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A Type-2 Fuzzy Image Processing Expert System for Diagnosing Brain Tumors

  • Systems-Level Quality Improvement
  • Published:
Journal of Medical Systems Aims and scope Submit manuscript

Abstract

The focus of this paper is diagnosing and differentiating Astrocytomas in MRI scans by developing an interval Type-2 fuzzy automated tumor detection system. This system consists of three modules: working memory, knowledge base, and inference engine. An image processing method with three steps of preprocessing, segmentation and feature extraction, and approximate reasoning is used in inference engine module to enhance the quality of MRI scans, segment them into desired regions, extract the required features, and finally diagnose and differentiate Astrocytomas. However, brain tumors have different characteristics in different planes, so considering one plane of patient’s MRI scan may cause inaccurate results. Therefore, in the developed system, several consecutive planes are processed. The performance of this system is evaluated using 95 MRI scans and the results show good improvement in diagnosing and differentiating Astrocytomas.

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Author information

Authors and Affiliations

  1. Department of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran

    M. Zarinbal & M. H. Fazel Zarandi

  2. TOBB Economics and Technology University, Ankara, Turkey

    I. B. Turksen

  3. Knowledge Intelligent Systems Laboratory, University of Toronto, Toronto, Canada

    M. H. Fazel Zarandi & I. B. Turksen

  4. Sub-special Neurosurgery, Milad Hospital, Tehran, Iran

    M. Izadi

Authors
  1. M. Zarinbal
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  2. M. H. Fazel Zarandi
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  3. I. B. Turksen
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  4. M. Izadi
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Corresponding author

Correspondence to M. H. Fazel Zarandi.

Additional information

This article is part of the Topical Collection on Systems-Level Quality Improvement

Appendix

Appendix

As mentioned before, horizontal and vertical modes are two important modes of collaborative clustering methods. IT2RECFC method (Eq.(9)) can be rewritten as Eq.(19) for horizontal mode (HIT2RECFC) and Eq.(20) for vertical mode (VIT2RECFC):

$$ \begin{array}{l} \min {J}_{HIT2 RECFC}\left[ii\right]={\displaystyle \sum_{k=1}^N{\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}^m\left[ii\right]{d}_{ik}^2\left[ii\right]}}+{\gamma}_H\left[ii\right]\left({\displaystyle \sum_{k=1}^N{\displaystyle \sum_{i=1}^{c\left[ii\right]}{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P{U}_{ik}\left[ii\right] \ln \left(\frac{U_{ik}\left[ii\right]}{U_{ik}\left[jj\right]}\right)}}}\right)\\ {}\left\{\begin{array}{c}\hfill {\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}}\left[ii\right]=\mathbf{1}\kern1.5em \forall k,ii=1,\dots, P\kern1em \hfill \\ {}\hfill \begin{array}{l}\mathbf{0}<{\displaystyle \sum_{k=1}^N{U}_{ik}}\left[ii\right]\kern1.5em \forall i,ii=1,\dots, P\\ {}\mathbf{0}\le {U}_{ik}\left[ii\right]\le \mathbf{1}\kern1.5em \forall i,k,ii=1,\dots, P\end{array}\hfill \end{array}\right.\end{array} $$
(19)
$$ \begin{array}{l} \min {J}_{VIT2 RECFC}\left[ii\right]={\displaystyle \sum_{k=1}^{N\left[ii\right]}{\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}^m\left[ii\right]{d}_{ik}^2\left[ii\right]}}+{\gamma}_V\left[ii\right]\left({\displaystyle \sum_{k=1}^{N\left[ii\right]}{\displaystyle \sum_{i=1}^{c\left[ii\right]}{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P{U}_{ik}\left[ii\right] \ln \left(\frac{U_{ik}\left[ii\right]}{{\tilde{U}}_{ik}\left[jj\right]}\right)}}}\right)\\ {}\left\{\begin{array}{c}\hfill {\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}}\left[ii\right]=\mathbf{1}\kern1.5em \forall k,ii=1,\dots, P\kern1em \hfill \\ {}\hfill \begin{array}{l}\mathbf{0}<{\displaystyle \sum_{k=1}^{N\left[ii\right]}{U}_{ik}}\left[ii\right]\kern1.5em \forall i,ii=1,\dots, P\\ {}\mathbf{0}\le {U}_{ik}\left[ii\right]\le \mathbf{1}\kern1.5em \forall i,k,ii=1,\dots, P\end{array}\hfill \end{array}\right.\end{array} $$
(20)

where, γ H [ii] and γ V [ii] are nonnegative coefficients of collaboration for HIT2RECFC and VIT2RECFC, respectively.

