发布时间: 2018-07-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.170575 2018 | Volume 23 | Number 7 图像处理和编码

 收稿日期: 2017-11-14; 修回日期: 2018-01-25 基金项目: 国家自然科学基金项目（11265007）；教育部回国人员科研启动基金项目（2010-1561）；云南省人培基金项目（KKSY201203030） 第一作者简介: 易三莉, 1977年生, 女, 讲师, 2011年于中南大学获生物医学工程专业博士学位, 主要研究方向为医学图像处理与分析。E-mail:152514845@qq.com;李思洁, 女, 硕士研究生, 主要研究方向为医学图像处理与分析。E-mail:596386364@qq.com;张桂芳, 女, 硕士研究生, 主要研究方向为图像处理与分析。E-mail:1439402142@qq.com. 中图法分类号: TP391 文献标识码: A 文章编号: 1006-8961(2018)07-1005-09

# 关键词

Application of weighted nuclear norm denoising algorithm in diffusion-weighted image
Yi Sanli, Li Sijie, He Jianfeng, Zhang Guifang
Institute of Information Engineering and Automation, Kunming University of Science and Technology, Kunming 650500, China
Supported by: National Natural Science Foundation of China (11265007); Scientific Research Starting Foundation for Returned Overseas Chinese Scholars, Ministry of Education, China (2010-1561)

# Abstract

Objective Diffusion-weighted imaging is a noninvasive method of detecting the diffusion of water molecules in living tissues and requires highly accurate data. Diffusion-weighted images have a high degree of self-similarity and rich feature details. The acquisition of diffusion-weighted images is often corrupted by noise and artifacts. Diffusion tensor images are calculated by the diffusion-weighted images. Meanwhile, diffusion tensor imaging is widely used in nerve fiber tracking in human brains. Noise affects the data accuracy of the diffusion tensor image and can cause erroneous tracking of fibers. Noise also affects subsequent processes. Therefore, the noise in the diffusion-weighted image should be reduced. Denoising is not only an important pre-processing step for many vision applications but also an ideal test bed for evaluating statistical image modeling methods. Method According to the characteristics of diffusion-weighted images, the weighted nuclear norm denoising algorithm is proposed for diffusion-weighted image denoising; this algorithm adopts the image nonlocal self-similarity. First, the diffusion-weighted image is divided into many target blocks, and the nonlocal similar blocks can be obtained from the entire image by block matching. The nonlocal similar blocks of the image can be obtained by a sufficiently large local window instead of the entire image. Second, the obtained nonlocal similar blocks are stacked into a similar block matrix, and then the similar block matrix is decomposed by a singular value decomposition. Large singular values are more important than small ones because they represent the energy of the major components of the image. Therefore, different singular values are assigned various weights. Third, the singular values obtained are shrunk by the soft-thresholding operator to acquire the denoised nonlocal similar blocks. The larger the singular values, the less they should be shrunk. By aggregating the denoised blocks, the target block can be estimated. Finally, by applying the above procedures to each target block and aggregating all blocks together, the denoised image can be reconstructed. Result The weighted nuclear norm denoising algorithm is compared with traditional diffusion-weighted image denoising algorithms, such as anisotropic algorithm and texture detection algorithm, by simulation and real data experiments. Simulation results show that the peak signal-to-noise ratio of the weighted nuclear norm denoising algorithm is at least 20 dB higher than those of the other traditional algorithms, and the structural similarity's value is 0.2~0.5 higher than those of the other algorithms. In the real data experiment, the neural fibers obtained by the tracking of the diffusion-weighted images denoised by different algorithms are compared. The use of the number of fibers or the length of the longest fiber to judge the effect of noise reduction fails to represent the noise reduction effect satisfactorily, according to our findings. Therefore, the average length of fibers is proposed to express the denoising effect. The longer the average length, the better the denoising effect and that the smoother the fibers. Results show that the average length and the texture of the nerve fibers obtained by denoising using the weighted nuclear denoising algorithm is sufficiently long and smooth, respectively. Conclusion An analysis of the experiment shows that the weighted nuclear norm denoising algorithm maximizes the self-similarity of the diffusion-weighted images and achieves image denoising through the processing of similar blocks. The weighted nuclear norm denoising algorithm can not only reduce the noise in the diffusion-weighted image and lead to visible peak signal-to-noise ratio improvements over state-of-the-art methods, such as texture detection, but also preserve the image's local structures better and generate less visual artifacts. The proposed algorithm can obtain improved results and DTI data accuracy and validity, which are helpful in the subsequent processing of images.

