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A new lattice Boltzmann algorithm for assembling local statistical information with MR brain imaging segmentation applications

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Abstract

Recently, the lattice Boltzmann (LB) model has been used in medical imaging segmentation as alternatives for level set methods. The advantages of the LB model include its simple programming and the fact that it is easily paralleled, which can shorten computing times during real-time processing. However, traditional LB algorithms often fail to segment magnetic resonance (MR) brain images, which usually contain low-contrast intensity levels, noise and bias field. To solve these problems, this paper proposes a new LB algorithm assembled by local statistical region information, which can increase the between-class variances of the foreground and background by reducing intra-class variations and can achieve a better anti-noise performance by smoothing the noise of neighborhood pixels. To test its effectiveness and efficiency, comparison experiments were carried out with other LB and non-LB algorithms. The results show that our algorithm was validated on synthetic images and real MR images with desirable performance in the presence of low-contrast gray levels and noise. It also achieved best segmentation performance (with Dice coefficient 97.9 %) compared to other algorithms (with Dice coefficient 80.33, 58.11, 88.2, 81.77, 96.1 % respectively). In addition, the computing speed of the new algorithm is acceptable (18.65–27.62 s).

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References

  • Al-Faris, A. Q., Ngah, U. K., Isa, N. A., & Shuaib, I. L. (2014). Computer-aided segmentation system for breast MRI tumour using modified automatic seeded region growing (BMRI-MASRG). Journal of Digital Imaging, 27(1), 133–144. doi:10.1007/s10278-013-9640-5.

    Article  Google Scholar 

  • Balla-Arabe, S., Gao, X. B., & Wang, B. (2013). A fast and robust level set method for image segmentation using fuzzy clustering and lattice Boltzmann method. IEEE Transactions on Systems Man & Cybernetics Part B Cybernetics A Publication of the IEEE Systems Man & Cybernetics Society, 43(3), 910–920. doi:10.1109/tsmcb.2012.2218233.

    Google Scholar 

  • Barkha, Bhansali, Sonam, T., & Savita, A. (2015). Hybrid method for image segmentation. International Journal of Computer Science and Information Technologies, 6(1), 514–518.

    Google Scholar 

  • Bereciartua, A., Picon, A., Galdran, A., & Iriondo, P. (2015). Automatic 3D model-based method for liver segmentation in MRI based on active contours and total variation minimization. Biomedical Signal Processing and Control, 20, 71–77. doi:10.1016/j.bspc.2015.04.005.

    Article  Google Scholar 

  • Bhatnagar, P. L., Gross, E. P., & Krook, M. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Physical Review, 94(3), 511–525.

    Article  MATH  Google Scholar 

  • Chen, J. H., Chai, Z. H., Shi, B. C., & Zhang, W. H. (2014). Lattice Boltzmann method for filtering and contour detection of the natural images. Computers & Mathematics with Applications, 68(3), 257–268. doi:10.1016/j.camwa.2014.05.023.

    Article  MATH  Google Scholar 

  • Chen, Y., Navarro, L., Wang, Y., & Courbebaisse, G. (2014). Segmentation of the thrombus of giant intracranial aneurysms from CT angiography scans with lattice Boltzmann method. Medical Image Analysis, 18(1), 1–8. doi:10.1016/j.media.2013.08.003.

    Article  Google Scholar 

  • Chupin, M., Hasboun, D., Poupon, F., Baillet, S., Garnero, L. (2002). Segmentation of the amygdalo-hippocampal complex by competitive region growing. In Proceedings of IEEE international symposium on biomedical imaging, (261–264).

  • Dakua, S. P., & Sahambi, J. S. (2011). Modified active contour model and random walk approach for left ventricular cardiac MR image segmentation. International Journal for Numerical Methods in Biomedical Engineering, 27(7), 1350–1361.

    MathSciNet  MATH  Google Scholar 

  • El-Dahshan, E. S. A., Mohsen, H. M., Revett, K., & Salem, A. B. M. (2014). Computer-aided diagnosis of human brain tumor through MRI: A survey and a new algorithm. Expert Systems with Applications, 41(11), 5526–5545. doi:10.1016/j.eswa.2014.01.021.

