## **Introduction**
This paper presents an active contour model to detect and segment objects in images whose boundaries may or may not be defined by their gradients. The curve evolution is based on the Mumford Shah energy functional and the level set methods, making it less susceptible to curve initialization errors.


## **Method**
Using an implicit shape representation, the boundary(s) C can be represented using the zero level set of an embedding function $\phi : \Omega \rightarrow$ **R** such that

$$
    C = \{x \in \Omega \:|\: \phi(x) = 0\}
$$

Such a representation of the boundary does not require a choice of parameterization.

The embedding function $\phi(.)$ is defined as 
$$
    \phi(x) = \pm \: distance(x,C)
$$
where the sign is either + or - depending upon whether the point is inside or outside the curve. This means that the boundary can be determined by finding out the point at which $\phi$ changes its sign.

According to the level set method [1], the embedding function is zero at all the points on the curve C at any time.

$$
    \phi(C(t),\:t) = 0 \:\: \forall t
$$

Consider the piecewise Mumford Shah energy functional model [2] with 2 regions $\Omega_{1}$ and $\Omega_{2}$ such that $\Omega = \Omega_{1} + \Omega_{2}$.

Let us define the Heaviside step function as 

$$
  H\phi \equiv H(\phi)\left\{
  \begin{array}{@{}ll@{}}
    1, & \text{if}\ \phi > 0 \: \Rightarrow x \in \Omega_{1} \\
    0, & \text{otherwise}\ \Rightarrow x \in \Omega_{2} \\
  \end{array}\right.
$$

Moving from an energy functional defined on the boundary C to one defined on the embedding function $\phi$, we get

$$
    E(\phi) = \int\limits_{\Omega_{1}} \: ({I}(x,y) - \mu_{1})^{2} \: dx + \int\limits_{\Omega_{2}} \: ({I}(x,y) - \mu_{2})^{2} \: dx + \nu \left| \partial \Omega_{1} \right|
$$

where we approximate the average brightness/intensity of all the pixels in the $\Omega_{1}$ and $\Omega_{2}$ regions to be $\mu_{1}$ and $\mu_{2}$ respectively. The term $\left| \partial \Omega_{1} \right|$ represents the length of the boundary and is added as a penalizer/regularizer.

Simplifying this expression, we get
$$
    \begin{align*}
    E(\phi) & =
    \int\limits_{\Omega} \: \Big[\big[({I}(x,y) - \mu_{1})^{2} - ({I}(x,y) - \mu_{2})^{2}\big]H\phi \\
    & + ({I}(x,y) - \mu_{2})^{2}\Big]\:dx + \nu \int\limits_{\Omega}\left| \nabla H\phi \right|\:dx
\end{align*}
$$

Since $H(\phi)$ is not differentiable everywhere, we assume a slightly smoothened Heaviside function such that

$$
    \frac{\mathrm{d} H(\phi)}{\mathrm{d} \phi} = \delta(\phi)
$$

One such choice of the smoothened delta function is

$$
    \delta_{\epsilon}(\phi) = \frac{1}{\pi} \frac{\epsilon}{\epsilon^{2} + \phi^{2}}, \: \epsilon > 0
$$

Now, the gradient descent equation can be computed as 
$$
    \frac{\partial \phi}{\partial t} =  \delta(\phi) \Big[\: \nu\:div\Big(\frac{\nabla \phi}{\left|\nabla \phi\right|}\Big) + ({I}(x,y) - \mu_{1})^{2} - ({I}(x,y) - \mu_{2})^{2}\Big]
$$

As the embedding function $\phi(.)$ evolves over time, the boundary points can be detected when it undergoes a sign change.


## **Discussion and Shortcomings**
As compared to the edge-based segmentation approaches such as geodesic active contours [3], this method uses region based segmentation, and can, therefore, result in curve evolution that is not just local.

The segmentation results are not dependent on the initialization of the curve. Because of the choice of the embedding function $\phi$ as a signed distance function, the curves can split and merge. Moreover, unlike geodesic active contours [3], new curves can also be formed. This is because of the numerical approximation of the delta function - since the delta function never actually goes to 0. This means that objects with interior contours such as a ring can also be segmented, which was not possible with previously available active contour models.

#### [1] S. Osher, and J. A. Sethian, "Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations,'' *Journal of Computational Physics*, vol. 79, no. 1, pp. 12-49, 1988.

#### [2] D. Mumford, and J. Shah, "Optimal approximations by piecewise smooth functions and associated variational problems,'' *Communications on Pure and Applied Mathematics*, vol. 42, no. 5, pp. 577-685, 2006.

#### [3] V. Caselles, R. Kimmel, and G. Sapiro, "Geodesic Active Contours,'' *International Journal of Computer Vision*, vol. 22, no. 1, pp. 61-79, February 1997.