LevelSets

Level sets, also known as level set methods, are a powerful numerical technique used to track and represent evolving interfaces or shapes. They are particularly useful in fields like computer graphics, image processing, and computational fluid dynamics. The core idea of a level set method is to represent a curve or surface implicitly as the zero level set of a higher-dimensional function. This higher-dimensional function is called the level set function, and its value at any point in space indicates the signed distance to the interface.

The evolution of the interface is then governed by the evolution of this higher-dimensional function. This is typically achieved by solving a partial differential equation, known as the Hamilton-Jacobi equation, that describes how the level set function changes over time. A key advantage of this approach is that it naturally handles topological changes, such as merging or splitting of interfaces, which can be very difficult to manage with explicit representations.

The level set method discretizes the level set function on a grid and uses numerical methods to approximate the solution of the Hamilton-Jacobi equation. Common discretization schemes include finite differences. While the level set function itself represents the interface implicitly, the actual interface is extracted by finding where the function's value is zero. This implicit representation avoids the need for complex mesh manipulation and simplifies the handling of complex geometric scenarios. The accuracy and efficiency of the method depend on the grid resolution and the numerical scheme employed.