## 1.

## Introduction

Active contours have been widely researched and applied in image processing and computer vision since they were first introduced by Kass ^{1} The applications of active contours range from image segmentation and shape modeling to visual tracking, and so on. This letter mainly addresses the application of active contours in image segmentation. Generally speaking, active contours can be classified into two kinds: parametric active contours^{1, 2, 3} and geometric active contours.^{4, 5, 6, 7, 8} Parametric active contours represented by point sets or B-splines are efficient in computation. On the contrary, geometric active contours represented by level set functions are a vast computational task, but they have the advantage of automatically handling changes in topology over parametric active contours. Unfortunately, both kinds of active contours suffer difficulties in choosing the appropriate initial contours. A dual active contour^{2} was proposed by Gunn and Nixon to relieve the problem in initialization for a parametric active contour. Inspired by their work, in this letter, we correspondingly propose a dual geometric active contour, of which the inner and outer contours evolve respectively from the interior and exterior of the segmented object to the desired boundary. The two contours interact with each other to avoid getting in local minima traps of the functional optimizations of the active contour during the evolutions of the level set functions that represent the contours. Promising results on image segmentation demonstrate the potentials of the proposed dual geometric active contour.

## 2.

## Dual Geometric Active Contour

We implement the dual geometric active contour on Chan-Vese active contour, which is a typical geometric active contour. Given a 2-D gray-value image
$f:\Omega \to {\mathbb{R}}^{+}$
. Chan-Vese active contour^{8} as a closed dynamic curve
$C$
is defined to minimize an energy functional, which is given by

## 1

$${E}_{\mathrm{CV}}({u}_{1},{u}_{2},\varphi )=\mu {\int}_{\Omega}\mid \nabla {H}_{\u03f5}\left(\varphi \right)\mid \phantom{\rule{0.2em}{0ex}}\mathrm{d}x+{\int}_{\Omega}{(f-{u}_{1})}^{2}{H}_{\u03f5}\left(\varphi \right)+{(f-{u}_{2})}^{2}[1-{H}_{\u03f5}\left(\varphi \right)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,$$^{8}The first term on the right-hand side of Eq. 1 measures the length of the curve $C$ . And the second term, in which ${u}_{1}$ and ${u}_{2}$ are the mean gray values of the image $f$ , respectively, inside and outside of the curve $C$ , measures the homogeneities in the different regions. The positive parameter $\mu $ weights the relative importance of the two terms.

As mentioned, active contours suffer difficulties in choosing the appropriate initial contours. Inspired by the work of Gunn and Nixon,^{2} we correspondingly propose a dual geometric active contour implemented on a Chan-Vese active contour. The proposed active contour consists of two contours initialized simultaneously inside and outside the object boundary. The inner and outer contours, which are represented by the level set functions
$\varphi $
and
$\psi $
, respectively, are defined to minimize the following energy functional

## 2

$${E}_{\mathrm{DCV}}({u}_{1},{u}_{2},\varphi ;{v}_{1},{v}_{2},\psi )={E}_{\mathrm{CV}}({u}_{1},{u}_{2},\varphi )+{E}_{\mathrm{CV}}({v}_{1},{v}_{2},\psi )+\tau {\int}_{\Omega}{[{H}_{\u03f5}\left(\varphi \right)-{H}_{\u03f5}\left(\psi \right)]}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x,$$^{9}the principles underlying the two terms are greatly different.

Minimizing energy functional ${E}_{\mathrm{DCV}}$ with respect to $\varphi $ and $\psi $ , respectively, we can obtain two associated Euler-Lagrange equations. Then, the evolutions of the level set functions $\varphi $ and $\psi $ are implemented by the following gradient descents:

## 3

$$\frac{\partial \varphi}{\partial t}={\delta}_{\u03f5}\left(\varphi \right)\{\mu \phantom{\rule{0.2em}{0ex}}\mathrm{div}\phantom{\rule{0.2em}{0ex}}\left(\frac{\nabla \varphi}{\mid \nabla \varphi \mid}\right)-[{(f-{u}_{1})}^{2}-{(f-{u}_{2})}^{2}]-2\tau [{H}_{\u03f5}\left(\varphi \right)-{H}_{\u03f5}\left(\psi \right)]\},$$## 4

$$\frac{\partial \psi}{\partial t}={\delta}_{\u03f5}\left(\psi \right)\{\mu \phantom{\rule{0.2em}{0ex}}\mathrm{div}\phantom{\rule{0.2em}{0ex}}\left(\frac{\nabla \psi}{\mid \nabla \psi \mid}\right)-[{(f-{v}_{1})}^{2}-{(f-{v}_{2})}^{2}]-2\tau [{H}_{\u03f5}\left(\psi \right)-{H}_{\u03f5}\left(\varphi \right)]\},$$^{10}The parameters ${u}_{1}$ and ${u}_{2}$ , and ${v}_{1}$ and ${v}_{2}$ required by the computation of the two gradient descents are updated by

