Name
Abstract | ||
|---|---|---|
Discrete filtering of information over triangulated surfaces has proved very useful in computer graphics applications. This technique is based on diffusion equations and has been extensively applied to image processing, harmonic map regularization and texture generating, etc. [C. L. Bajaj and G. Xu, ACM Trans. Graph., 22 (2003), pp. 4-32], [C. Wu, J. Deng, and F. Chen, IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 666-679]. However, little has been done on analysis (especially quantitative analysis) of the behavior of these filtering procedures. Since in applications mesh surfaces can be of arbitrary topology and the filtering can be nonlinear and even anisotropic, the analysis of the quantitative behavior is a very difficult issue. In this paper, we first present the discrete linear, nonlinear, and anisotropic filtering schemes via discretizing diffusion equations with appropriately defined differential operators on triangulated surfaces, and then use concepts of discrete scale-spaces to describe these filtering procedures and analyze their properties respectively. Scale-space properties such as existence and uniqueness, continuous dependence on initial value, discrete semigroup property, grey level shift invariance and conservation of total grey level, information reduction (also known as topology simplification), and constant limit behavior have been proved. In particular, the information reduction property is analyzed by eigenvalue and eigenvector analysis of matrices. Different from the direct observation of the local filtering to the diffusion equations and other interpretation methods based on wholly global quantities such as energy and entropy, this viewpoint helps us understand the filtering both globally (information reduction as image components shrink) and locally (how the image component contributes to its shrink rate). With careful consideration of the correspondence between eigenvalues and eigenvectors and their features, differences between linear and nonlinear filtering, as well as between isotropic and anisotropic filtering, are discussed. We also get some stability results of the filtering schemes. Several examples are provided to illustrate the properties. |
Year | DOI | Venue |
|---|---|---|
2009 | 10.1137/080722758 | SIAM J. Imaging Sciences |
Keywords | Field | DocType |
eigenvector analysis,information reduction property,diffusion equation,scale-space analysis,information reduction,constant limit behavior,discrete semigroup property,triangulated surface,image component,quantitative analysis,arbitrary triangulated surfaces,discrete filtering,discrete scale-spaces,scale space | Discretization,Mathematical optimization,Nonlinear system,Matrix (mathematics),Mathematical analysis,Filter (signal processing),Anisotropic filtering,Scale space,Trilinear filtering,Eigenvalues and eigenvectors,Mathematics | Journal |
Volume | Issue | ISSN |
2 | 2 | 1936-4954 |
Citations | PageRank | References |
10 | 0.54 | 32 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Chunlin Wu | 1 | 172 | 4.51 |
| jiansong deng | 2 | 458 | 38.59 |
| Falai Chen | 3 | 403 | 32.47 |
| Xue-Cheng Tai | 4 | 2090 | 131.53 |