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Superpixel-based brain tumor segmentation in MR images using an extended local fuzzy active contour model

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Abstract

In this paper, to deal with poor boundaries in the presence of noise and heterogeneity of magnetic resonance (MR) images, a new region-based fuzzy active contour model based on techniques of curve evolution is introduced for the brain tumor segmentation. On the other hand, since brain MR images intrinsically contain significant amounts of dark areas such as cerebrospinal fluid, therefore for properly declining the heterogeneity of classes and better segmentation results, the proposed fuzzy energy-based function has been extended to consider three distinct regions; target, dark tissues with a dark background and the rest of the foreground. Moreover, due to the inevitable dependency of pixel-based models on the initial contour, artifact, and inhomogeneity of MR images, we have used superpixels as basic atomic units not only to reduce the sensitivity to the mentioned factors but also to reduce the computational cost of the algorithm. Results show that the proposed method outperforms the accuracy of the state-of-the-art models in both real and synthetic brain MR images.

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Authors and Affiliations

  1. Department of Electrical Engineering, University of Guilan, Rasht, Iran

    Niloufar Alipour & Reza P. R. Hasanzadeh

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  1. Niloufar Alipour
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  2. Reza P. R. Hasanzadeh
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Correspondence to Reza P. R. Hasanzadeh.

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Appendix

Appendix

This appendix contains the proof of energy differences (16) used by the proposed algorithm. Let us Consider a pixel, P, with intensity value I(P) and membership degrees \( {u}_{o_1} \) and \( {u}_{o_2} \). If we calculate the new membership degrees \( {u}_{n_1} \) and \( {u}_{n_2} \) using (15) for the point P and change its old membership degrees to the new values, the values of vi, i = 1,2,3 will be changed to new ones: \( \tilde{v}_{i},i \) = 1,2,3. The new values of vi, i = 1,2,3 can be calculated as:

$$ \tilde{v}_{i}(x)=\frac{\int_{\varOmega_y}W\left(x,y\right).{u}_i{(y)}^m\ I(y)\ dy}{\int_{\varOmega_y}W\left(x,y\right).{u}_i{(y)}^m\ dy}=\kern0.5em \frac{\sum_{\varOmega_y}W\left(x,y\right).{\left[{u}_i(y)\right]}^mI(y)+\left({u_{n_i}}^m\ I(P)-{u_{o_i}}^m\ I(P)\right)}{\sum_{\varOmega_y}W\left(x,y\right).{\left[{u}_i(y)\right]}^m+\left({u_{n_i}}^m-{u_{o_i}}^{\mathrm{m}}\right)} = \frac{s_i(x)\ {v}_i(x)+I(P)\left({u_{n_i}}^m-{u_{o_i}}^m\right)}{s_i(x)+\left({u_{n_i}}^m-{u_{o_i}}^m\right)}=\kern0.5em \frac{v_i(x)\ \left({s}_i(x)+{u_{n_i}}^m-{u_{o_i}}^m\ \right)-{v}_i(x)\ \left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)+I(P)\left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)}{s_i(x)+\left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)}=\kern0.5em {v}_i(x)+\frac{{u_{n_i}}^m-{u_{o_i}}^m}{s_i(x)+\left({u_{n_i}}^m-\kern0.5em {u_{o_i}}^m\right)}\ \left(I(P)-{v}_i(x)\right) $$
(21)

where \( {s}_i(x)={\sum}_{\varOmega_y}W\left(x,y\right).{\left[{u}_i(y)\right]}^m,\kern0.5em i=1,2 \). In a similar way, it is proven that the new \( {\overset{\sim }{v}}_3 \) is given by:

$$ {\overset{\sim }{v}}_3(x)={v}_3(x)+\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{o_1}-{u}_{o_2}\right)}^m}{s_3(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{o_1}-{u}_{o_2}\right)}^m}\ \left(I(P)-{v}_3(x)\right) $$
(22)

where \( {s}_3(x)={\sum}_{\varOmega_y}W\left(x,y\right).{\left[1-{u}_1(y)-{u}_2(y)\right]}^m \).

