A Review on Level Set Method for Image Segmentation
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paper_title: A Variational Level Set Approach to Multiphase Motion

paper_content:

A coupled level set method for the motion of multiple junctions (of, e.g., solid, liquid, and grain boundaries), which follows the gradient flow for an energy functional consisting of surface tension (proportional to length) and bulk energies (proportional to area), is developed. The approach combines the level set method of S. Osher and J. A. Sethian with a theoretical variational formulation of the motion by F. Reitich and H. M. Soner. The resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions. The gradient projection method leads to a coupled system of perturbed (by curvature terms) Hamilton?Jacobi equations. The coupling is enforced using a single Lagrange multiplier associated with a constraint which essentially prevents (a) regions from overlapping and (b) the development of a vacuum. The numerical implementation is relatively simple and the results agree with (and go beyond) the theory as given in 12. Other applications of this methodology, including the decomposition of a domain into subregions with minimal interface length, are discussed. Finally, some new techniques and results in level set methodology are presented.
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paper_title: Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations

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We devise new numerical algorithms, called PSC algorithms, for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front also can be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand sides, by using techniques from hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be also used for more general Hamilton-Jacobi-type problems. We demonstrate our algorithms by computing the solution to a variety of surface motion problems.
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paper_title: Snakes: Active contour models

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A snake is an energy-minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges. Snakes are active contour models: they lock onto nearby edges, localizing them accurately. Scale-space continuation can be used to enlarge the capture region surrounding a feature. Snakes provide a unified account of a number of visual problems, including detection of edges, lines, and subjective contours; motion tracking; and stereo matching. We have used snakes successfully for interactive interpretation, in which user-imposed constraint forces guide the snake near features of interest.
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paper_title: Level set evolution without re-initialization: a new variational formulation

paper_content:

In this paper, we present a new variational formulation for geometric active contours that forces the level set function to be close to a signed distance function, and therefore completely eliminates the need of the costly re-initialization procedure. Our variational formulation consists of an internal energy term that penalizes the deviation of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired image features, such as object boundaries. The resulting evolution of the level set function is the gradient flow that minimizes the overall energy functional. The proposed variational level set formulation has three main advantages over the traditional level set formulations. First, a significantly larger time step can be used for numerically solving the evolution partial differential equation, and therefore speeds up the curve evolution. Second, the level set function can be initialized with general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function. Third, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and is computationally more efficient. The proposed algorithm has been applied to both simulated and real images with promising results.
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paper_title: Level set evolution without re-initialization: a new variational formulation

paper_content:

In this paper, we present a new variational formulation for geometric active contours that forces the level set function to be close to a signed distance function, and therefore completely eliminates the need of the costly re-initialization procedure. Our variational formulation consists of an internal energy term that penalizes the deviation of the level set function from a signed distance function, and an external energy term that drives the motion of the zero level set toward the desired image features, such as object boundaries. The resulting evolution of the level set function is the gradient flow that minimizes the overall energy functional. The proposed variational level set formulation has three main advantages over the traditional level set formulations. First, a significantly larger time step can be used for numerically solving the evolution partial differential equation, and therefore speeds up the curve evolution. Second, the level set function can be initialized with general functions that are more efficient to construct and easier to use in practice than the widely used signed distance function. Third, the level set evolution in our formulation can be easily implemented by simple finite difference scheme and is computationally more efficient. The proposed algorithm has been applied to both simulated and real images with promising results.
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paper_title: Distance Regularized Level Set Evolution and Its Application to Image Segmentation

paper_content:

Level set methods have been widely used in image processing and computer vision. In conventional level set formulations, the level set function typically develops irregularities during its evolution, which may cause numerical errors and eventually destroy the stability of the evolution. Therefore, a numerical remedy, called reinitialization, is typically applied to periodically replace the degraded level set function with a signed distance function. However, the practice of reinitialization not only raises serious problems as when and how it should be performed, but also affects numerical accuracy in an undesirable way. This paper proposes a new variational level set formulation in which the regularity of the level set function is intrinsically maintained during the level set evolution. The level set evolution is derived as the gradient flow that minimizes an energy functional with a distance regularization term and an external energy that drives the motion of the zero level set toward desired locations. The distance regularization term is defined with a potential function such that the derived level set evolution has a unique forward-and-backward (FAB) diffusion effect, which is able to maintain a desired shape of the level set function, particularly a signed distance profile near the zero level set. This yields a new type of level set evolution called distance regularized level set evolution (DRLSE). The distance regularization effect eliminates the need for reinitialization and thereby avoids its induced numerical errors. In contrast to complicated implementations of conventional level set formulations, a simpler and more efficient finite difference scheme can be used to implement the DRLSE formulation. DRLSE also allows the use of more general and efficient initialization of the level set function. In its numerical implementation, relatively large time steps can be used in the finite difference scheme to reduce the number of iterations, while ensuring sufficient numerical accuracy. To demonstrate the effectiveness of the DRLSE formulation, we apply it to an edge-based active contour model for image segmentation, and provide a simple narrowband implementation to greatly reduce computational cost.
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Title: A Review on Level Set Method for Image Segmentation

Section 1: INTRODUCTION
Description 1: Introduce the fundamental concepts of image processing and segmentation, highlighting the significance and various applications of image segmentation, and give an overview of methods including the Level Set Method.

Section 2: VARIOUS LEVEL SET METHODS. 2.1 A New Variational Formulation.
Description 2: Discuss Chunming Li and Chenyang Xu's 2005 variational formulation for level set functions that eliminates the costly re-initialization procedure, enhancing computational efficiency.

Section 3: Image Segmentation Using Level Set
Description 3: Review Tenn Francis Chen's 2008 method for medical image segmentation using the level set approach, emphasizing its versatility in handling complex shapes without predefined knowledge of the target shape.

Section 4: Image Segmentation Using Active Contour
Description 4: Examine the 2009 proposal by M. Airouche, L. Bentabet, and M. Zelmat on using active contour model and Level Set Methods for detecting oil spills, focusing on the method's robust handling of topological changes and complex regions.

Section 5: Distance Regularized Level Set Evolution And Its Application to Image Segmentation
Description 5: Analyze Chunming Li and Chenyang Xu's 2010 method incorporating a distance regularized term in the level set evolution to maintain regularity and computational efficiency.

Section 6: CONCLUSION
Description 6: Summarize the reviewed methods, compare their advantages and limitations, and suggest the need for improvements or the development of a new robust method combining the strengths of existing approaches.

Section 7: ACKNOWLEDGMENTS
Description 7: Express gratitude to contributors and mentors who provided support and guidance.