the local variance based on the two formulas mentioned above. and Then, the weighting coefficients are calculated based on these two types of local variance, using the following formula:

[0071]

[0072] in Let be the local variance of the t-th sub-image of the same frequency in the n-th densely packed image at position (i,j) calculated using the variance formula k. This formula can be used to derive... and Two weighting coefficients.

[0073] Step 1.7 Note The LSB plane flipped image of the dense image is used to calculate the weighted residual using two residual formulas. Residual formula 1 is as follows:

[0074]

[0075] Residual Formula 2 is as follows:

[0076]

[0077] Where w n S(t) is the weight coefficient matrix of the t-th sub-image with the same frequency as the n-th densely packed image. n (t), These are the original image matrix and the carrier estimation coefficient matrix, respectively. Four types of residuals can be calculated here, denoted as follows: and

[0078] Step 1.8 Since "0" is an invalid embedding position in the DCT domain embedding rule, the number of times the coefficient at position (i,j) in the t-th frequency band is non-zero in all n images is counted and denoted as N. (i,j) (t) represents the number of valid statistics for that position.

[0079] Step 1.9 normalizes the four types of residuals using two different formulas to obtain the final 8-dimensional features. The formula for normalizing the effective mean is as follows:

[0080]

[0081] The formula for weighted average normalization is as follows:

[0082]

[0083] Where w k,(i,j) (t) represents the weight coefficient of the t-th sub-image of the same frequency in the k-th densely packed image at position (i,j), rs k,(i,j) (t) represents the corresponding residual.

[0084] The final calculated 8-dimensional features are used as the residual feature set.

[0085] Step 2 calculates the SPAM feature set, such as Figure 5 As shown, the specific steps are as follows:

[0086] Step 2.1 Label the variables of the image in 8 specific directions using the following symbols: ←, →, ↑, ↓, ↗, ↖, ↘, ↙, with the direction of the arrow in the image being the positive direction.

[0087] All differential pixel and probability transition calculations are always performed in the same direction. The following explanation uses the horizontal positive (→) direction as an example.

[0088] Step 2.2 Calculate the difference image matrix D, denoted as Di. n,(i,j) (t) represents the adjacent DCT coefficient difference at the t-th dense co-frequency sub-image (i,j) of the n-th image. Its calculation formula is as follows:

[0089]

[0090] Where I represents the quantized DCT coefficient at the corresponding location in the image.

[0091] Step 2.3 Calculate the first-order SPAM feature F 1st The differential adjacency DCT coefficient matrix is ​​modeled using the following formula:

[0092]

[0093] Where u∈{-T,···,T}, and T is the selected difference threshold. Specifically, if Then let

[0094] Step 2.4 To reduce the feature dimension, only differences with absolute values ​​no greater than 4 are counted. Therefore, the difference threshold T = 4 is chosen. Then, the number of squared differences in the → direction is counted, taking the values: -4, -3, ..., 0, ..., 3, 4.

[0095] Step 2.5 Count the number of times the coefficient at position (i,j) in the t-th frequency band is non-zero in all n images, and denote it as N. (i,j) (t).

[0096] Step 2.6 Calculate the effective average of the differential adjacency DCT coefficients at each location for all co-frequency sub-images of the n images, using the formula as the feature.

[0097]

[0098] "·" is a punctuation mark. The value in F represents the values ​​of the nine differentially adjacent pixels in that direction. · → The value in the middle represents the effective average of the nine differentially adjacent pixels. These nine values ​​are used as the 1- to 9-dimensional differential mean features of the adjacent DCT coefficients at that location.

[0099] Step 2.7 Perform the same statistical analysis on the other 7 directions ←, ↑, ↓, ↗, ↖, ↘, ↙ respectively, and use the results as the 10 to 72-dimensional neighboring DCT coefficient difference mean features for that position. Use these 72-dimensional features as the SPAM feature set.

[0100] Step 3 as follows Figure 6 and Figure 7 The model shown is a fusion model trained based on residual feature sets and SPAM feature sets:

[0101] Step 3.1 Take the 72-dimensional SPAM features as the first to 72-dimensional features of the multidimensional features, and take the 8-dimensional residual features as the 73rd to 80th-dimensional features of the multidimensional features. According to this rule, combine the residual feature set and the SPAM feature set into an 80-dimensional multidimensional feature set, and denote it as x.

[0102] Step 3.2: Construct a classifier for each frequency band, resulting in a total of 64 classifiers. Each classifier uses the following... Figure 6 The fusion model shown:

[0103] Step 3.3 uses four classifiers—Lightweight Gradient Boosting Tree (LGBM), Support Vector Machine (SVM), Gradient Boosting Tree (GBM), and Logistic Regression (LR)—as base learners, and Logistic Regression (LR) as the meta-learner. Model fusion is performed using stacking combined with K-fold cross-validation. The specific operation method is as follows:

[0104] Step 3.4 defines the input dataset containing m features x and corresponding labels y as... The base learner is denoted as h b The meta-learner is denoted as h. m The fusion classifier is denoted as H.

