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

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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.

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Correspondence to Hamid Soltanian-Zadeh.



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:

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

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:

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

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} $$
$$\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} $$

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.

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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 (2009). https://doi.org/10.1007/s11265-008-0196-4

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  • Image segmentation
  • Brain structures
  • Shape modeling
  • Constrained deformation
  • Medical image processing
  • Magnetic resonance imaging (MRI)