Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2017 / Article

Research Article | Open Access

Volume 2017 |Article ID 6581279 |

Jian Zhao, Shuaishuai Fan, Jian Jia, Shunli Zhang, Bo Jiang, Wensheng Xu, Wanru Zhang, "Texture Directionality-Based Digital Watermarking in Nonsubsample Shearlet Domain", Mathematical Problems in Engineering, vol. 2017, Article ID 6581279, 14 pages, 2017.

Texture Directionality-Based Digital Watermarking in Nonsubsample Shearlet Domain

Academic Editor: Alessandro Lo Schiavo
Received23 Feb 2017
Revised23 Apr 2017
Accepted10 May 2017
Published12 Jun 2017


Digital watermarking is a technique used to protect an author’s copyright and has become widespread due to the rapid development of multimedia technologies. In this paper, a novel watermarking algorithm using the nonsubsample shearlet transform is proposed, which combines the directional edge features of an image. A shearlet provides an optimal multiresolution and multidirectional representation of an image based on distributed discontinuities such as edges, which ensures that the embedded watermark does not blur the image. In the proposed algorithm, the nonsubsample shearlet transform is used to decompose the cover image into directional subbands, where different directional subbands represent different directional and textured features. The subband whose texture directionality is strongest is selected to carry the watermark and is thus suitable for the human visual system. Next, singular value decomposition is performed on the selected subband image. Finally, the watermark is embedded in the singular value matrix, which is beneficial for the watermarking robustness and invisibility. In comparison with related watermarking algorithms based on discrete wavelet transforms and nonsubsample contourlet transform domains, experimental results demonstrate that the proposed scheme is highly robust against scaling, cropping, and compression.

1. Introduction

Given the rapid development of digital data and associated technologies, copyright protection of multimedia has become a challenging problem. In order to protect an author’s ownership, the method of digital watermarking (a digital data hiding technique) provides an effective solution. A good digital watermarking algorithm must ensure that the embedded watermark is visually imperceptible, the reconstructed watermarked image has good visual quality, and the watermark is resilient to attacks. In other words, a watermarking technique should resist watermark degradation due to common signal processing and geometric attacks.

A watermarking algorithm is regarded optimal if the scheme can balance the quality index of watermark robustness, imperceptibility, and capacity as proposed by Fridrich [1]. Watermarking can be performed in either the spatial domain (e.g., least significant bit) [2] or the transform domain (e.g., discrete Fourier transform (DFT) or discrete cosine transform (DCT)) [3, 4]. Moreover, the discrete wavelet transform (DWT) [5, 6] has the ability to capture 1D signal characteristics at different scales or resolutions, which makes the method widely applicable for image processing applications such as denoising and compression. However, DWT is unable to effectively represent multidimensional signals, and its limited directional filtering restricts watermarking capacity.

In order to overcome the limitations of the wavelet transform, several multiscale geometric analysis (MGA) tools such as ridgelet, directionlet, and contourlet transforms have been developed and successfully applied in image processing applications [712]. Shearlet, a multiresolution method, provides a tremendously effective representation of images containing edges [13, 14] and is widely used in watermarking algorithms [1518]. Spread spectrum image watermarking in the discrete shearlet domain [15] employs the multiresolution representation characteristics of the DST to facilitate texture modeling for a given image. Zhao et al. [16] proposed a watermarking scheme based on the extended shearlet, where the watermark was embedded into the subband with the largest information entropy. Based on previous studies [17], a watermarking algorithm was proposed using the shearlet transform and bidiagonal singular value to achieve balance between robustness and imperceptibility.

In this paper, a nonsubsample shearlet transform (NSST), along with a combination of Laplace and shear filters, is used to decompose an image into different scales and directional subbands. The watermark can then be embedded into a special directional subband that achieves the best result. The proposed watermarking scheme has high robustness and imperceptibility depending on texture directionality in the NSST domain. The texture of an image with strong directionality reflects the contours and directional edges of the image, and the directional filter bank (DFB) of the NSST can effectively capture high-frequency content. The selection of a subband with strong texture directionality plays a significant role in the embedding and extraction steps of the watermarking algorithm.