# Key words

diffusion weighted imaging; weighted nuclear norm denoising algorithm; image denoising; peak signal-to-noise ratio; nerve fiber tracking

# 1.1 扩散张量成像技术

 $\mathit{\boldsymbol{D = }}\left[{\begin{array}{*{20}{c}} {{D_{xx}}}&{{D_{xy}}}&{{D_{xz}}} \\ {{D_{xy}}}&{{D_{yy}}}&{{D_{yz}}} \\ {{D_{xz}}}&{{D_{yz}}}&{{D_{zz}}} \end{array}} \right]$ (1)

 $\left\{ \begin{gathered} AD{C_i} =-\frac{1}{{{b_i}}}\ln \frac{{{S_i}}}{{{S_0}}} \hfill \\ \mathit{\boldsymbol{G}}*\mathit{\boldsymbol{D}} = ADC \hfill \\ \end{gathered} \right.$ (2)

 ${\mathit{\boldsymbol{\hat X}}_j} = \arg \;\mathop {\min }\limits_{{\mathit{\boldsymbol{X}}_j}} \frac{1}{{\sigma _n^2}}\left\| {{\mathit{\boldsymbol{Y}}_j}-{\mathit{\boldsymbol{X}}_j}} \right\|_{\text{F}}^2 + {\left\| {{\mathit{\boldsymbol{X}}_j}} \right\|_{\omega, *}}$ (5)

 ${\left\| {{\mathit{\boldsymbol{X}}_j}} \right\|_{\omega, *}} = \sum\limits_i {{{\left| {{\omega _i}{\sigma _i}\left( \mathit{\boldsymbol{X}} \right)} \right|}_1}}$ (6)

 ${\omega _i} = c\sqrt n /\left( {{\sigma _i}\left( {{\mathit{\boldsymbol{X}}_j}} \right) + \varepsilon } \right)$ (7)

 ${\hat \sigma _i}\left( {{\mathit{\boldsymbol{X}}_j}} \right) = \sqrt {\max \left( {\sigma _i^2\left( {{\mathit{\boldsymbol{Y}}_j}} \right)-n\sigma _n^2, 0} \right)}$ (8)

 ${S_\omega }{\left( \mathit{\boldsymbol{ \boldsymbol{\varSigma} }} \right)_{ii}} = \max \left( {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_{ii}}-{\omega _i}, 0} \right)$ (9)

WNNM算法利用了图像本身非局部块之间的相似性，通过对含噪图像中的每个块进行块匹配处理得到相似块矩阵，再根据奇异值的特点，对相似块矩阵的奇异值进行收缩处理得到降噪块估计，该算法能够更好的保持图像局部结构，也就是能够保留图像中较多的纹理细节信息同时也能够较大程度的减少噪声。而DWI图像中含有较多的纹理细节，且其自相似性程度高，噪声对DWI图像产生的破坏则会对DTI数据的准确性造成一定的影响。因此，本文根据DWI图像的特点提出将WNNM算法应用于DWI图像的降噪，该算法能够尽可能多的保存DWI图像中的纹理细节信息，且能够较好的减少噪声，具有较好的降噪效果，提高了数据的准确性。

# 2.1 模拟数据

Table 1 Gradient coding for six coding directions

 $i$ ${x_i}$ ${y_i}$ ${z_i}$ 1 1.000 1.000 0.000 2 0.000 1.000 1.000 3 1.000 0.000 1.000 4 0.000 1.000 -1.000 5 1.000 -1.000 0.000 6 -1.000 0.000 1.000

Table 2 Evaluation values of the denoised images by using various denoising algorithms