Theorem 1-HIT2RECFC: U ik [ii] that optimizes Eq.(19) is obtained by:

$$ {U}_{ik}\left[ii\right]={\left({\left(\frac{\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]}}{W_0\left[\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]} \exp \left(-\left(m-1\right)\left(\frac{\gamma_H\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P\left(1- \ln \left({U}_{ik}\left[jj\right]\right)\right)}-{\varLambda}_k}{\gamma_H\left[ii\right]}\right)\right)\right]}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m-1$}\right.}\right)}^{-1} $$
(21)

or,

$$ \begin{array}{l}{U}_{ik}\left[ii\right]=\left[{\underline{u}}_{ik}\left[ii\right],{\overline{u}}_{ik}\left[ii\right]\right]=\\ {}\left[{\left({\left(\frac{\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]}}{W_0\left[\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]} \exp \left(-\left(m-1\right)\left(\frac{\gamma_H\left[ii\right]-{\overline{\lambda}}_k-{\gamma}_H\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({\underline{u}}_{ik}\left[jj\right]\right)}}{\gamma_H\left[ii\right]}\right)\right)\right]}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m-1$}\right.}\right)}^{-1}\right.,\\ {}\kern1em \left.{\left({\left(\frac{\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]}}{W_0\left[\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]} \exp \left(-\left(m-1\right)\left(\frac{\gamma_H\left[ii\right]-{\underline{\lambda}}_k-{\gamma}_H\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({\overline{u}}_{ik}\left[jj\right]\right)}}{\gamma_H\left[ii\right]}\right)\right)\right]}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m-1$}\right.}\right)}^{-1}\right]\end{array} $$
(22)

where, W 0(.) is the principle branch of Lambert-W function and

$$ {\gamma}_H\left[ii\right]= \max \left\{0,\frac{m{d}_{ik}^2\left[ii\right] \exp \left(\frac{m-1}{P}{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({U}_{ik}\left[jj\right]\right)}\right)-{\varLambda}_k}{-1+\left(1-\frac{1}{P}\right){\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({U}_{ik}\left[jj\right]\right)}}\right\} $$
(23)
$$ 0\le {\varLambda}_k\le \frac{1}{c\left[ii\right]}\left({\gamma}_H\left[ii\right]-\frac{\gamma_H\left[ii\right]}{e\left(m-1\right)}-{\gamma}_H\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({\overline{u}}_{ik}\left[jj\right]\right)}\right) $$
(24)

By considering d ik [ii] as Euclidean distance, the prototypes are updated by:

$$ {V}_i\left[ii\right]=\frac{{\displaystyle \sum_{k=1}^N{U}_{ik}^m\left[ii\right]{x}_k\left[ii\right]}}{{\displaystyle \sum_{k=1}^N{U}_{ik}^m\left[ii\right]}} $$
(25)

Proof:

Using Lagrangian multiplier, \( {\varLambda}_k=\left[{\underline{\lambda}}_k,{\overline{\lambda}}_k\right] \), the first constraint term in Eq.(19), \( \left({\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}}\left[ii\right]=\mathbf{1}\right) \), could be included in the objective function. Thus, minimization of the following function with respect to U[ii] is our concern:

$$ {J}_{HIT2 RECFC}\left[ii\right]={\displaystyle \sum_{k=1}^N{\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}^m\left[ii\right]{d}_{ik}^2}}\left[ii\right]+{\gamma}_H\left[ii\right]{\displaystyle \sum_{k=1}^N{\displaystyle \sum_{i=1}^{c\left[ii\right]}{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P{U}_{ik}\left[ii\right] \ln \left(\frac{U_{ik}\left[ii\right]}{U_{ik}\left[jj\right]}\right)}}}-{\displaystyle \sum_{k=1}^N{\varLambda}_k\left({\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}\left[ii\right]}-1\right)} $$
(26)

Minimizing Eq.(26) with respect to U[ii] is equivalent to minimizing individual objective function with respect to each U ik [ii], as the membership values of each observation in each cluster are independent of the other observations. The three necessary conditions leading to the local minimum are \( \frac{\partial {J}_{HIT2 RECFC}\left[ii\right]}{\partial {U}_{ik}\left[ii\right]}=\mathbf{0},\kern0.5em \frac{\partial {J}_{HIT2 RECFC}\left[ii\right]}{\partial {\gamma}_H\left[ii\right]}=\mathbf{0},\kern0.5em \frac{\partial {J}_{HIT2 RECFC}\left[ii\right]}{\partial {\varLambda}_k}=\mathbf{0}\kern1em \forall i,k \).