## 5

$${u}_{1}=\frac{\int f\left(x\right){H}_{\u03f5}\left(\varphi \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x}{\int {H}_{\u03f5}\left(\varphi \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x},\phantom{\rule{1em}{0ex}}{u}_{2}=\frac{\int f\left(x\right)[1-{H}_{\u03f5}\left(\varphi \right)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x}{\int [1-{H}_{\u03f5}\left(\varphi \right)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x},$$## 6

$${v}_{1}=\frac{\int f\left(x\right){H}_{\u03f5}\left(\psi \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x}{\int {H}_{\u03f5}\left(\psi \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x},\phantom{\rule{1em}{0ex}}{v}_{2}=\frac{\int f\left(x\right)[1-{H}_{\u03f5}\left(\psi \right)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x}{\int [1-{H}_{\u03f5}\left(\psi \right)]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x},$$Based on the preceding description of the proposed active contour, we summarize the complete algorithm as follows:

**Step 1**. Initialize the level set functions $\varphi $ and $\psi $ .**Step 2**. Compute ${u}_{1}$ and ${u}_{2}$ , and ${v}_{1}$ and ${v}_{2}$ by Eqs. 5, 6, respectively.**Step 3**. Evolve $\varphi $ according Eq. 3, then redistance it.**Step 4**. Evolve $\psi $ according Eq. 4, then redistance it. Go to step 2.

## 3.

## Experimental Results

We validated the performance of the dual geometric active contour on many real images by comparing it with the Chan-Vese active contour. In all our experiments, we use $\u03f5=0.05$ , $\mu =0.005\times {255}^{2}$ , and $\mathrm{\Delta}t=0.02$ (the step of the artificial time $t$ ) for both Chan-Vese active contour and the proposed active contour, and we use $\tau =2.5\times {255}^{2}$ especially for the proposed active contour.

Figure 1 shows an example of our experiments. The size of the tested hand image is $320\times 240\phantom{\rule{0.3em}{0ex}}\mathrm{pixels}$ . Figure 1a shows the initialization of the Chan-Vese active contour from the inside of the hand boundary, and Fig. 1d shows its corresponding result at the steady state. It is shown that the contour at the steady state excludes two regions that belong to the hand area, since the gray values of the pixels in the two regions are closer to the mean gray value of the image in the background than that in the hand area. Contrarily, Fig. 1b shows the initialization of the Chan-Vese active contour from the outside of hand boundary. As shown in Fig. 1e, the contour at the steady state includes several undesired regions in which the gray values of the pixels are closer to the mean gray value of the image in the hand area, such as the regions of the pen and the USB disk. In Fig. 1c, a dual geometric active contour is initialized, and the solid lines denote the initialization of the outer contour and the dashed lines denote the initialization of the inner contour. Figure 1f shows that the two contours completely superpose on each other and exactly conform to the hand boundary at the steady state.

Similarly, another example of our experiments is shown in Fig. 2. The size of the tested cup image is $320\times 240\phantom{\rule{0.3em}{0ex}}\mathrm{pixels}$ . It is also shown that the dual geometric active contour outperforms the Chan-Vese active contour in the segmentation.

## 4.

## Conclusions

We proposed a dual geometric active contour to overcome the difficulties in initialization that are suffered by the conventional geometric active contours. The dual geometric active contour simultaneously initializes two contours, respectively, inside and outside of desired boundary. And the two contours interact with each other in their evolving procedures to avoid the local optimizations of the energy functional. The experiments conducted on real images show that the proposed dual active contour is robust to bad contour initializations, while it is still, to some extent, dependent on the choice of the initial contour. In the future work, a shape prior can be incorporated into the proposed active contour model to further relieve the problem in initialization.

## Acknowledgments

The authors wish to thank the anonymous reviewers and associate editor Giordano Beretta for their useful comments.