 标准差 评价标准 PM算法 TV算法 各向同性 Wiener 纹理检测 WNNM $\sigma$=25 PSNR/dB 77.346 9 80.102 6 75.000 9 74.968 0 82.718 9 107.213 2 SSIM 0.513 3 0.776 1 0.404 6 0.481 8 0.818 1 0.998 2 MSE 0.001 7 0.001 2 0.002 3 0.002 2 0.000 7 0.000 4 $\sigma$=40 PSNR/dB 77.696 9 79.799 5 75.090 4 74.731 3 82.759 4 107.113 1 SSIM 0.516 4 0.763 0 0.403 9 0.491 3 0.839 4 0.992 6 MSE 0.001 8 0.001 4 0.002 7 0.002 2 0.000 9 0.000 6 $\sigma$=50 PSNR/dB 77.993 1 80.384 5 75.849 7 74.492 5 83.330 6 107.698 9 SSIM 0.531 0 0.784 3 0.404 1 0.465 4 0.832 6 0.989 9 MSE 0.002 6 0.002 2 0.003 2 0.002 2 0.001 9 0.001 1 $\sigma$=60 PSNR/dB 78.087 9 79.464 5 75.483 1 74.334 1 82.674 8 106.802 5 SSIM 0.520 3 0.713 2 0.414 1 0.430 8 0.786 6 0.972 4 MSE 0.003 3 0.003 3 0.004 8 0.002 4 0.003 1 0.003 1 注：加粗字体为在同一个评价标准下得到评价值的最优值。

# 2.2 真实数据

 $av{e_{{\text{length}}}} = \frac{{\sum\limits_{i = 1}^n {length\left( i \right)} }}{n}$ (10)

Table 3 Nerve fiber tracking data

 TV PM 各向同性 Wiener 纹理检测 WNNM 纤维数量/条 190 205 219 219 209 228 最长纤维长度/体素 43 34 52 48 47 43 平均长度/体素 20.7 20.502 4 21.109 6 20.429 2 20.411 5 21.330 4 注：加粗字体为纤维平均长度中的最大值。

