Brain MR image segmentation for tumor detection based on Riesz probability distributions

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Algorithm 1. The EM algorithm for the Riesz mixture model
1: begin
2: Initialisation: ${\mathrm{\Phi }}^{(0)}=\left({{\pi }_k}^{(0)},{s_k}^{(0)},{{\mathrm{\Sigma }}_k}^{(0)}\right)$
3: Iteration (t+$1):$
4: Expectation:
5: ${\tilde{p}}_{\mathit{ik}}=\frac{{\pi }_k^{(t)}{f}_k\left(X_i|{s_k}^{(0)},{{\mathrm{\Sigma }}_k}^{(0)}\right)}{\underset{\mathit{j}=1}{\sum ^K}{\pi }_j^{(t)}{f}_j\left(X_i|{s_j}^{(0)},{{\mathrm{\Sigma }}_j}^{(0)}\right)}.$
6: Maximization:
7: ${\hat{\pi }}_{\mathit{k}}=\frac{1}{N}\underset{\mathit{i}=1}{\sum ^N}{\tilde{p}}_{\mathit{ik}}.$
8: ${\hat{s}}_{\mathit{k},\mathit{j}}=\left\{\begin{array}{ccc}{\psi }^{-1}\left(-log(2{\mathrm{\Delta }}_1({\mathrm{\Sigma }}_k))+\frac{\underset{\mathit{i}=1}{\sum ^N}{\tilde{p}}_{\mathit{ik}}log\left({\mathrm{\Delta }}_1(X_i)\right)}{\underset{\mathit{i}=1}{\sum ^N}{\tilde{p}}_{\mathit{ik}}}\right)& ;& \mathit{j}=1\\ \frac{\mathit{j}-1}{2}+{\psi }^{-1}\left(log\left(\frac{{\mathrm{\Delta }}_{\mathit{j}-1}({\mathrm{\Sigma }}_k)}{2{\mathrm{\Delta }}_j({\mathrm{\Sigma }}_k)}\right)+\frac{\underset{\mathit{i}=1}{\sum ^N}{\tilde{p}}_{\mathit{ik}}log\left(\frac{{\mathrm{\Delta }}_j(X_i)}{{\mathrm{\Delta }}_{\mathit{j}-1}(X_i)}\right)}{\underset{\mathit{i}=1}{\sum ^N}{\tilde{p}}_{\mathit{ik}}}\right)& ;& 2\le \mathit{j}\le \mathit{r}\end{array}\right.$
9: $\underset{\mathit{l}=1}{\sum ^r}({\hat{s}}_{\mathit{k},\mathit{l}}-{\hat{s}}_{\mathit{k},\mathit{l}+1})\widehat{{\mathrm{\Sigma }}_k}{\left(P_l(\widehat{{\mathrm{\Sigma }}_k})\right)}^{-1}\widehat{{\mathrm{\Sigma }}_k}=\frac{1}{2\underset{\mathit{i}=1}{\sum ^N}{\tilde{p}}_{\mathit{ik}}}\underset{\mathit{i}=1}{\sum ^N}{\tilde{p}}_{\mathit{ik}}{X}_i.$
10: If $||{\mathrm{\Phi }}^{(\mathit{t}+1)}-{\mathrm{\Phi }}^{(\mathit{t})}||<\mathit{\epsilon }$ is not satisfied return to step $2.$
11: end

Figure 1. The flowchart of the proposed image segmentation method.

Table 1. Other riesz mixture models.

Riesz mixture model 2	${\mathrm{M}}_1^{\prime }\sim {\mathcal{R}}_3(s_1^{\prime },{\mathrm{\Sigma }}_1^{\prime })$ and ${\mathrm{M}}_2^{\prime }\sim {\mathcal{R}}_3(s_2^{\prime },{\mathrm{\Sigma }}_2^{\prime })$
$s_1^{\prime }$	$(4.4151,2.1242,6.7157)$
$s_2^{\prime }$	$(4.3197,2.1438,7.3523)$
${\mathrm{\Sigma }}_1^{\prime }$	$\left(\begin{array}{ccc}1.1346& 0.0658& 0.0519\\ 0.0658& 1.3122& -0.0304\\ 0.0519& -0.0304& 2.5979\end{array}\right)$
${\mathrm{\Sigma }}_2^{\prime }$	$\left(\begin{array}{ccc}1.1606& 0.0607& 0.0771\\ 0.0607& 1.2805& 0.1208\\ 0.0771& 0.1208& 2.3083\end{array}\right)$
Riesz mixture model 3	${\mathrm{M}}_1^{{ }^{\prime \prime }}\sim {\mathcal{R}}_3(s_1^{{ }^{\prime \prime }},{\mathrm{\Sigma }}_1^{{ }^{\prime \prime }})$ and ${\mathrm{M}}_2^{{ }^{\prime \prime }}\sim {\mathcal{R}}_3(s_2^{{ }^{\prime \prime }},{\mathrm{\Sigma }}_2^{{ }^{\prime \prime }})$
$s_1^{{ }^{\prime \prime }}$	$(10,17,19)$
$s_2^{{ }^{\prime \prime }}$	$(15,17,18)$
${\mathrm{\Sigma }}_1^{{ }^{\prime \prime }}$	$\left(\begin{array}{ccc}2.8255& 1.6921& -2.9052\\ 1.6921& 2.6183& -2.2620\\ -2.9052& -2.2620& 3.7281\end{array}\right)$
${\mathrm{\Sigma }}_2^{{ }^{\prime \prime }}$	$\left(\begin{array}{ccc}2.9232& 1.8945& -0.4002\\ 1.8945& 2.7291& 0.7391\\ -0.4002& 0.7391& 1.4830\end{array}\right).$

Table 2. The resulting estimated parameters using the EMR and their distances toward the actual parameters.