The rest of this paper is organized as follows. Basic shearlet theory and related data are presented in Section 2. The proposed novel watermarking embedding and detection scheme is presented in Section 3. Experimental results are provided and discussed in Section 4, while important findings and conclusions are summarized in Section 5.

2. Correlation with Basic Theory

2.1. Shearlet Transform

Shearlets, or composite wavelets, provide an effective way to achieve a sparse directional image representation based on affine transformations and a combination of geometry and multiple scales. For dimension , the affine system with composite dilations is described bywhere , A and B are invertible matrices, and . When A is the set , it is related to the scale transform, and the shear matrices are related to the area-preserving geometrical transform. The transform is described as [14]We select such thatwhere is a continuous wavelet for which with and is selected to satisfy with , on , and . Under this assumption, the associated continuous shearlet transform of is defined asSampling on the discrete set of , , and is done as follows [13, 16]:The associated discrete shearlet transform is described by

2.2. Nonsubsample Shearlet Transform (NSST)

The nonsubsample shearlet transform (NSST) is composed of the nonsubsample Laplacian pyramid transform and several different combinations of shearing filters. Figure 1(a) shows the flowchart for a nonsubsample shearlet where the Laplacian pyramid (LP) is used to capture point discontinuities and the directional filter bank (DFB) is used to link point discontinuities into linear structures. The DFB is designed to capture high-frequency content such as directional edges. An example of a frequency partition is shown in Figure 1(b).

A test image (zoneplate) is decomposed by NSST as shown in Figure 2. For LP decomposition, an approximate image is generated, and after DFB, 8 and 16 different directional subbands are obtained in the respective scale.

As observed in Figure 2, image directionality is strongly apparent after the shearlet transform. Different directional subbands represent the directional information related to different image details. Finally, the subband with the maximum texture directionality is selected for watermark embedding.

2.3. Singular Value Decomposition (SVD)

Singular value decomposition (SVD) is a numerical statistical analysis tool that is used to effectively deal with matrices. Assume that an image is a nonnegative matrix. The singular value decomposition of matrix is defined aswhere and are orthogonal matrices, is a diagonal matrix, and is the transpose of V. The diagonal elements of S satisfy , where is the rank of and is equal to the number of nonnegative singular values.

An important property of SVD is that the singular values (SVs) of an image possess significant stability because these values do not vary rapidly with small image perturbations. Therefore, a watermark embedded in a singular matrix exhibits high robustness [1923].

3. Proposed Watermarking Scheme

3.1. Texture Directionality

Texture is a property that exists in many areas. In the scope of image processing, texture denotes a trend or pattern that reflects a change in the intensity and direction of image pixels. Directionality is one of the most significant characteristic features. This paper proposes a method based on statistical measurements for calculating image directionality from the directional histogram [24, 25].

For a given image pixel, if the outputs of the horizontal and vertical edge operators are and , respectively, then the corresponding gradient vector isThus, the modulus and angle of the vector are obtained. The direction edge histogram is then calculated by quantizing and counting the ratio of effective pixels. A direction edge histogram for the Lena test image is shown in Figure 3(a).

Assuming that the variation trend denotes peaks in the histogram, Figure 3(b) shows one of these peaks. Let be a collection of bins from the previous valley to the next valley for each peak , and let be the angle position of the peak. is the bin height at angle position . Texture directionality is calculated as [24]The image texture direction is computed as a numerical value based on the above description. Figure 4 shows reconstructed images for eight shearlet direction coefficients, and the corresponding texture directionality values are shown in Table 1. The fourth subband with the largest value represents strong directionality and is selected to carry the watermark. The texture of an image with strong directionality reflects the contours and directional edges, and the directional filter bank of NSST can effectively capture high-frequency content. Further, due to its scale-invariant nature, stability does not change with scaling, rotation, or noise attacks. Thus, this property is an advantage for watermark hiding and robustness.



3.2. Proposed Watermarking Algorithm

Consider that the cover (host) image is a grayscale image with size , while the watermark is a binary image W with size and elements . The selection of a suitable location, effective coefficients to embed, and the embedding mechanism is of crucial importance in any watermarking scheme. In the proposed scheme, we select the eighth directional subband of level 3 with maximum directionality. This suggests that the subband has significant values to indicate the presence of directional edges, which contribute to watermark imperceptibility. Details on the embedding and extraction process are provided below.