    Article  Google Scholar 

  • Frisch, U., d’Humieres, D., Hasslacher, B., Lallemand, P., Pomeau, Y., & Rivet, J. P. (1987). Lattice gas hydrodynamics in two and three dimensions. Complex Systems, 1(2), 649–707.

    MathSciNet  MATH  Google Scholar 

  • Gordillo, N., Montseny, E., & Sobrevilla, P. (2013). State of the art survey on MRI brain tumor segmentation. Magnetic Resonance Imaging, 31(6), 1426–1438. doi:10.1016/j.mri.2013.05.002.

    Article  Google Scholar 

  • Grady, L. (2006). Random walks for image segmentation. IEEE Transactions on Pattern Analysis & Machine Intelligence, 28(11), 1768–1783.

    Article  Google Scholar 

  • Guo, Z., & Zheng, C. (2009). Theory and applications of lattice Boltzmann method. Beijing: Science.

    Google Scholar 

  • Hagan, A., & Zhao, Y. (2009). Parallel 3D image segmentation of large data sets on a GPU cluster. In Proceedings of advances in visual computing, (pp. 960–969): Springer.

  • Hu, S., Coupe, P., Pruessner, J. C., & Collins, D. L. (2011). Appearance-based modeling for segmentation of hippocampus and amygdala using multi-contrast MR imaging. Neuroimage, 58(2), 549–559. doi:10.1016/j.neuroimage.2011.06.054.

    Article  Google Scholar 

  • Huang, R., Wu, H., & Cheng, P. (2013). A new lattice Boltzmann model for solid–liquid phase change. International Journal of Heat and Mass Transfer, 59(1), 295–301.

    Article  Google Scholar 

  • Li, C. M., Huang, R., Ding, Z. H., Gatenby, J. C., Metaxas, D. N., & Gore, J. C. (2011). A level set method for image segmentation in the presence of intensity inhomogeneities with application to MRI. IEEE Transactions on Image Processing A Publication of the IEEE Signal Processing Society, 20(5), 2007–2016. doi:10.1109/tip.2011.2146190.

    MathSciNet  Google Scholar 

  • Li, X., Jiang, D., Shi, Y., & Li, W. (2015). Segmentation of MR image using local and global region based geodesic model. Biomedical Engineering Online, 14(1), 1–16.

    Article  Google Scholar 

  • Ma, Z., Tavares, J. M. R., & Jorge, R. M. N. (2009). A review on the current segmentation algorithms for medical images. In Proceedings of the first international conference on computer imaging theory and applications, (pp. 135–140).

  • Mahmood, Q., Chodorowski, A., & Persson, M. (2015). Automated MRI brain tissue segmentation based on mean shift and fuzzy c-means using a priori tissue probability maps. IRBM, 36(3), 185–196. doi:10.1016/j.irbm.2015.01.007.

    Article  Google Scholar 

  • Pingen, G., Waidmann, M., Evgrafov, A., & Maute, K. (2009). A parametric level-set approach for topology optimization of flow domains. Structural and Multidisciplinary Optimization, 41(1), 117–131. doi:10.1007/s00158-009-0405-1.

    Article  MathSciNet  MATH  Google Scholar 

  • Rajendran, A., & Dhanasekaran, R. (2012). Fuzzy clustering and deformable model for tumor segmentation on MRI brain image: A combined approach. Procedia Engineering, 30(2), 327–333. doi:10.1016/j.proeng.2012.01.868.

    Article  Google Scholar 

  • Sachdeva, J., Kumar, V., Gupta, I., Khandelwal, N., & Ahuja, C. K. (2012). A novel content-based active contour model for brain tumor segmentation. Magnetic Resonance Imaging, 30(3), 694–715. doi:10.1016/j.mri.2012.01.006.

