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Typically, the energy minimization or snakes based object detection frameworks evolve a parametrized curve guided by some form of image gradient information. However due to heavy reliance on gradients, the approaches tend to fail in scenarios where this information is misleading or unavailable. This cripples the snake and renders it unusable as it gets stuck in a localminima away from the actual object. Moreover, the parametrized snake lacks the ability to model multiple evolving curves in a single run. In order to address these issues, Chan and Vese introduced a new framework which utilized region based information to guide a spline, and tries to solve the minimal partition problem formulated by Mumford and Shah. The framework is built upon the following energy equation, where $C$ is a levelset formulation of the curve:$F(c1, c2, C) = \mu . \text{Length}(C) + v . \text{Area}(inside(C))\\ \lambda_1 \int_{inside(C)}u_0(x,y)  c_1^2 dxdy + \lambda_2 \int_{outside(C)}u_0(x,y)  c_2^2 dxdy$ The framework essentially divides the image into two regions (per curve), which are referred to as inside and outside of the curve. The first two terms of the equation control the physical aspects of the curve, particularly the length, and area inside the curve, with their contributions controlled by two parameters $\mu$ and $v$. The image forces in this equation correspond to the third and fourth terms, which are identical but work in respective regions identified by the curve. The terms use $c_1$ and $c_2$, which are the mean intensity values inside and outside the curve respectively to guide the curve towards a minima where the both regions are consistent with respect to the mean intensity values. The proposed framework also utilizes an improved representation of the curve in the form of a level set function $\phi(x, y)$, which has many numerical advantages and naturally supports multiple curves evolved during a single run, as compared to the traditional snakes model where only one curve can be evolved in a single run. The unknown function $\phi$ is computed using EulerLagrange equations formulated using modified Heaviside function $H$, and Dirac measure $\delta$. The proposed framework was applied on numerous challenging 2D images with varying degree of difficulties. The framework was also capable of segmenting point clouds decomposed into 2D images, objects with blurred boundaries, and contours without gradients, all without requiring image denoising. Due to the formulation of $\phi$ approximation routine, the framework has tendencies to find actual global minima independent of the initial position of the curve. However the choice of multiple parameters namely $\lambda_{1,2}, \mu, v$ is done heuristically, and seem to be problem dependent. Also, the framework's implicit dependency on absolute image intensities in regions inside and outside of curve sometimes fail in very specific cases where the averages tend to be zero, though the authors proposed to use image curvature and orientation information from the initial image $u_0$.
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## **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 edgebased 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 curvaturedependent speed: Algorithms based on HamiltonJacobi formulations,'' *Journal of Computational Physics*, vol. 79, no. 1, pp. 1249, 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. 577685, 2006. #### [3] V. Caselles, R. Kimmel, and G. Sapiro, "Geodesic Active Contours,'' *International Journal of Computer Vision*, vol. 22, no. 1, pp. 6179, February 1997. 