3.2.1. Watermark Embedding

Step 1. In order to increase watermarking security, the encrypted watermark is obtained through the Arnold scrambling transform:where key represents scrambling times and is held to the decrypted watermark as the secret key.

Step 2. A host image with size is decomposed by 2-level NSST. An approximation image and directional subbands with size are obtained.

Step 3. The subband with maximum texture directionality is divided into nonoverlapping blocks of size , equal to the size of the watermark.

Step 4. SVD is performed on the block that has the largest energy , and the singular value matrix of size is obtained to achieve watermark embedding based on the rulewhere represents the singular value matrix containing watermarking information and is the embedding factor used to control imperceptibility. With an increasing value of , the amount of the embedded watermark is increased, which means that the values of SSIM (see (14)) become larger and that the distortion is more pronounced. Table 2 shows the relationship between values of and the watermark imperceptibility in terms of SSIM.


Step 5. The watermarked image is obtained by the recombination of modified singular values and inverse NSST.

3.2.2. Watermark Extraction

Watermark extraction is the inverse process of watermark embedding. Note that the original image is used here to extract the watermark. The steps of watermark extraction are provided below.

Step 1. A 2-level NSST is applied to the watermarked image.

Step 2. The subband with maximum directionality is divided into nonoverlapping blocks.

Step 3. SVD is applied to the block with the largest energy , and the singular value matrix is obtained.

Step 4. The singular value matrix is used to acquire the encrypted watermark as follows:

Step 5. The extracted watermark is obtained by the inverse Arnold transform in combination with key:

4. Experimental Analysis

A large number of varied test images that reflect different scenes were used to test the effectiveness of the scheme. Lena, Snow, Snow2, Plane, Baboon, Woman, Lake, Crowd, Peppers, Scenery, Man, and Bridge (each with size ) were selected as host images and a binary logo with size was selected as the watermark (Figure 5). The “pkva” filters of NSST are used with the number of directional subbands set as 8 and 16. Also, the values of and the secret key are set to 0.7 and 10, respectively, after numerous experimental trials. Structural similarity index measurement (SSIM) is used to measure visual quality and watermark robustness is assessed by normalized correlation (NC) and bit error rate (BER).where and are the averages of and , and are the variances of and , and is the covariance of and .where represents the elements of the original watermark and represents the elements of the extracted watermark.where is the number of erroneously detected bits for a watermark image.

4.1. Watermarking Invisibility

The performance of the proposed algorithm was tested under no attacks. Watermark imperceptibility in terms of SSIM is shown in Table 3. SSIM values between the host and watermarked images for the proposed algorithm are greater than 0.99 and reached 1 in some images, which shows good watermark invisibility. Further, the watermark can be extracted with little distortion. NC values of the extracted watermarks are greater than 0.99 and BER values are near 0, which illustrates effective watermark extraction.

Lena SSIM = 0.9997Snow SSIM = 0.9960Snow2 SSIM = 0.9998Plane SSIM = 0.9990

NC/BER = 0.9961/0.1953NC/BER = 0.9990/0.0488NC/BER = 0.9990/0.0488NC/BER = 0.9996/0.0268

Baboon SSIM = 0.9998Woman SSIM = 0.9995Lake SSIM = 0.9998Crowd SSIM = 0.9996

NC/BER = 0.9995/0.0244NC/BER = 1.0000/0.0000NC/BER = 1.0000/0.0000NC/BER = 0.9971/0.1465

Peppers SSIM = 0.9998Scenery SSIM = 0.9997Man SSIM = 0.9990Bridge SSIM = 0.9997


4.2. Watermarking Robustness

Tables 49 list the extracted watermarks for various attacks (scaling, JPEG compression, noise, and filtering) and the corresponding NC and BER values. Results show high robustness against various attacks, where the average NC and BER values are found to reach 0.985 and 0.37, respectively. The watermarking algorithm does not exhibit strong robustness for average filtering attack () and rotation attack, for which the average NC values are 0.96 and 0.95, respectively. The performance of the watermark for various attacks is analyzed in detail in the following subsections.