Parameter	initial parameter value	Estimated value	Distance
$\widehat{s_1}$	$(1.5,3.2,4)$	$(1.8023,3.9629,4.5907)$	$\mathit{d}(s_1,\widehat{s_1})=1.7609$
$\widehat{s_2}$	$(2.3,2.9,4.5)$	$(2.7748,3.3752,4.9554)$	$\mathit{d}(s_2,\widehat{s_2})=2.1117$
$\widehat{{\mathrm{\Sigma }}_1}$	$\left(\begin{array}{ccc}1.5& 0.1& 0.11\\ 0.1& 1.8& 0.15\\ 0.11& 0.15& 2\end{array}\right)$	$\left(\begin{array}{ccc}1.9737& 0.1515& 0.1356\\ 0.1515& 2.0178& 0.1723\\ 0.1356& 0.1723& 2.1898\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_1,\widehat{{\mathrm{\Sigma }}_1})=0.3147$
$\widehat{{\mathrm{\Sigma }}_2}$	$\left(\begin{array}{ccc}2.5& 0.2& 0.21\\ 0.2& 3& 0.23\\ 0.21& 0.23& 4\end{array}\right)$	$\left(\begin{array}{ccc}2.9821& 0.2478& 0.2579\\ 0.2478& 3.9992& 0.2555\\ 0.2579& 0.2555& 4.2887\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_2,\widehat{{\mathrm{\Sigma }}_2})=1.3566$
$\widehat{s_1^{\prime }}$	$\left(4.2,2,6.5\right)$	$\left(4.6827,2.1274,6.8422\right)$	$\mathit{d}(s_1^{\prime },\widehat{s_1^{\prime }})=0.2960$
$\widehat{s_2^{\prime }}$	$\left(4,2,7\right)$	$\left(4.5869,2.1513,7.5185\right)$	$\mathit{d}(s_2^{\prime },\widehat{s_2^{\prime }})=0.3147$
$\widehat{{\mathrm{\Sigma }}_1^{\prime }}$	$\left(\begin{array}{ccc}1& 0.08& 0.1\\ 0.08& 1.5& 0.1\\ 0.1& 0.1& 2.1\end{array}\right)$	$\left(\begin{array}{ccc}1.0686& 0.0633& 0.0297\\ 0.0633& 1.2998& -0.0600\\ 0.0297& -0.0600& 2.5521\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_1^{\prime },\widehat{{\mathrm{\Sigma }}_1^{\prime }})=0.0967$
$\widehat{{\mathrm{\Sigma }}_2^{\prime }}$	$\left(\begin{array}{ccc}1& 0.02& 0.07\\ 0.02& 1.5& 0.5\\ 0.07& 0.5& 2\end{array}\right)$	$\left(\begin{array}{ccc}1.0910& 0.0557& 0.0591\\ 0.0557& 1.2638& 0.1191\\ 0.0591& 0.1191& 2.2518\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_2^{\prime },\widehat{{\mathrm{\Sigma }}_2^{\prime }})=0.0950$
$\widehat{s_1^{{ }^{\prime \prime }}}$	$\left(9,15,18.5\right)$	$\left(9.7831,16.8324,19.1857\right)$	$\mathit{d}(s_1^{{ }^{\prime \prime }},\widehat{s_1^{{ }^{\prime \prime }}})=0.3311$
$\widehat{s_2^{{ }^{\prime \prime }}}$	$\left(13,16.5,19\right)$	$\left(14.0143,16.9423,19.0237\right)$	$\mathit{d}(s_2^{{ }^{\prime \prime }},\widehat{s_2^{{ }^{\prime \prime }}})=1.4223$
$\widehat{{\mathrm{\Sigma }}_1^{{ }^{\prime \prime }}}$	$\left(\begin{array}{ccc}2.3& 1.5& 0.5\\ 1.5& 2& -2\\ 0.5& -2& 3\end{array}\right)$	$\left(\begin{array}{ccc}2.6961& 1.8729& -2.9499\\ 1.8729& 2.8549& -2.6202\\ -2.9499& -2.6202& 3.5011\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_1^{{ }^{\prime \prime }},\widehat{{\mathrm{\Sigma }}_1^{{ }^{\prime \prime }}})=0.6710$.
$\widehat{{\mathrm{\Sigma }}_2^{{ }^{\prime \prime }}}$	$\left(\begin{array}{ccc}2& 1.5& -0.2\\ 1.5& 3& 0.7\\ -0.2& 0.7& 1.8\end{array}\right)$	$\left(\begin{array}{ccc}2.5720& 1.9035& -0.5124\\ 1.9035& 2.9114& 0.5881\\ -0.5124& 0.5881& 1.6921\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_2^{{ }^{\prime \prime }},\widehat{{\mathrm{\Sigma }}_2^{{ }^{\prime \prime }}})=0.5208$.