    Article  Google Scholar 

  • Shi, B., Deng, B., Du, R., & Chen, X. (2008). A new scheme for source term in LBGK model for convection-diffusion equation. Computers & Mathematics with Applications, 55(5), 1568–1575. doi:10.1016/j.camwa.2007.08.016.

    Article  MathSciNet  MATH  Google Scholar 

  • Sun, X. Y., Wang, Z., George, C. (2012). Parallel active contour with lattice Boltzmann scheme on morden GPU. In Proceedings of IEEE international conference on image processing.

  • Tsutahara, M. (2012). The finite-difference lattice Boltzmann method and its application in computational aero-acoustics. Fluid Dynamics Research, 44(2), 859–874.

    MathSciNet  MATH  Google Scholar 

  • Vn, P. R. (2012). Denoising of magnetic resonance and X-ray images using variance stabilization and patch based algorithms. International Journal of Multimedia & its Applications, 4(4), 53–71. doi:10.5121/ijma.2012.4605.

    Article  Google Scholar 

  • Wang, Z., Yan, Z., & Chen, G. (2011). Lattice Boltzmann method of active contour for image segmentation. In Proceedings of sixth international conference on image & graphics, (pp. 338–343). doi:10.1109/icig.2011.138.

  • Wen, J. L., Yan, Z. Z., & Jiang, J. H. (2014). Novel lattice Boltzmann method based on integrated edge and region information for medical image segmentation. Biomed Materials Engineering, 23(1), 1247–1252. doi:10.3233/BME-130926.

    Google Scholar 

  • Whittaker, E. T. (1967). On the partial difference equations of mathematical physics. LBM Journal Research, Development, 11(2), 215–234.

    Article  MathSciNet  Google Scholar 

  • Zhang, K. H., Zhang, L., & Zhang, S. (2010). A variational multiphase level set approach to simultaneous segmentation and bias correction. In Proceedings of image processing (ICIP), (pp. 4105–4108).

  • Zhang, Y., Matuszewski, B. J., Shark, L. -K., & Moore, C. J. (2008). Medical image segmentation using new hybrid level-set method. In Proceeding of fifth international conference biomedical visualization: Information visualization in medical & biomedical informatics, (pp. 71–76). doi:10.1109/MediVis.2008.12.

  • Zhu, S. C., & Yuille, A. (1996). Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(7), 884–900.

    Google Scholar 

Download references

Acknowledgments

This work was supported by the National Science Foundation of China under Grant No. 61171146, “Science foundation for outstanding young researchers” and “SRF for ROCS, SEM”, and in part by the Science and Technology Commission of Shanghai Municipality under Grants Nos. 13DZ1941203 and 15441905400. The authors would also like to express their appreciation to Mr. Xinghui Shu, for insightful input regarding the proposed research.

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Correspondence to Zhuangzhi Yan.

Appendix

Appendix

We use the maximum likelihood theory to estimate the variables \(c_{i}, \sigma _{i}^{2}\) and \(b({{\varvec{x}}})\). Suppose the image I will be segmented into \(\{{{\Omega }_{i}}:i=1,2,\cdots ,N\}\), so that \({\Omega =}\bigcap \nolimits _{{i=1}}^N {{\Omega }_{i}} , \bigcup \nolimits _{{i=1}}^N {\Omega } _{i}=\emptyset ,\) if \(i\ne j\). In this paper, \(N=2\).

Suppose the likelihood function of \({{\Omega }_{i}}\) is \(L\left( {\bar{I}\left( {{\varvec{x}}} \right) } \vert \theta _{i}\right) =\prod \nolimits _{x\in {{\Omega }_{i}}} {p(\bar{I}({{\varvec{x}}})\vert \theta _{i})} \), because \(p\left( {\bar{I}\left( {{\varvec{x}}} \right) } \vert \theta _{i}\right) =\prod \nolimits _{O_{x}\cap {{\Omega }_{i}}} {p(I({{\varvec{y}}})\vert \theta _{i})} \), so we can get \(L\left( {\bar{I}\left( {{\varvec{x}}} \right) } \vert \theta _{i}\right) =\prod \nolimits _{x\in {{\Omega }_{i}}} {p(\bar{I}({{\varvec{x}}})\vert \theta _{i})} =\prod \nolimits _{x\in {{\Omega }_{i}}} \prod \nolimits _{O_{x}\cap {{\Omega }_{i}}} {p(I({{\varvec{y}}})\vert \theta _{i})} \), then the logarithmic:

$$\begin{aligned} \ln \left( L\left( {\bar{I}\left( {{\varvec{x}}} \right) } \vert \theta _{i}\right) \right)= & {} \int _{{\Omega }_{i}} {\int _{{{O}_{x}}\cap {{\Omega }_{i}}}} {p(I({{\varvec{y}}})\vert \theta _{i})d{{\varvec{y}}}d{{\varvec{x}}}} \\= & {} -\int _{{\Omega }_{i}} \int _{O_{x}\cap {{\Omega }_{i}}} {\left( \mathrm{ln}\sqrt{2\uppi } \sigma _{i}+\frac{\left( I\left( {{\varvec{y}}} \right) -{b({{\varvec{x}}})c}_{i} \right) ^{2}}{2\sigma _{i}^{2}}\right) d{{\varvec{y}}}d{{\varvec{x}}}} \end{aligned}$$

Using this formula to get the derivation of \(c_{i}\) and \(\sigma _{i}^{2}\) formula, let them be zero:

$$\begin{aligned}&\frac{\partial \mathrm{ln} L}{\partial c_{i}}=\sum \nolimits _{i=1}^N \int _{{\Omega }_{i}} \int _{O_{x}\cap {\Omega }_{i}} {\left( \frac{\left( I\left( {{\varvec{y}}} \right) -{b({{\varvec{x}}})c}_{i} \right) }{2\sigma _{i}^{2}}\right) b({{\varvec{x}}})d{{\varvec{y}}}d{{\varvec{x}}}} =0\\&\frac{\partial \mathrm{ln} L}{\partial \sigma _{i}}=-\sum \nolimits _{i=1}^N \int _{{\Omega }_{i}} \int _{O_{x}\cap {{\Omega }_{i}}} {\left( -\frac{1}{2\sigma _{i}^{2}}-\frac{\left( I\left( {{\varvec{y}}} \right) -{b({{\varvec{x}}})c}_{i} \right) ^{2}}{2{(\sigma _{i}^{2})}^{2}}\right) d{{\varvec{y}}}d{{\varvec{x}}}}=0 \end{aligned}$$

Suppose \(K_{\rho }({{\varvec{x}}} , {{\varvec{y}}})\) to be the indicator function of the sampling window, \(M_{i}(\varphi )\) is the phase indicators of \({{\Omega }_{i}}\), which is defined as \(M_{i}\left( \varphi \right) =\left\{ \begin{array}{ll} H\left( \varphi \right) &{}i=1\\ 1-H\left( \varphi \right) &{}i=2\\ \end{array}\right. . H\left( \varphi \right) \) is a Heaviside functional, so we can get the maximum likelihood estimators as follows:

$$\begin{aligned} \left\{ {\begin{array}{ll} \tilde{c}_{i}&{}=\frac{\int _{{\Omega }_{i}} {K_{\rho }*b\left( {{\varvec{x}}} \right) I({{\varvec{y}}})M_{i}(\varphi ){\mathrm {d}}{} \mathbf{x}}}{\int _{{\Omega }_{i}} {K_{\rho }*b^{2}\left( {{\varvec{x}}} \right) M_{i}(\varphi ){\mathrm {d}}{} \mathbf{x}}} \\ \tilde{\sigma }_{i}^{2}&{}=\frac{\int _{{\Omega }_{i}} {K_{\rho }*{(I\left( {{\varvec{y}}} \right) -b\left( {{\varvec{x}}} \right) \tilde{c}_{i})}^{2}M_{i}(\varphi ){\mathrm {d}}{} \mathbf{x}}}{\int _{{\Omega }_{i}} {K_{\rho }*M_{i}(\varphi ){\mathrm {d}}{} \mathbf{x}}}\\ \end{array} } \right. \end{aligned}$$
(11)