# 参考文献

• [1] Zhang X Y, Peng J, Xu M, et al. Denoise diffusion-weighted images using higher-order singular value decomposition[J]. NeuroImage, 2017, 156: 128–145. [DOI:10.1016/j.neuroimage.2017.04.017]
• [2] Liu M Z, Vemuri B C, Deriche R. A robust variational approach for simultaneous smoothing and estimation of DTI[J]. NeuroImage, 2013, 67: 33–41. [DOI:10.1016/j.neuroimage.2012.11.012]
• [3] Haldar J P, Wedeen V J, Nezamzadeh M, et al. Improved diffusion imaging through SNR-enhancing joint reconstruction[J]. Magnetic Resonance in Medicine, 2013, 69(1): 277–289. [DOI:10.1002/mrm.24229]
• [4] Bao L J, Robini M, Liu W Y, et al. Structure-adaptive sparse denoising for diffusion-tensor MRI[J]. Medical Image Analysis, 2013, 17(4): 442–457. [DOI:10.1016/j.media.2013.01.006]
• [5] Lam F, Babacan S D, Haldar J P, et al. Denoising diffusion-weighted MR magnitude image sequences using low rank and edge constraints[C]//9th IEEE International Symposium on Biomedical Imaging. Barcelona, Spain: IEEE, 2012: 1401-1404. [DOI:10.1109/ISBI.2012.6235830]
• [6] Lam F, Babacan S D, Haldar J P, et al. Denoising diffusion-weighted magnitude MR images using rank and edge constraints[J]. Magnetic Resonance in Medicine, 2014, 71(3): 1272–1284. [DOI:10.1002/mrm.24728]
• [7] Lam F, Liu D, Song Z, et al. A fast algorithm for denoising magnitude diffusion-weighted images with rank and edge constraints[J]. Magnetic Resonance in Medicine, 2016, 75(1): 433–440. [DOI:10.1002/mrm.25643]
• [8] Becker S M A, Tabelow K, Voss H U, et al. Position-orientation adaptive smoothing of diffusion weighted magnetic resonance data (POAS)[J]. Medical Image Analysis, 2012, 16(6): 1142–1155. [DOI:10.1016/j.media.2012.05.007]
• [9] Becker S M A, Tabelow K, Mohammadi S, et al. Adaptive smoothing of multi-shell diffusion weighted magnetic resonance data by msPOAS[J]. NeuroImage, 2014, 95: 90–105. [DOI:10.1016/j.neuroimage.2014.03.053]
• [10] McGraw T, Vemuri B C, Chen Y, et al. DT-MRI denoising and neuronal fiber tracking[J]. Medical Image Analysis, 2004, 8(2): 95–111. [DOI:10.1016/j.media.2003.12.001]
• [11] Zhang X F, Ye H, Tian W F. Restoration of DTI images based on anisotropic diffusion[J]. Chinese Medical Equipment Journal, 2007, 28(5): 25–26, 31. [张相芬, 叶宏, 田蔚风. 基于各向异性扩散的DTI图像恢复[J]. 医疗卫生装备, 2007, 28(5): 25–26, 31. ] [DOI:10.3969/j.issn.1003-8868.2007.05.009]
• [12] Basser P J, Mattiello J, LeBihan D. MR diffusion tensor spectroscopy and imaging[J]. Biophysical Journal, 1994, 66(1): 259–267. [DOI:10.1016/S0006-3495(94)80775-1]
• [13] Stejskal E O, Tanner J E. Spin diffusion measurements:spin echoes in the presence of a time-dependent field gradient[J]. The Journal of Chemical Physics, 1965, 42(1): 288–292. [DOI:10.1063/1.1695690]
• [14] Maximov I I, Grinberg F, Shah N J. Robust tensor estimation in diffusion tensor imaging[J]. Journal of Magnetic Resonance, 2011, 213(1): 136–144. [DOI:10.1016/j.jmr.2011.09.035]
• [15] Gu S H, Zhang L, Zuo W M, et al. Weighted nuclear norm minimization with application to image Denoising[C]//Proceedings of 2014 IEEE Conference on Computer Vision and Pattern Recognition. Columbus, OH, USA: IEEE, 2014: 2862-2869. [DOI:10.1109/CVPR.2014.366]
• [16] Candès E J, Recht B. Exact matrix completion via convex optimization[J]. Communications of the ACM, 2012, 55(6): 111–119. [DOI:10.1145/2184319.2184343]
• [17] Martin-Fernandez M, Muñoz-Moreno E, Cammoun L, et al. Sequential anisotropic multichannel Wiener filtering with Rician bias correction applied to 3D regularization of DWI data[J]. Medical Image Analysis, 2009, 13(1): 19–35. [DOI:10.1016/j.media.2008.05.004]
• [18] Rafsanjani H K, Sedaaghi M H, Saryazdi S. An adaptive diffusion coefficient selection for image denoising[J]. Digital Signal Processing, 2017, 64: 71–82. [DOI:10.1016/j.dsp.2017.02.004]
• [19] Kim K H, Ronen I, Formisano E, et al. Robust fiber tracking method by vector selection criterion in diffusion tensor images[C]//Proceedings of the 26th Annual International Conference of Engineering in Medicine and Biology Society. San Francisco, CA, USA: IEEE, 2004: 1080-1083. [DOI:10.1109/IEMBS.2004.1403351]
• [20] Lai Y. Research on white matter fiber tracking in diffusion tensor imaging[D]. Suzhou: Soochow University, 2013. [赖昀. 基于弥散张量成像的脑白质纤维追踪算法研究[D]. 苏州: 苏州大学, 2013.] http://cdmd.cnki.com.cn/Article/CDMD-10285-1013229567.htm
• [21] Zhang X F. Study on DTI image denoising[D]. Shanghai: Shanghai Jiao Tong University, 2008. [张相芬. DTI图像去噪方法研究[D]. 上海: 上海交通大学, 2008.] http://www.wanfangdata.com.cn/details/detail.do?_type=degree&id=Y1415551