Active contours without edges Active contours without edges
Paper summary 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 local-minima 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 level-set formulation of the curve: \begin{equation} \begin{split} 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 \end{split} \end{equation} 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 Euler-Lagrange 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 $u_0$.

Summary by Anmol Sharma 2 months ago
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