Skip to main content
Log in

Effect of Number of Coupled Structures on the Segmentation of Brain Structures

Journal of Signal Processing Systems Aims and scope Submit manuscript

Abstract

This paper reports the effect of the coupling information on the performance of model-based segmentation of the brain structures from magnetic resonance images (MRI). We have developed a three-dimensional, nonparametric, entropy-based, and multi-shape method that benefits from coupling of the shapes. The proposed method uses principal component analysis (PCA) to develop shape models that capture structural variability and integrates geometrical relationship among different structures into the algorithm by coupling them (limiting their independent deformations). At the same time, to allow variations of the coupled structures, it registers each structure individually when building the shape models. It defines an entropy-based energy function which is minimized using quasi-Newton algorithm. Probability density functions (pdf) are estimated iteratively using nonparametric Parzen window method. In the optimization algorithm, analytical derivatives are used for maximum speed and accuracy. Sample results are given for the segmentation of caudate, thalamus, putamen, pallidum, hippocampus, and amygdala illustrating superior performance of the proposed method compared to the most similar method in the literature. The similarity of the results obtained by the proposed method with the expert segmentation is 4% to 12% higher than that of the most similar method. Experimental studies show that the proposed coupling method, which regulates shape variability during segmentation, improves accuracy of the results of the proposed method by 6% and those of the other method by 1%. In addition, the more the structures are used in the coupling process, the more accurate the results are obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5

Similar content being viewed by others

References

  1. Suri, J. S., Setarehdan, S. K., & Singh, S. (2002). Advanced algorithmic approaches to medical image segmentation, state of the art applications in cardiology, neurology, mammography and pathology. New York: Springer, February.

    Google Scholar 

  2. Macovski, A. (1996). Noise in MRI. Magnetic Resonance in Medicine, 36(3), 494–497.

    Article  Google Scholar 

  3. Kass, M., Witkin, A., & Terzopoulos, D. (1987). Snakes: Active contour models. International Journal of Computer Vision, 1(4), 259–268.

    Google Scholar 

  4. Angelini, E. D., Jin, Y., & Laine, A. F. (2005). State-of-the-art of level set methods in segmentation and registration of medical imaging modalities. The handbook of medical image analysis—volume III: Registration models. New York: Kluwer.

    Google Scholar 

  5. Caselles, V., Kimmel, R., & Sapiro, G. (1997). Geodesic active contours. International Journal of Computer Vision, 22(1), 61–79.

    Article  MATH  Google Scholar 

  6. Jacob, M., Blu, T., & Unser, M. (2004). Efficient energies and algorithms for parametric snakes. IEEE Transactions on Image Processing, 13(9), 1231–1244.

    Article  Google Scholar 

  7. Bresson, X. (2005). Image segmentation with variational active contours. Ph.D. thesis, University of Lausanne, July.

  8. Woolrich, M. W., & Behrens, T. E. (2006). Variational Bayes inference of spatial mixture models for segmentation. IEEE Transactions on Medical Imaging, 25(10), 1380–1391, October.

    Article  Google Scholar 

  9. Huang, A., Nielson, G. M., Razdan, A., Farin, G. E., Baluch, D. P., & Capco, D. G. (2006). Thin structure segmentation and visualization in three-dimensional biomedical images: a shape-based approach. IEEE Transactions on Visualization and Computer Graphics, 12(1), 93–102, January/February.

    Article  Google Scholar 

  10. Paragios, N. A. (2000) Geodesic active regions and level set methods. Ph.D. thesis, University of Nice, January.

  11. Leventon, M. E., Grimson, W. E. L., & Faugeras, O. (2000). Statistical shape influence in geodesic active contours. IEEE International Conference on Computer Vision and Pattern Recognition, 1, 1316–1323.

    Google Scholar 

  12. Jehan-Besson, S., Herbulot, A., Barlaud, M., & Aubert, G. (2005). Shape gradient for image and video segmentation, mathematical models in computer vision: The handbook. Berlin: Springer.