[0105] Step 3.5 Use cross-validation on the base learners of the first layer to generate a new dataset as input to the meta-learners of the second layer. Randomly divide the original dataset d into K equal parts (each part is called a "fold") d = {d1, d2, ..., d...} K}

[0106] Step 3.6, taking the k-th iteration as an example, select one fold that has not been selected in previous iterations from the dataset as the test set, and the remaining K-1 folds as the training set. Train the four base learners of the first layer using K-1 folds to obtain...

[0107] Step 3.7 Input the untrained test set into the trained base learner to obtain the prediction results, and use the prediction results along with the original data as a new data subset d`. k ={x` i ,y i},in

[0108] Step 3.8 ends after K rounds of looping, merging the newly generated data subsets in the loop into a new dataset d`={d`1,d`2,···,d` K}, and use the new dataset to train the second-layer meta-learner to obtain

[0109] Step 3.9 Retrain the base learner of the first layer based on the original dataset d to obtain

[0110] Step 3.10 The final prediction result of the fusion model H based on feature x on the dataset d is:

[0111] Step 4 uses the same steps 1 and 2 as above to extract features from the test data, and inputs them into the fusion model trained in step 3. This allows the DCT coefficients at all positions in the test data to be classified and the embedded position keys to be restored.

Claims

1. An integrated learning based method for locating a payload in a JPEG image, the method comprising: The method comprises the following steps: Step 1, collecting N stego JPEG images embedded based on the same embedding position key and using the same DCT domain steganography algorithm; Step 2, performing Huffman decoding on all images to obtain quantized DCT coefficient matrices; combining the coefficients at the same position of all coefficient blocks in each image to obtain 64 stego sub-images of the same frequency; Step 3, extracting 8-dimensional residual features for the stego sub-images of the same frequency in step 2; Step 3.1, obtaining a corresponding carrier estimate using a wavelet filter, and obtaining a residual by subtracting the corresponding carrier estimate from the original stego sub-image of the same frequency; Step 3.2, obtaining 4 weights using 2 different local variance formulas and 2 different weight calculation methods; Step 3.3, multiplying the residual in step 3.1 by the 4 weights in step 3.2 to obtain 4 weighted residuals, and then using 2 different normalization methods to obtain a final 8-dimensional residual feature vector; Step 4, extracting 72-dimensional SPAM features for the stego sub-images of the same frequency in step 2; Step 4.1, calculating the adjacent DCT coefficient difference matrix in 8 directions for each position; Step 4.2, Markov modeling the adjacent DCT coefficient difference matrix, and taking the difference threshold as 4: the number of difference values of -4, -3,..., 0,..., 3, and 4 is counted, and 72 transition probabilities can be obtained for each position; Step 4.3, counting the number of non-zero DCT coefficients for each position in all N images as the effective embedding number of the position; Step 4.4, dividing the transition probability in step 4.2 by the corresponding effective embedding number in step 4.3 to obtain a 72-dimensional SPAM feature vector; Step 5, concatenating the 8-dimensional residual feature vector in step 3 and the 72-dimensional SPAM feature vector in step 4 into an 80-dimensional feature vector; Step 6, using a light gradient boosting tree, a support vector machine, a gradient boosting tree and a logistic regression as a base learner, using a logistic regression as a meta-learner, using a stacking method combined with K-fold cross-validation to perform model fusion; and using the 80-dimensional feature vector in step 5 to train a final model.

2. The method of claim 1, wherein, The 2 different local variance formulas in step 3.2 comprise: The local variance formula 1 is as follows: wherein DCT coefficients within a 3x3 neighborhood of the t-th co-located sub- image of the n-th carrier image at position (i,j), are the carrier estimated coefficients for the corresponding positions; The local variance formula 2 is as follows: Different from formula 1, the calculation method of formula 2 is to subtract a same fixed coefficient from the neighborhood DCT coefficient, and the coefficient is a carrier estimate coefficient corresponding to the center position of the neighborhood; The local variance is obtained by Equation 1 and Equation 2, respectively and 3. The method of claim 1, wherein, The step 6 comprises the following sub-steps: Step 6.1, Define an input dataset containing m features x and corresponding labels y as The base learner is denoted as h b The meta-learner is denoted as h m The fusion classifier is denoted as H; Step 6.

2. Use cross-validation on the base learners of the first layer to generate a new dataset as input for the second layer meta-learner; randomly split the original dataset d into K equal parts, each part is called a fold, d = {d1, d2, ···, d K}. Step 6.

3. Select one fold in the dataset that has not been selected in the previous loop as the test set, and the remaining K-1 folds as the training set. Train the first layer of four base learners using the K-1 folds to obtain Step 6.

4. Input the test set not involved in training into the trained base learner to obtain the prediction result, and input the prediction result together with the original data as a new data subset d k = {x i , y i}, wherein Step 6.

5. After K rounds of the loop are completed, the newly generated data subsets in the loop are combined in a stacked manner into a new data set d` = {d`1, d`2, ···, d`K} and the meta-learner of the second layer is trained using the new data set to obtain K} and the meta-learner of the second layer is trained using the new data set to obtain Step 6.

6. Re-training the base learner of the first layer based on the original dataset d yields Step 6.

7. The final fused model H has a prediction result on the data set d based on the features x as

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