Zernike moment and local distribution fitting fuzzy energy-based active contours for image segmentation

Abstract

This paper presents a new region-based active contour model for extracting the object boundaries in an image, based on techniques of curve evolution. The proposed model introduces an energy functional that involves intensity distributions in local image regions and fuzzy membership functions. The local image intensity distribution information used to guide the motion of the contour, in the paper, is derived by Hueckel operator in the neighborhood of each image point. The parameters of Hueckel operator are estimated by a set of orthogonal Zernike moments before curve evolution. Meanwhile, the fuzzy membership functions are used to measure the association degree of each image pixel to the region outside and inside the contour. To minimize the energy functional, instead of solving the Euler–Lagrange equation of the underlying problem, the paper employs a direct method to compute the energy alterations. As a result, the model can deal with images with intensity inhomogeneity. In addition, the model effectively alleviates the sensitivity to contour initialization. Moreover, the model reduces computational cost, avoids problems associated with choosing time steps as well as allows fast convergence to the segmentation solutions. Experimental results on synthetic, real images and comparisons with other models show the desired performances of the proposed model.

Appendix 1

This section presents the numerical approximation of the alternations in the old and new energy, given in Eq. 17.

Let us consider an image point $P$in the given image $I$. Assume that the intensity value of point $P$ is $I_o$, and the corresponding degree of membership for this image point to outside of contour $C$ is $u_o$. Suppose that we change the degree of membership of point $P$ to the new value $u_n$, which is calculated from Eq. 16. From Eq. (13), we observe that a change in the degree of membership of point $P$ will cause a change in the model energy. The new energy of the model $\tilde{F}$ is given as:

$$\begin{aligned} \tilde{F}&\!=\!&\lambda _1 \underbrace{\sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_1 (\mathbf{x})} \right]^{2}} } [\tilde{u}(\mathbf{y})]^{m}}_{\tilde{C}} \nonumber \\&+\,\lambda _2 \underbrace{\sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_2 (\mathbf{x})} \right]^{2}} } [1\!-\!\tilde{u}(\mathbf{y})]^{m}}_{\tilde{D}} \end{aligned}$$

(19)

where $\tilde{C}$and $\tilde{D}$are calculated as:

$$\begin{aligned} \tilde{C}&\!=\!&\sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_1 (\mathbf{x})} \right]^{2}} } [\tilde{u}(\mathbf{y})]^{m} \nonumber \\&= \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_1 (\mathbf{x})} \right]^{2}} } [u(\mathbf{y})]^{m}\nonumber \\&+\sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I_o\!-\!f_1 (\mathbf{x})} \right]^{2}} } \left[ {u_n^m -u_o^m } \right] \end{aligned}$$

(20)

$$\begin{aligned} \tilde{D}&= \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_2 (\mathbf{x})} \right]^{2}} } [1\!-\!\tilde{u}(\mathbf{y})]^{m} \nonumber \\&= \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_2 (\mathbf{x})} \right]^{2}} } [1\!-\!u(\mathbf{y})]^{m} \nonumber \\&+\sum _\Omega {\!\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[\! {I_o\!-\!f_2 (\mathbf{x})}\! \right]^{2}} } \left[\!{(1\!-\!u_n )^{m}\!-\!(1\!-\!u_o )^{m}}\!\right]\nonumber \\ \end{aligned}$$

(21)

Combining (19), (20), and (21), the new energy of the local model is given by:

$$\begin{aligned} \tilde{F}&\!=\!&\lambda _1 \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_1 (\mathbf{x})} \right]^{2}} } [u(\mathbf{y})]^{m}\nonumber \\&+\,\lambda _2 \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I(\mathbf{y})\!-\!f_2 (\mathbf{x})} \right]^{2}} } [1\!-\!u(\mathbf{y})]^{m} \nonumber \\&+\lambda _1 \sum _\Omega {\sum _\Omega {\!\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I_o\!-\!f_1 (\mathbf{x})} \right]^{2}} } \left[ {u_n^m\!-\!u_o^m } \right]\nonumber \\&+\lambda _2 \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}\!-\!\mathbf{y})\left[ {I_o\!-\!f_2 (\mathbf{x})} \right]^{2}} } \left[ (1\!-\!u_n )^{m}\right.\nonumber \\&\left.-\,(1-u_o)^{m} \right] \nonumber \\&= F + \Delta F \end{aligned}$$

(22)

where

$$\begin{aligned} F&= \lambda _1 \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}-\mathbf{y})\left[ {I(\mathbf{y})-f_1 (\mathbf{x})} \right]^{2}} } [u(\mathbf{y})]^{m}\nonumber \\&+\,\lambda _2 \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}-\mathbf{y})\left[ {I(\mathbf{y})-f_2 (\mathbf{x})} \right]^{2}} } [1-u(\mathbf{y})]^{m}\nonumber \\ \end{aligned}$$

(23)

and

$$\begin{aligned} \Delta F&= \lambda _1 \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}-\mathbf{y})\left[ {I_o -f_1 (\mathbf{x})} \right]^{2}} } \left[ {u_n^m -u_o^m } \right] \nonumber \\&+\lambda _2 \sum _\Omega {\sum _\Omega {\omega (\mathbf{x}-\mathbf{y})\left[ {I_o -f_2 (\mathbf{x})} \right]^{2}} } \left[ (1-u_n )^{m}\right.\nonumber \\&\left.-(1-u_o )^{m} \right] \end{aligned}$$

(24)

$\Delta F$ is the change energy when changing the degree of membership in the energy functional in Eq.(13).