    Google Scholar 

  13. Pohl, K. M., Fisher, J., Kikinis, R., Grimson, W. E. L., & Wells, W. M. (2005). Shape based segmentation of anatomical structures in magnetic resonance images. International Conference on Computer Vision, 3765, 489–498.

    Google Scholar 

  14. Yang, J., Staib, L. H., & Duncan, J. S. (2004). Neighbor-constrained segmentation with level set based 3D deformable models. IEEE Transactions on Medical Imaging, 23(8), 940–948, August.

    Article  Google Scholar 

  15. Litvin, A., & Karl, W. C. (2005). Coupled shape distribution-based segmentation of multiple objects. Technical Report ECE-2005-01, March, Boston University, Boston, USA.

  16. Tsai, A., Wells, W., Tempany, C., Grimson, E., & Willsky, A. (2004). Mutual information in coupled multi-shape model for medical image segmentation. Medical Image Analysis, 8(4), 429–445, December.

    Article  Google Scholar 

  17. Kim, J., Fisher, J. W., Yezzi, A., Çetin, M., & Willsky, A. S. (2005). A nonparametric statistical method for image segmentation using information theory and curve evolution. IEEE Transactions on Image Processing, 14(10), 1486–1502, October.

    Article  MathSciNet  Google Scholar 

  18. Yan, P., & Kassim, A. A. (2006). Medical image segmentation using minimal path deformable models with implicit shape priors. IEEE Transactions on Information Technology in Biomedicine, 10(4), 677–684, October.

    Article  Google Scholar 

  19. Hong, B. W., prados, E., Soatto, S., & vese, L. (2006). Shape representation based on integral kernels: application to image matching and segmentation. Proceedings of the 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 1, 833–840.

    Google Scholar 

  20. Pizer, S., Fritsch, D., Yushkevich, P., Johnson, V., & Chaney, E. (1999). Segmentation, registration, and measurement of shape variation via image object shape. IEEE Transactions on Medical Imaging, 18, 851–865, October.

    Article  Google Scholar 

  21. Cootes, T. F., Taylor, C. J., Cooper, D. H., & Graham, J. (1995). Active shape models-their training and application. Computer Vision and Image Understanding, 61(1), 38–59 January.

    Article  Google Scholar 

  22. Colliot, O., Camara, O., & Bloch, I. (2006). Integration of fuzzy spatial relations in deformable models—Application to brain MRI segmentation. Pattern Recognition, 39(8), 1401–1414, August.

    Article  Google Scholar 

  23. Bijari, P. B., Akhoundi-Asl, A. R., & Soltanian-Zadeh, H. (2006). Interactive coupled object segmentation using symmetry and distance constraint. Presented to the Third Cairo International Biomedical Engineering Conference, Cairo.

  24. Maintz, B. A., & Viergever, M. A. (1998). A survey of medical image registration. Medical Image Analysis, 2(1), 1–36, April.

    Article  Google Scholar 

  25. Nelder, J. A., & Mead, R. (1965). A simplex method for function minimization. The Computer Journal, 7, 306–313.

    Google Scholar 

  26. Kim, J., Fisher, J., Yezzi, A., Cetin, M., & Willsky, A. (2002). Nonparametric methods for image segmentation using information theory. IEEE International Conference on Image Processing, 3, 797–800.

    Google Scholar 

  27. Parzen, E. (1962). On the estimation of a probability density function and the mode. Annals of Mathematical Statistics, 33, 1065–1076.

    Article  MATH  MathSciNet  Google Scholar 

  28. Bathe, K. J., & Cimento, A. P. (1980). Some practical procedures for the solution of nonlinear finite element equations. Computer Methods in Applied Mechanics and Engineering, 22, 59–85.

    Article  MATH  Google Scholar 

  29. Dice, L. R. (1945). Measures of the amount of ecologic association between species. Ecology, 26, 297–302.

    Article  Google Scholar 

  30. Guerra, C., & Pascucci, V. (1999). 3D segment matching using the Hausdorff distance. Seventh International Conference on Image Processing and Its Applications, 1, 18–22.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hamid Soltanian-Zadeh.