Test imageScale = 1/4
Scale = 1/2
Scale = 2
Scale = 4


Test imageJPEG = 30
JPEG = 40
JPEG = 75
JPEG = 90


Test imageVar = 0.001
Var = 0.005
Var = 0.01
Var = 0.05


Test imageVar = 0.001
Var = 0.005
Var = 0.01
Var = 0.05


Test imageAverage filtering
Average filtering
Median filtering
Median filtering


Test imageAngle = 5° 
Angle = 10° 
Angle = 15° 
Angle = 30° 


4.2.1. Robustness to Scaling Attacks

Scaling operations can be used to attack a watermarked image. Table 3 shows the values of SSIM, NC, and BER for scaling attacks with scaling parameters 1/4, 1/2, 2, and 4. Figure 6 shows the extracted watermark from the attacked images (Snow, Plane, and Crowd) for a scaling parameter of 1/2.

For these test images, the extracted watermarks are clearly distinguishable. NC is greater than 0.98 and average BER is 0.8. In comparison with the other images, the result for Crowd exhibits limited robustness, but its NC value reaches 0.96, and the watermark can be extracted clearly. Therefore, the proposed algorithm can resist scaling attacks.

4.2.2. Robustness to JPEG Compression Attacks

JPEG compression is a commonly used attack in image processing. Table 5 shows the values of NC and BER for JPEG compression with different quality factors. Results suggest that, irrespective of compression factor, NC values are greater than 0.94. Therefore, the proposed algorithm is robust against JPEG compression. Three perfectly extracted watermarks from a JPEG attack (quality factor ) are shown in Figure 7.

4.2.3. Robustness to Noise Attacks

In order to evaluate the robustness of the proposed scheme against noise, every watermarked image is attacked by “Gaussian” and “Salt and Pepper” noise with variances of 0.001, 0.005, 0.01, and 0.05. Results are shown in Tables 6 and 7. For a “Gaussian” noise attack, noise points are found to appear in the extracted watermarks when the noise variance increases. However, the extracted watermark is still distinguishable in all test images.

Similarly, for a “Salt and Pepper” noise attack, the algorithm shows good robustness. BER is 0 for Woman and Lake with a noise variance of 0.001, which illustrates tremendous robustness to noise attacks. Figures 8 and 9 show watermarks extracted from images attacked using “Gaussian” and “Salt and Pepper” noise with a variance of 0.001.

4.2.4. Robustness to Filtering Attacks

As shown in Table 8, for both average and median filtering, the extracted watermarks are distinguishable in the test images. Three examples are presented in Figure 10 for a median filtering attack. SSIM values of the watermarked image suggest that the image is easily distorted by the filtering attack. However, average NC is 0.975 and BER is 0.8, which illustrates that the proposed algorithm can resist the filtering attack.

4.2.5. Robustness to Rotation Attacks

The watermark is embedded into the most obvious directional subband. As shown by the SSIM values in Table 9, the loss of information is significant when the image is rotated. In order to illustrate the ability to resist a rotation attack, watermarked images are rotated by 5°, 10°, 15°, and 30°. With an increase in rotation angle, the quality of the extracted watermark is reduced. The average NC value is 0.95. Compared with other forms of image attacks, the quality of the extracted watermark is poorer. Figure 11 shows watermarks extracted from images attacked using rotation attack with 15°.

4.3. Comparative Analysis

In this section, the proposed algorithm is compared with methods based on the discrete wavelet transform domain (DWT-SVD) [20], nonsubsample contourlet transform domain (NSCT-SVD) [12], and shearlet transform domain (ST-SVD) [17]. Tables 10 and 11 and Figures 12 and 13 show a comparison of robustness for certain attacks on the four methods. The line chart indicates that NC values of the proposed NSST-SVD technique are the largest among the three methods for noise and filtering attacks. This enhancement in performance highlights the optimal nature of the shearlet transform for digital watermarking. NC for JPEG and scaling attacks is only marginally lower for the proposed algorithm compared to the others. Overall, the proposed algorithm displays noteworthy performance.

