Prior distribution-based statistical active contour model

Abstract

Employing prior information can greatly improve the segmentation result of many image segmentation problems. For example, a commonly used prior information is the shape of the object. In this paper, we introduce a different kind of prior information called the prior distribution. On the basis of non-parametric statistical active contour model, we add prior distribution energy to build a novel prior active contour model. During the convergence of contour curve, distribution difference between the inside and outside of the active contour is maximized while the distribution difference between the inside/outside of contour and the prior object/background is minimized. Furthermore, in order to improve the computation speed, a method to accelerate the computation speed is also proposed, which significantly relieves the burden of estimating probability density functions. As the experimental results suggest, satisfactory effects can be achieved in the segmentation of synthetic images and natural images via the our algorithm. Compared with the traditional non-parametric statistical active contour model without prior information, our method achieves a distinct improvement in both accuracy and computation efficiency.

Appendix: Derivation of the gradient flow formula

In this section, we accomplished the derivation of gradient flow of the prior distribution-based statistical active contour model energy function. The proposed energy function in (23) shown as

$$ \begin{array}{lll}E(\phi(x))&=E_{\text{SACM}}(\phi(x))+\lambda_{1}E_{\text{prior}+}(p_{+},p_{\text{prior}+},\phi(x))\\ &+\lambda_{2}E_{\text{prior}-}(p_{-},p_{\text{prior}-},\phi(x)). \end{array} $$

(34)

To minimize the energy function, it is required to calculate the first derivative with respect to ϕ(x), i.e.,

$$ \begin{array}{lll}\frac{\partial E(\phi(x))}{\partial \phi(x)}=&\frac{\partial E_{\text{SACM}}(\phi(x))}{\partial \phi(x)}\\ &-\frac{\lambda_{1}}{2}{\int}_{R}\frac{\partial p_{-}(z,\phi(x))}{\partial \phi(x)}\sqrt{\frac{p_{\text{prior}-}(z)}{p_{-}(z,\phi(x))}}\mathrm{d}z\\ &-\frac{\lambda_{2}}{2}{\int}_{R}\frac{\partial p_{+}(z,\phi(x))}{\partial \phi(x)}\sqrt{\frac{p_{\text{prior}+}(z)}{p_{+}(z,\phi(x))}}\mathrm{d}z \end{array} $$

(35)

It can be proved that [24],

$$ \frac{\partial p_{-}(z,\phi(x))}{\partial \phi(x)}=\frac{\delta(\phi(x))}{|R_{-}|}(p_{-}(z,\phi(x))-K_{\sigma}(z-I(x))) $$

(36)

$$ \frac{\partial p_{+}(z,\phi(x))}{\partial \phi(x)}=\frac{\delta(\phi(x))}{|R_{+}|}(K_{\sigma}(z-I(x))-p_{+}(z,\phi(x))) $$

(37)

where |R₋|and |R₊| respectively represent the area of the inside and outside regions that can be determined according to ${\int \limits }_{\varOmega } H(-\phi (x))\mathrm {d}x$ and ${\int \limits }_{\varOmega } H(\phi (x))\mathrm {d}x$.

Based on the equations above, the following equation is given:

$$ \frac{\partial E(\phi(x))}{\partial \phi(x)}=\frac{\partial E_{\text{SACM}}(\phi(x))}{\partial \phi(x)}+\delta(\phi(x))V(x) $$

(38)

where

$$ \begin{array}{lll}V(x)&=\frac{1}{2}\left( \frac{\lambda_{2}}{|R_{+}|}{\int}_{R}\sqrt{p_{+}(z)p_{\text{prior}+}(z)}\mathrm{d}z-\frac{\lambda_{1}}{|R_{-}|}{\int}_{R}\sqrt{p_{-}(z)p_{\text{prior}-}(z)}\mathrm{d}z\right)\\ &+\frac{1}{2}{\int}_{R}K_{\sigma}(z-I(x))(\frac{\lambda_{1}}{|R_{-}|}\sqrt{\frac{p_{\text{prior}-}(z)}{p_{-}(z,\phi(x))}}-\frac{\lambda_{2}}{|R_{+}|}\sqrt{\frac{p_{\text{prior}+}(z)}{p_{+}(z,\phi(x))}})\mathrm{d}z. \end{array} $$

(39)

Finally, the time parameter t is brought into the level set function in order to obtain the gradient vector flow of the minimum level set function:

$$ \frac{\partial \phi}{\partial t}=-\frac{\partial E(\phi(x))}{\partial\phi(x)} $$

(40)