Thus, the changed values \( \Delta {v}_i=\tilde{v}_{i}-{v}_i \) for the point P can be easily computed using formulation (21) for i = 1, 2 and (22) for i = 3. Furthermore, changing the membership degrees of point P to the new values will lead to a change in the model energy. Assuming that the new energy is denoted by \( \overset{\sim }{F} \):

(23)

We will separately examine \( {\overset{\sim }{A}}_1 \) and \( {\overset{\sim }{B}}_1 \) to formulate our result. Therefore,

$$ {\overset{\sim }{A}}_1={\sum}_{\varOmega_y}W\left(x,y\right).{\left[\tilde{u}_{i}(y)\right]}^m\ {\left(I(y)-\tilde{v}_{i}(x)\right)}^2={\sum}_{\varOmega_y}W\left(x,y\right)\ \left[{\left[{u}_i(y)\right]}^m\ {\left(I(y)-\tilde{v}_{i}(x)\right)}^2+\left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(I(P)-\tilde{v}_{i}(x)\right)}^2\right] $$
(24)

By substituting (21) into (24), we obtain following equation:

$$ {\overset{\sim }{A}}_1=\sum \limits_{\varOmega_y}W\left(x,y\right).\left[{\left[{u}_i(y)\right]}^m\ {\left(I(y)-{v}_i(x)-\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\ \left(I(P)-{v}_i(x)\right)\right)}^2+\kern0.5em \left({u_{n_i}}^m-{u_{0_i}}^m\right){\left(I(P)-{v}_i(x)-\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2\right]=\sum \limits_{\varOmega_y}W\left(x,y\right).\left[{\left[{u}_i(y)\right]}^m\ \left[{\left(I(y)-{v}_i(x)\right)}^2+{\left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2\ {\left(I(P)-{v}_i(x)\right)}^2-2\ \left(I(y)-{v}_i(x)\right)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)\ \left(I(P)-{v}_i(x)\right)\right]\right]+\sum \limits_{\varOmega_y}W\left(x,y\right).\left[\left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(I(P)-{v}_i(x)\right)}^2\ \frac{{\left[{s}_i(x)\right]}^2}{{\left({s}_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)\right)}^2}\right]={A}_1+{s}_i(x){\left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\ \left(I(P)-{v}_i(x)\right)\right)}^2-2\left({s}_i(x){v}_i(x)-{s}_i(x){v}_i(x)\right)\left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)\left(I(P)-{v}_i(x)\right)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(\frac{s_i(x)\ \left(I(P)-{v}_i(x)\right)}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2={A}_1+{s}_i(x)\ \left({u_{n_i}}^m-{u_{0_i}}^m\right)\ {\left(\frac{I(P)-{v}_i(x)}{s_i(x)+\left({u_{n_i}}^m-{u_{0_i}}^m\right)}\right)}^2\ \left[\left({u_{n_i}}^m-{u_{0_i}}^m\right)+{s}_i(x)\right]={A}_1+{s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2 $$
(25)

For the image domain Ωx:

$$ \overset{\sim }{A}=\sum \limits_{\varOmega_x}{\overset{\sim }{A}}_1=\sum \limits_{\varOmega_x}\left({A}_1+{s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)=A+\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right) $$
(26)

In a similar way, it can be shown that:

$$ \overset{\sim }{B}=B+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_i(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right) $$
(27)

Combining (23), (26) and (27), the new total energy functional is given by

$$ \overset{\sim }{F}=\sum \limits_{i=1}^2\left(A+\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)\right)+B+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_3(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)\overset{\sim }{F}=F+\sum \limits_{i=1}^2\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_i(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i\right)}^2\right) $$
(28)

Therefore,

$$ \Delta F=\overset{\sim }{F}-F=\sum \limits_{i=1}^2\sum \limits_{\varOmega_x}\left({s}_i(x)\ \left(\frac{{u_{n_i}}^m-{u_{0_i}}^m}{s_i(x)+{u_{n_i}}^m-{u_{0_i}}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right)+\sum \limits_{\varOmega_x}\left({s}_3(x)\ \left(\frac{{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}{s_3(x)+{\left(1-{u}_{n_1}-{u}_{n_2}\right)}^m-{\left(1-{u}_{0_1}-{u}_{0_2}\right)}^m}\right)\ {\left(I(P)-{v}_i(x)\right)}^2\right) $$
(29)

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Alipour, N., Hasanzadeh, R.P.R. Superpixel-based brain tumor segmentation in MR images using an extended local fuzzy active contour model. Multimed Tools Appl 80, 8835–8859 (2021). https://doi.org/10.1007/s11042-020-10122-1

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