Comparison of NC values for four schemes

JPEG ()0.99711.00000.99380.9751
Gaussian noise (Var = 0.001)0.76730.96830.98590.9960
Salt and Pepper (Var = 0.001)0.82680.95540.96220.9961
Scaling (scale = 1/4)0.93540.98660.94270.9829
Average filtering ()0.91560.97310.91820.9888
Median filtering ()0.99800.93210.97180.9927

Comparison of NC values for four schemes

JPEG ()0.98800.99830.99260.9976
Gaussian noise (Var = 0.001)0.89890.97390.97180.9858
Salt and Pepper (Var = 0.001)0.84160.97470.99340.9995
Scaling (scale = 1/4)0.94410.99680.87420.9961
Average filtering ()0.92220.98010.84670.9858
Median filtering ()0.98030.95570.89360.9951

5. Conclusions

A novel watermarking algorithm based on texture directionality was proposed by combining the advantages of the nonsubsample shearlet and singular value decomposition. The strongest directionality subband, which provides the most significant texture information of the image, was selected for embedding the watermark since it preserves perceptual image quality. The proposed scheme is found to exhibit noteworthy robustness for most image processing attacks such as noise, rotation, cropping, and filtering. A comparison with two existing hybrid watermarking techniques showed better performance of the proposed method for many of the tested attacks.

Conflicts of Interest

The authors declare no competing financial interests regarding the publication of this paper.


This study was financially supported by the National Natural Science Foundation of China (nos. 61379010, 61572400, and 41601353), Natural Science Basic Research Plan in Shaanxi Province of China (no. 2015JM6293), and Scientific Research Program Funded by Shaanxi Provincial Education Department (16JK1765).


