Level set and PDE methods for computer graphics

Abstract

Course AbstractLevel set methods, an important class of partial differential equation (PDE) methods, define dynamic surfaces implicitly as the level set (isosurface) of a sampled, evolving nD function. The course begins with preparatory material that introduces the concept of using partial differential equations to solve problems in computer graphics, geometric modeling and computer vision. This will include the structure and behavior of several different types of differential equations, e.g. the level set equation and the heat equation, as well as a general approach to developing PDE-based applications. The second stage of the course will describe the numerical methods and algorithms needed to actually implement the mathematics and methods presented in the first stage. The course closes with detailed presentations on several level set/PDE applications, including image/video inpainting, pattern formation, image/volume processing, 3D shape reconstruction, image/volume segmentation, image/shape morphing, geometric modeling, anisotropic diffusion, and natural phenomena simulation. PrerequisitesKnowledge of calculus, linear algebra, computer graphics, geometric modeling, image processing and computer vision. Some familiarity with differential geometry, differential equations, numerical computing and image processing is strongly recommended, but not required. participates in the reviewing process for a number of journals and funding agencies. He has published over 45 research papers in computational physics, computer graphics and vision, as well as a new book on level set methods. Additionally, he has been a consultant for Industrial Light + Magic for the last three years.

Efficient simulation of large bodies of water by coupling two and three dimensional techniques

“…Luckily, we can leverage recent work on Run-Length-Encoded (RLE) grids by Breen et al 2004;Wiebe and Houston 2004;Nielsen and Museth 2005;Houston et al 2005;Houston et al 2006] (note that [Curless and Levoy 1996] applied RLE to volumes and [Bridson 2003] suggested RLE level sets). Their fluid simulator solves the level set on a narrow band near the interface, but uses a uniform MAC discretization inside the liquid for the fluid solver with long cells allowed only in the air.…”

Simplex Free Adaptive Tree Fast Sweeping and Evolution Methods for Solving Level Set Equations in Arbitrary Dimension

“…In addition, for the eikonal equation used to maintain the signed distance property of the level set function, [19] used some special treatment at T-junctions in the context of the fast marching spirit [37]. There have been other local level set methods [20], [35], [5], [14], [4] proposed which range from variants of AMR to using tubes of uniformly spaced grid points near the interface. Some of the methods approach the complexity of [26], eliminating the need to store the unused grid points away from the interface of interest.…”

Efficient simulation of large bodies of water by coupling two and three dimensional techniques

“…Luckily, we can leverage recent work on Run-LengthEncoded (RLE) grids by Breen et al 2004;Wiebe and Houston 2004;Nielsen and Museth 2005;Houston et al 2005;Houston et al 2006] (note that [Curless and Levoy 1996] applied RLE to volumes and [Bridson 2003] suggested RLE level sets). Their fluid simulator solves the level set on a narrow band near the interface, but uses a uniform MAC discretization inside the liquid for the fluid solver with long cells allowed only in the air.…”