Appendix

Appendix

Here, we present details of finding the derivatives of the energy function with respect to the optimization parameters. First, details of computation of derivatives with respect to the principal shape coefficients are presented. Energy function has m + 1 regions but region m + 1 is constructed from the other m regions. Therefore, the derivative with respect to region m + 1 depends on the other m regions:

$$\begin{array}{*{20}l} {J\left( {\Omega _1 , \ldots ,\Omega _{m + 1} } \right) = J\left( P \right) = \sum\limits_{j = 1}^{m + 1} { - \int_{\Omega _j } {\log \hat p\left( {I\left( {\mathbf{x}} \right),\Omega _j } \right)d{\mathbf{x}}} } } \hfill \\ {\nabla _{w_i } J = \sum\limits_{j = 1}^m {\nabla _{w_i } \int_\Omega { - H\left( { - \Phi ^j \left( {\mathbf{x}} \right)} \right)\log \left( {\hat p_j \left( {\mathbf{x}} \right)} \right)d{\mathbf{x}}} } } \hfill \\ { + \nabla _{w_i } \int_\Omega { - \prod\limits_{k = 1}^m {H\left( {\Phi ^k \left( {\mathbf{x}} \right)} \right)\log \left( {\hat p_{m + 1} \left( {\mathbf{x}} \right)} \right)d{\mathbf{x}}} } } \hfill \\ { = \sum\limits_{j = 1}^m {\int_\Omega {\delta \left( {\Phi ^j \left( {\mathbf{x}} \right)} \right)\nabla _{w_i } \left( {\Phi ^j \left( {\mathbf{x}} \right)} \right)\log \left( {\hat p_j \left( {\mathbf{x}} \right)} \right)d{\mathbf{x}}} } } \hfill \\ { - \sum\limits_{j = 1}^m {\int_\Omega {H\left( { - \Phi ^j \left( {\mathbf{x}} \right)} \right)\nabla _{w_i } \left( {\log \left( {\hat p_j \left( {\mathbf{x}} \right)} \right)} \right)d{\mathbf{x}}} } } \hfill \\ { - \int_\Omega {\nabla _{w_i } \left( {\prod\limits_{k = 1}^m {H\left( {\Phi ^k \left( {\mathbf{x}} \right)} \right)} } \right)\log \left( {\hat p_{m + 1} \left( {\mathbf{x}} \right)} \right)d{\mathbf{x}}} } \hfill \\ { - \int_\Omega {\prod\limits_{k = 1}^m {H\left( {\Phi ^k \left( {\mathbf{x}} \right)} \right)} \nabla _{w_i } \log \left( {\hat p_{m + 1} \left( {\mathbf{x}} \right)} \right)d{\mathbf{x}}} } \hfill \\ \end{array} .$$
(11)

We assume that different regions have no intersection (which is generally correct). Therefore, we use the following equation for the derivative of region m + 1 with respect to the principal shape coefficients:

$$\nabla _{w_i } \left( {\prod\limits_{k = 1}^m {H\left( {\Phi ^k \left( {\mathbf{x}} \right)} \right)} } \right) = \sum\limits_{k = 1}^m {\delta \left( {\Phi ^k \left( {\mathbf{x}} \right)} \right)\nabla _{w_i } \Phi ^k \left( {\mathbf{x}} \right)} .$$
(12)

In addition, we have the following relations for the derivatives of the estimated pdf’s with respect to the parameters:

$$\begin{array}{*{20}l} {{\nabla _{{w_{i} }} \ifmmode\expandafter\hat\else\expandafter\^\fi{p}_{j} {\left( x \right)} = \nabla _{{w_{i} }} \frac{1}{{{\left| {\Omega _{j} } \right|}}}{\int_{\Omega _{j} } {K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} = } }} \hfill} \\ {{\nabla _{{w_{i} }} \frac{{{\int_\Omega {H{\left( { - \Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }}}{{{\int_\Omega {H{\left( { - \Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }}} = } \hfill} \\ {{\frac{{{\int_\Omega \delta }{\left( {\Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}\nabla _{{W_{i} }} {\left( {\Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} \times {\int_\Omega H }{\left( { - \Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}}}{{{\left( {{\int_\Omega H }{\left( { - \Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}^{2} }} - } \hfill} \\ {{\frac{{{\int_\Omega H }{\left( { - \Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} \times {\int_\Omega \delta }{\left( {\Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}\nabla _{{w_{i} }} {\left( {\Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}}}{{{\left( {{\int_\Omega H }{\left( { - \Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}^{2} }} = } \hfill} \\ {{\frac{{{\oint_{\Gamma _{j} } {\nabla _{{W_{i} }} {\left( { - \Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} \times {\int_{\Omega _{j} } {K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }} }}}{{{\left( {{\left| {\Omega _{j} } \right|}} \right)}^{2} }}} \hfill} \\ {{\frac{{{\oint_{\Gamma _{j} } {\nabla _{{W_{i} }} {\left( {\Phi _{j} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}\,\,} }K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}}}{{{\left| {\Omega _{j} } \right|}}},} \hfill} \\ \end{array} $$
(13)
$$\begin{array}{*{20}l} {{\nabla _{{w_{i} }} \ifmmode\expandafter\hat\else\expandafter\^\fi{p}_{{m + 1}} {\left( x \right)} = \nabla _{{w_{i} }} \frac{1}{{{\left| {\Omega _{{m + 1}} } \right|}}}{\int_{\Omega m + 1} {K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} = } }} \hfill} \\ {{\nabla _{{w_{i} }} \frac{{{\int_\Omega {{\prod\limits_{k = 1}^m {H{\left( {\Phi _{K} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}} }K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }}}{{{\int_\Omega {{\prod\limits_{k = 1}^m {H{\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }} }}} = } \hfill} \\ {{\frac{{{\int_\Omega {{\prod\limits_{k = 1}^m {H{\left( {\Phi _{K} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}} }d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} \times {\int_\Omega {{\sum\limits_{k = 1}^m \delta }{\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}\nabla _{{W_{i} }} {\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }} }}}{{{\left( {{\int_\Omega {{\prod\limits_{k = 1}^m {H{\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }} }} \right)}^{2} }} - } \hfill} \\ {{\frac{{{\int_\Omega {{\sum\limits_{k = 1}^m \delta }{\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}\nabla _{{W_{i} }} {\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} \times {\int_\Omega {{\prod\limits_{k = 1}^m {H{\left( {\Phi _{K} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}} }K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }} }}}{{{\left( {{\int_\Omega {{\prod\limits_{k = 1}^m {H{\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }} }} \right)}^{2} }}} \hfill} \\ {{ = {\sum\limits_{k = 1}^m {\frac{{{\oint_{\Gamma _{k} } {\nabla _{{W_{i} }} {\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }}}{{{\left| {\Omega _{{m + 1}} } \right|}}}} }} \hfill} \\ {{{\sum\limits_{k = 1}^m {\frac{{{\oint_{\Gamma _{k} } {\nabla _{{W_{i} }} {\left( {\Phi _{k} {\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x} \times {\int_{\Omega _{{m + 1}} } {K{\left( {I{\left( x \right)} - I{\left( {\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} \right)}} \right)}d\ifmmode\expandafter\hat\else\expandafter\^\fi{x}} }} }}}{{{\left| {\Omega _{{m + 1}} } \right|}^{2} }}.} }} \hfill} \\ \end{array} $$
(14)

Inserting Eqs. (13) and (14) into Eq. (11), Eq. (6) is obtained. For the computation of the derivatives with respect to\(p_i^k \), it is sufficient to consider the fact that these parameters depend on a single region.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akhondi-Asl, A., Soltanian-Zadeh, H. Effect of Number of Coupled Structures on the Segmentation of Brain Structures. J Sign Process Syst Sign Image Video Technol 54, 215–230 (2009). https://doi.org/10.1007/s11265-008-0196-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11265-008-0196-4

Keywords

Navigation