Cause \(b({{\varvec{x}}})\) is related with the whole image, so we compute the joint likelihood function \(L\left( {\bar{I}\left( {{\varvec{x}}} \right) } \vert \theta _{i}\right) =\prod \nolimits _{i=1}^N \prod \nolimits _{x\in {{\Omega }_{i}}} {p(\bar{I}({{\varvec{x}}})\vert \theta _{i})} =\prod \nolimits _{i=1}^N \prod \nolimits _{x\in {{\Omega }_{i}}} \prod \nolimits _{O_{x}\cap {{\Omega }_{i}}} {p(I({{\varvec{y}}})\vert \theta _{i})} \), then the logarithmic

$$\begin{aligned} \ln \left( L\left( {\bar{I}\left( {{\varvec{x}}} \right) } \vert \theta _{i}\right) \right)= & {} \sum \limits _{i=1}^N \int _{{\Omega }_{i}} \int _{O_{x}\cap {{\Omega }_{i}}} {p(I({{\varvec{y}}})\vert \theta _{i})d{{\varvec{y}}}d{{\varvec{x}}}}\\= & {} -\sum \limits _{i=1}^N \int _{{\Omega }_{i}} \int _{O_{x}\cap {{\Omega }_{i}}} {\left( \mathrm{ln}\sqrt{2\uppi } \sigma _{i}+\frac{\left( I\left( {{\varvec{y}}} \right) -{b({{\varvec{x}}})c}_{i} \right) ^{2}}{2\sigma _{i}^{2}}\right) d{{\varvec{y}}}d{{\varvec{x}}}} \end{aligned}$$

So, we get the derivation of \(b({{\varvec{x}}})\)

$$\begin{aligned}&\frac{\partial \mathrm{ln} L}{\partial b({{\varvec{x}}})}=-\sum \nolimits _{i=1}^N \int _{{\Omega }_{i}} \int _{O_{x}\cap {{\Omega }_{i}}} {\left( \frac{I\left( {{\varvec{y}}} \right) -{b({{\varvec{x}}})c}_{i}}{\sigma _{i}^{2}}\right) c_{i}d{{\varvec{y}}}d{{\varvec{x}}}} =0\nonumber \\&\tilde{b}\left( {{\varvec{x}}} \right) =\frac{\sum \nolimits _{i=1}^N {K_{\rho }*I({{\varvec{y}}})M_{i}(\varphi ){\cdot }\frac{\tilde{c}_{i}}{\tilde{\sigma }_{i}^{2}}} }{\sum \nolimits _{i=1}^N {K_{\rho }*M_{i}(\varphi ){\cdot }\frac{\tilde{c}_{i}}{\tilde{\sigma }_{i}^{2}}} } \end{aligned}$$
(12)

By substituting, Eqs. (11) and (12) into (5), the local statistic force \(\vec {F}_{\mathrm {local}}\) is described as:

$$\begin{aligned} \vec {F}_{\mathrm {local}}=\left[ \sum \nolimits _{{y\in O}_{x}\cap {{\Omega }_{i}}} \left( \log \frac{\tilde{\sigma }_{2}}{\tilde{\sigma }_{1}}+\frac{\left( I\left( {{\varvec{y}}} \right) -\tilde{\mu }_{2} \right) ^{2}}{2\tilde{\sigma }_{2}^{2}}-\frac{\left( I\left( {{\varvec{y}}} \right) -\tilde{\mu }_{1} \right) ^{2}}{2\tilde{\sigma }_{1}^{2}} \right) \right] \vec {n} \end{aligned}$$

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Wen, J., Jiang, J. & Yan, Z. A new lattice Boltzmann algorithm for assembling local statistical information with MR brain imaging segmentation applications. Multidim Syst Sign Process 28, 1611–1627 (2017). https://doi.org/10.1007/s11045-016-0436-x

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