Improved segmentation of low-contrast lesions using sigmoid edge model

Abstract

Purpose

The intensity profile of an image in the vicinity of a tissue’s boundary is modeled by a step/ramp function. However, this assumption does not hold in cases of low-contrast images, heterogeneous tissue textures, and where partial volume effect exists. We propose a hybrid algorithm for segmentation of CT/MR tumors in low-contrast, noisy images having heterogeneous/homogeneous or hyper-/hypo-intense abnormalities. We also model a smoothed noisy intensity profile by a sigmoid function and employ it to find the true location of boundary more accurately.

Methods

A novel combination of the SVM, watershed, and scattered data approximation algorithms is employed to initially segment a tumor. Small and large abnormalities are treated distinctly. Next, the proposed sigmoid edge model is fitted to the normal profile of the border. The estimated parameters of the model are then utilized to find true boundary of a tissue.

Results

We extensively evaluated our method using synthetic images (contaminated with varying levels of noise) and clinical CT/MR data. Clinical images included 57 CT/MR volumes consisting of small/large tumors, very low-/high-contrast images, liver/brain tumors, and hyper-/hypo-intense abnormalities. We achieved a Dice measure of $0.83\,(\pm 0.07)$ and average symmetric surface distance of $2.56\,(\pm 6.31)$ mm. Regarding IBSR dataset, we fulfilled Jaccard index of $0.85\,(\pm 0.07)$. The average run-time of our code was $154\,(\pm 71)$ s.

Conclusion

Individual treatment of small and large tumors and boundary correction using the proposed sigmoid edge model can be used to develop a robust tumor segmentation algorithm which deals with any types of tumors.

Appendix

We evaluate Eq. (9) for regions R1–R3 (Fig. 6b). It can be assumed that variance of the Gaussian filter is smaller than important structures present in the image, i.e. $\sigma {<}T$. Therefore, the lower and upper limits of the integral in Eq. (9) may be changed into $\left[ {t-4\sigma ,t+4\sigma } \right]$ since the surface under a normal distribution has $<$0.01 % outside this range. To evaluate Eq. (9) for R1, it is rewritten as Eq. (17).

$$\begin{aligned} x_\mathrm{smth} (t)= & {} {\int }_{t-4\sigma }^{t+4\sigma } \frac{A\tau }{T}\times \frac{\hbox {exp}\left( {\frac{-\left( {t-\tau } \right) ^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}\tau \nonumber \\= & {} {\int }_{-4\sigma }^{+4\sigma } \frac{A\left( {w+t} \right) }{T}\times \frac{\hbox {exp}\left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w \nonumber \\= & {} \frac{A}{T}{\int }_{-4\sigma }^{+4\sigma } w\times \frac{\hbox {exp}\left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w\nonumber \\&+\,\frac{At}{T}{\int }_{-4\sigma }^{+4\sigma } \frac{\exp \left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w \nonumber \\= & {} \frac{A\left( {-\sigma ^{2}} \right) }{T\sqrt{2\pi }\sigma }\hbox {exp}\left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) _{-4\sigma }^{+4\sigma } +\frac{At}{T}\simeq \frac{At}{T} \end{aligned}$$

(17)

Regarding region R2, Eq. (9) at $t=T$ is calculated by Eqs. (18) and (19).

$$\begin{aligned} x_\mathrm{smth} (t)= & {} {\int }_{t-4\sigma }^T \frac{A\tau }{T}\times \frac{\exp \left( {\frac{-\left( {t-\tau } \right) ^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}\tau \nonumber \\&+{\int }_T^{t+4\sigma } A\times \frac{\exp \left( {\frac{-\left( {t-\tau } \right) ^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}\tau \nonumber \\= & {} {\int }_{-4\sigma }^{T-t} \frac{A\left( {w+t} \right) }{T}\times \frac{\exp \left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w\nonumber \\&+\,{\int }_{T-t}^{+4\sigma } A\times \frac{\exp \left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w.\end{aligned}$$

(18)

$$\begin{aligned} x_\mathrm{smth} ( T )= & {} {\int }_{-4\sigma }^0 \frac{A\left( {w+T} \right) }{T}\times \frac{\hbox {exp}\left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w\nonumber \\&+{\int }_0^{+4\sigma } A\times \frac{\hbox {exp}\left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}sw \nonumber \\= & {} {\int }_{-4\sigma }^0 \frac{Aw}{T}\times \frac{\hbox {exp}\left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w\nonumber \\&+{\int }_{-4\sigma }^{+4\sigma } A\times \frac{\exp \left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) }{\sqrt{2\pi }\sigma }\mathrm{d}w\nonumber \\= & {} \frac{A\left( {-\sigma ^{2}} \right) }{T\sqrt{2\pi }\sigma }\hbox {exp}\left( {\frac{-w^{2}}{2\sigma ^{2}}} \right) _{-4\sigma }^0 +A\nonumber \\= & {} \frac{A\left( {-\sigma } \right) }{T\sqrt{2\pi }}+A=A\left( {1-\frac{\sigma }{\sqrt{2\pi }T}} \right) . \end{aligned}$$

(19)