The first necessary condition, \( \frac{\partial {J}_{HIT2 RECFC}\left[ii\right]}{\partial {U}_{ik}\left[ii\right]}=\mathbf{0} \), results in:

$$ m{U}_{ik}^{\left(m-1\right)}\left[ii\right]{d}_{ik}^2+{\gamma}_H\left( \ln \left({U}_{ik}\left[ii\right]\right)-{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({U}_{ik}\left[jj\right]\right)}+1\right)-{\varLambda}_k=\mathbf{0} $$
(27)

Thus, for ii th data site, U ik [ii] is obtained by:

$$ {U}_{ik}\left[ii\right]={\left({\left(\frac{\frac{m\left(m-1\right){d}_{ik}^2}{\gamma_H}}{W_0\left[\frac{m\left(m-1\right){d}_{ik}^2}{\gamma_H} \exp \left(-\left(m-1\right)\left(\frac{\gamma_H-{\varLambda}_k-{\gamma}_H{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({U}_{ik}\left[jj\right]\right)}}{\gamma_H}\right)\right)\right]}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m-1$}\right.}\right)}^{-1} $$
(28)

The second necessary condition is \( \frac{\partial {J}_{HIT2 RECFC}\left[ii\right]}{\partial {\gamma}_H\left[ii\right]}=\mathbf{0} \), so:

$$ {\gamma}_H\left[ii\right]= \max \left\{0,\frac{m{d}_{ik}^2\left[ii\right] \exp \left(\frac{m-1}{P}{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({U}_{ik}\left[jj\right]\right)}\right)-{\varLambda}_k}{-1+\left(1-\frac{1}{P}\right){\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({U}_{ik}\left[jj\right]\right)}}\right\} $$
(29)

For the third necessary condition, \( \frac{\partial {J}_{RECFC}\left[ii\right]}{\partial {\varLambda}_k}=\mathbf{0} \), solving \( {\displaystyle \sum_{i=1}^{c\left[ii\right]}{U}_{ik}}\left[ii\right]=\mathbf{1} \) with respect to Λ k would not result in an exact solution, so the bounds have to be found. This problem could be studied from two viewpoints, (1) U ik [ii] ≥ 0 ∀ i, k and (2) U ik [ii] ≤ 1 ∀ i, k. As

$$ \frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]} \exp \left(-\left(m-1\right)\left(\frac{\gamma_H\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P\left(1- \ln \left({U}_{ik}\left[jj\right]\right)\right)}-{\varLambda}_k}{\gamma_H\left[ii\right]}\right)\right)>0 $$

U ik [ii] ≥ 0 ∀ i, k is always true. Now consider U ik [ii] ≤ 1 ∀ i, k:

$$ \begin{array}{l}-\frac{1}{e}\le -\left(m-1\right)\left(\frac{\gamma_H\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P\left(1- \ln \left({U}_{ik}\left[jj\right]\right)\right)}-{\varLambda}_k}{\gamma_H\left[ii\right]}\right)\le \\ {}\kern2em {W}_0\left[\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_H\left[ii\right]} \exp \left(-\left(m-1\right)\left(\frac{\gamma_H\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P\left(1- \ln \left({U}_{ik}\left[jj\right]\right)\right)}-{\varLambda}_k}{\gamma_H\left[ii\right]}\right)\right)\right]\end{array} $$

So, the bounds for \( {\varLambda}_k=\left[{\underline{\lambda}}_k,{\overline{\lambda}}_k\right] \) would be:

$$ 0\le {\underline{\lambda}}_k\le {\overline{\lambda}}_k\le \frac{1}{c\left[ii\right]}\left({\gamma}_H-\frac{\gamma_H}{e\left(m-1\right)}-{\gamma}_H{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({\overline{u}}_{ik}\left[jj\right]\right)}\right) $$
(30)