  1. J. Fridrich, “Applications of data hiding in digital images,” in Proceedings of the Fifth International Symposium on Signal Processing and Its Applications, Queensland, Australia. View at: Publisher Site | Google Scholar
  2. N. Bansal, V. K. Deolia, A. Bansal, and P. Pathak, “Digital image watermarking using least significant bit technique in different bit positions,” in Proceedings of the 2014 6th International Conference on Computational Intelligence and Communication Networks, CICN 2014, pp. 813–818, Bhopal, India, 2014. View at: Publisher Site | Google Scholar
  3. M. Cedillo-Hernandez, F. Garcia-Ugalde, M. Nakano-Miyatake, and H. Perez-Meana, “Robust object-based watermarking using SURF feature matching and DFT domain,” Radioengineering, vol. 22, no. 4, pp. 1057–1071, 2013. View at: Google Scholar
  4. Q. Su, G. Wang, S. Jia, X. Zhang, Q. Liu, and X. Liu, “Embedding color image watermark in color image based on two-level DCT,” Signal, Image and Video Processing, vol. 9, no. 5, pp. 991–1007, 2015. View at: Publisher Site | Google Scholar
  5. A. Mishra, C. Agarwal, A. Sharma, and P. Bedi, “Optimized gray-scale image watermarkingusing DWT-SVD and firefly algorithm,” Expert Systems with Applications, vol. 41, no. 17, pp. 7858–7867, 2014. View at: Publisher Site | Google Scholar
  6. M. Benouaret, A. Sahour, and S. Harize, “Real time implementation of a signal denoising approach based on eight-bits DWT,” AEU—International Journal of Electronics and Communications, vol. 66, no. 11, pp. 937–943, 2012. View at: Publisher Site | Google Scholar
  7. H. Sadreazami and M. Amini, “A robust spread spectrum based image watermarking in ridgelet domain,” AEU—International Journal of Electronics and Communications, vol. 66, no. 5, pp. 364–371, 2012. View at: Publisher Site | Google Scholar
  8. T. Wang and H. Li, “A novel scrambling digital image watermark algorithm based on double transform domains,” Mathematical Problems in Engineering, vol. 2015, Article ID 937432, 13 pages, 2015. View at: Publisher Site | Google Scholar
  9. H. Bi, Y. Liu, M. Wu, and Y. Ge, “NSCT domain additive watermark detection using RAO hypothesis test and cauchy distribution,” Mathematical Problems in Engineering, vol. 2016, Article ID 4065215, 18 pages, 2016. View at: Publisher Site | Google Scholar
  10. J. Liu, G. Liu, W. He, and Y. Li, “A new digital watermarking algorithm based on WBCT,” Procedia Engineering, vol. 29, pp. 1559–1564, 2012. View at: Publisher Site | Google Scholar
  11. F. Rahimi and H. Rabani, “A visually imperceptible and robust image watermarking scheme in contourlet domain,” in Proceedings of the 2010 IEEE 10th International Conference on Signal Processing, ICSP2010, pp. 1817–1820, Beijing, China, 2010. View at: Publisher Site | Google Scholar
  12. S.-M. Zhu and J.-M. Liu, “Novel blind watermarking scheme in contourlet domain based on singular value decomposition,” in Proceedings of the 2nd International Workshop on Knowledge Discovery and Data Mining (WKKD '09), pp. 672–675, Moscow, Russia, 2009. View at: Publisher Site | Google Scholar
  13. G. Easley, D. Labate, and W.-Q. Lim, “Sparse directional image representations using the discrete shearlet transform,” Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, vol. 25, no. 1, pp. 25–46, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  14. P. Kittipoom, G. Kutyniok, and W.-Q. Lim, “Construction of compactly supported shearlet frames,” Constructive Approximation. An International Journal for Approximations and Expansions, vol. 35, no. 1, pp. 21–72, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  15. B. Ahmederahgi, F. Kurugollu, P. Milligan, and A. Bouridane, “Spread spectrum image watermarking based on the discrete shearlet transform,” in Proceedings of the 4th European Workshop on Visual Information Processing (EUVIP '13), pp. 178–183, Paris, France, June 2013. View at: Google Scholar
  16. J. Zhao, M. Sun, J. Jia, L. Huang, F. Han, and S. Liu, “Image watermark based on extended shearlet and insertion using the largest information entropy on horizontal cone,” Mathematical Problems in Engineering, vol. 2015, Article ID 450819, 10 pages, 2015. View at: Publisher Site | Google Scholar
  17. M. Mardanpour and M. A. Z. Chahooki, “Robust transparent image watermarking with Shearlet transform and bidiagonal singular value decomposition,” AEU—International Journal of Electronics and Communications, vol. 70, no. 6, pp. 790–798, 2016. View at: Publisher Site | Google Scholar
  18. X. Wang, Y. Liu, S. Li, H. Yang, and P. Niu, “Robust image watermarking approach using polar harmonic transforms based geometric correction,” Neurocomputing, vol. 174, pp. 627–642, 2016. View at: Publisher Site | Google Scholar
  19. C.-C. Lin, C.-C. Chang, and Y.-H. Chen, “A novel SVD-based watermarking scheme for protecting rightful ownership of digital images,” Journal of Information Hiding and Multimedia Signal Processing, vol. 5, no. 2, pp. 124–143, 2014. View at: Google Scholar
  20. R. Mehta and N. Rajpal, “A hybrid semi-blind gray scale image watermarking algorithm based on DWT-SVD using human visual system model,” in Proceedings of the 6th International Conference on Contemporary Computing, (IC3 '13), pp. 163–168, Noida, India, 2013. View at: Publisher Site | Google Scholar
  21. B. L. Gunjal and S. N. Mali, “MEO based secured, robust, high capacity and perceptual quality image watermarking in DWT-SVD domain,” SpringerPlus, vol. 4, no. 1, pp. 1–16, 2015. View at: Publisher Site | Google Scholar
  22. A. Benhocine, L. Laouamer, L. Nana, and A. C. Pascu, “New images watermarking scheme based on singular value decomposition,” Journal of Information Hiding and Multimedia Signal Processing, vol. 4, no. 1, pp. 9–18, 2013. View at: Google Scholar
  23. K. Loukhaoukha, “On the security of digital watermarking scheme based on SVD and tiny-GA,” Journal of Information Hiding and Multimedia Signal Processing, vol. 3, no. 2, pp. 135–141, 2012. View at: Google Scholar
  24. M. M. Islam, D. Zhang, and G. Lu, “A geometric method to compute directionality features for texture images,” in Proceedings of the IEEE International Conference on Multimedia and Expo, (ICME '08), pp. 1521–1524, Hannover, Germany, 2008. View at: Publisher Site | Google Scholar
  25. Y.-W. Kim and I.-S. Oh, “Watermarking text document images using edge direction histograms,” Pattern Recognition Letters, vol. 25, no. 11, pp. 1243–1251, 2004. View at: Publisher Site | Google Scholar

Copyright © 2017 Jian Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.