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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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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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:
Using this formula to get the derivation of \(c_{i}\) and \(\sigma _{i}^{2}\) formula, let them be zero:
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:
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
So, we get the derivation of \(b({{\varvec{x}}})\)
By substituting, Eqs. (11) and (12) into (5), the local statistic force \(\vec {F}_{\mathrm {local}}\) is described as:
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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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DOI: https://doi.org/10.1007/s11045-016-0436-x