By differentiating Eq.(27) with respect to the prototypes, V i [ii] are obtained. Here, the d ik [ii] is considered Euclidean distance function, so the prototypes are updated by:

$$ {V}_i\left[ii\right]=\frac{{\displaystyle \sum_{k=1}^N{U}_{ik}^m\left[ii\right]{x}_k\left[ii\right]}}{{\displaystyle \sum_{j=1}^N{U}_{ik}^m\left[ii\right]}} $$
(31)

or,

$$ \left[{\underline{v}}_i\left[ii\right],{\overline{v}}_i\left[ii\right]\right]=\left[ \min \left({H}_i\left[ii\right]\right), \max \left({H}_i\left[ii\right]\right)\right] $$
(32)

where, \( {H}_i=\left(\frac{{\displaystyle \sum_{k=1}^N{\underline{u}}_{ik}^m\left[ii\right]{x}_k\left[ii\right]}}{{\displaystyle \sum_{k=1}^N{\underline{u}}_{ik}^m\left[ii\right]}},\frac{{\displaystyle \sum_{k=1}^N{\underline{u}}_{ik}^m\left[ii\right]{x}_k\left[ii\right]}}{{\displaystyle \sum_{k=1}^N{\overline{u}}_{ik}^m\left[ii\right]}},\frac{{\displaystyle \sum_{k=1}^N{\overline{u}}_{ik}^m\left[ii\right]{x}_k\left[ii\right]}}{{\displaystyle \sum_{k=1}^N{\underline{u}}_{ik}^m\left[ii\right]}},\frac{{\displaystyle \sum_{k=1}^N{\overline{u}}_{ik}^m\left[ii\right]{x}_k\left[ii\right]}}{{\displaystyle \sum_{k=1}^N{\overline{u}}_{ik}^m\left[ii\right]}}\right) \).

■End of proof.■

Theorem 2-VIT2RECFC: U ik [ii] that optimizes Eq.(20) is obtained by:

$$ {U}_{ik}\left[ii\right]={\left({\left(\frac{\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_V\left[ii\right]}}{W_0\left[\frac{m\left(m-1\right){d}_{ik}^2\left[ii\right]}{\gamma_V\left[ii\right]} \exp \left(-\left(m-1\right)\left(\frac{\gamma_V\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P\left(1- \ln \left({\tilde{U}}_{ik}\left[jj\right]\right)\right)}-{\varGamma}_k}{\gamma_V\left[ii\right]}\right)\right)\right]}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$m-1$}\right.}\right)}^{-1} $$
(33)

where,

$$ {\gamma}_V\left[ii\right]= \max \left\{0,\frac{m{d}_{ik}^2\left[ii\right] \exp \left(\frac{m-1}{P}{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({\tilde{U}}_{ik}\left[jj\right]\right)}\right)-{\varGamma}_k}{-1+\left(1-\frac{1}{P}\right){\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({\tilde{U}}_{ik}\left[jj\right]\right)}}\right\} $$
(34)
$$ 0\le {\varGamma}_k\le \frac{1}{c\left[ii\right]}\left({\gamma}_V\left[ii\right]-\frac{\gamma_V\left[ii\right]}{e\left(m-1\right)}-{\gamma}_V\left[ii\right]{\displaystyle \sum_{\begin{array}{l}jj=1\\ {}jj\ne ii\end{array}}^P \ln \left({\tilde{\overline{u}}}_{ik}\left[jj\right]\right)}\right) $$
(35)
$$ {\tilde{U}}_{ik}\left[jj\right]=\frac{1}{{\displaystyle \sum_{l=1}^c{\left(\frac{\left\Vert {x}_k\left[ii\right]-{V}_i\left[jj\right]\right\Vert }{\left\Vert {x}_k\left[ii\right]-{V}_l\left[jj\right]\right\Vert}\right)}^2}} $$
(36)

By considering d ik [ii] as Euclidean distance, the prototypes are updated by:

$$ {V}_i\left[ii\right]=\frac{{\displaystyle \sum_{k=1}^{N\left[ii\right]}{U}_{ik}^m\left[ii\right]{x}_k\left[ii\right]}}{{\displaystyle \sum_{k=1}^{N\left[ii\right]}{U}_{ik}^m\left[ii\right]}} $$
(37)

Proof: This theorem can be proved in the same manner as Theorem 1. ■

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Zarinbal, M., Fazel Zarandi, M.H., Turksen, I.B. et al. A Type-2 Fuzzy Image Processing Expert System for Diagnosing Brain Tumors. J Med Syst 39, 110 (2015). https://doi.org/10.1007/s10916-015-0311-6

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