Table 3. The resulting estimated parameters using the EMW and their distances toward the actual parameters.

Parameter	Estimated value	Distance
$\widehat{{\mathrm{\Sigma }}_1}$	$\left(\begin{array}{ccc}1.9097& 1.0476& 1.2616\\ 1.0476& 2.3129& 0.7130\\ 1.2616& 0.7130& 2.4455\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_1,\widehat{{\mathrm{\Sigma }}_1})=2.1165$
$\widehat{{\mathrm{\Sigma }}_2}$	$\left(\begin{array}{ccc}2.9905& 0.2688& 0.7079\\ 0.2688& 2.8791& 0.8410\\ 0.7079& 0.8410& 3.0745\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_2,\widehat{{\mathrm{\Sigma }}_2})=2.4811$
$\widehat{{\mathrm{\Sigma }}_1^{\prime }}$	$\left(\begin{array}{ccc}1.0696& 0.0563& 0.0487\\ 0.0563& 1.3157& 0.0193\\ 0.0487& 0.0193& 2.5297\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_1^{\prime },\widehat{{\mathrm{\Sigma }}_1^{\prime }})=0.1184$
$\widehat{{\mathrm{\Sigma }}_2^{\prime }}$	$\left(\begin{array}{ccc}1.0934& 0.0507& 0.0736\\ 0.0507& 1.2819& 0.1843\\ 0.0736& 0.1843& 2.2260\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_2^{\prime },\widehat{{\mathrm{\Sigma }}_2^{\prime }})=0.1399$
$\widehat{{\mathrm{\Sigma }}_1^{{ }^{\prime \prime }}}$	$\left(\begin{array}{ccc}2.8907& 2.0150& -1.9834\\ 2.0150& 2.9114& -1.7598\\ -1.9834& -1.7598& 3.8902\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_2^{{ }^{\prime \prime }},\widehat{{\mathrm{\Sigma }}_2^{{ }^{\prime \prime }}})=1.5902$
$\widehat{{\mathrm{\Sigma }}_2^{{ }^{\prime \prime }}}$	$\left(\begin{array}{ccc}2.9971& 1.0032& -0.4358\\ 1.0032& 3.0415& 0.8120\\ -0.4358& 0.8120& 3.8785\end{array}\right)$	$d_{\mathit{HS}}({\mathrm{\Sigma }}_2^{\prime {\mathit{ }}^{\mathrm{\prime }}},\widehat{{\mathrm{\Sigma }}_2^{{ }^{\prime \prime }}})=1.4223$.

Figure 2. The original synthetic image.

Figure 3. Synthetic noisy images for the different riesz mixture models.

Figure 4. Simulated segmented images by the mixture model 1.

Figure 5. Simulated segmented images by the mixture model 2.

Figure 6. Simulated segmented images by the mixture model 3.

Table 4. Different evaluation criteria values with Riesz and Wishart distributions.

Mixture model	$\mathit{MCR}$	$\mathit{PSNR}$	$\mathit{ARI}$	$\mathit{RI}$	$\mathit{MI}$	$\mathit{HI}$
Mixture Riesz model 1	$0.0035$	$24.3128$	$0.0610$	$0.6927$	$0.2587$	$0.3767$
Mixture Wishart model 1	$0.0052$	$22.1668$	$0.0579$	$0.6884$	$0.2642$	$0.3854$
Mixture Riesz model 2	$0.0198$	$21.0030$	$0.7049$	$0.8577$	$0.1423$	$-0.4183$
Mixture Wishart model 2	$0.1132$	$18.7668$	$0.0574$	$0.3648$	$0.6352$	$-0.2703$
Mixture Riesz model 3	$0.0455$	$21.6783$	$0.7028$	$0.8986$	$0.1014$	$0.2139$
Mixture Wishart model 3	$0.0628$	$20.9922$	$0.2471$	$0.6070$	$0.3930$	$0.4021$

Figure 7. Real brain MR image and its blocks.

Figure 8. Segmentation results of the brain image. (a) the segmented image by the riesz distribution, (b) the segmented image by the Wishart distribution.

Figure 9. Segmentation results of the brain image. (c) the segmented image by the riesz distribution, (d) the segmented image by the Wishart distribution.

Table 5. The MCR and PSNR values by varying mixture with brain images.

Figures	Mixture Model	$\mathit{MCR}$	$\mathit{PSNR}$
$(8)$	Mixture Riesz	$0.023$	$29.42$
	Mixture Wishart	$0.042$	$25.56$
$(9)$	Mixture Riesz	$0.002$	$27.54$
	Mixture Wishart	$0.014$	$22.13$