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Multiresolution methods and hierarchical data organization have become powerful tools for the representation of surfaces in computer graphics. Their power lies in the fact that they combine a lot of useful properties, such as level of detail, local support, smoothness, error bounds and fast computations. Interestingly, many popular multiresolution representations or processing methods can be considered as extensions of concepts that have their origins in signal processing.
In this talk I will discuss three prominent examples of multiresolution geometric signal processing: The first one introduces wavelets with an emphasis on B-spline wavelets. Topics comprise boundary conditions, level of detail control, local and global oracles, and extensions to multiple dimensions. Applications include progressive compression and rendering of volume data. As a second example I will present subdivision algorithms that smooth and refine surfaces using a sequence of subdivision steps. I will describe subdivision surfaces, their origins, the most popular subdivision rules, their properties, and applications. The last example introduces the concept of surface filtering based on so-called fairing operators. They smooth a mesh using iterative low-pass filtering to attenuate high geometric frequencies from surfaces. Such operators are very effective for the removal of noise from geometry.
Colloquium - 16:00
Abstract: The boundary of a 3-dimensional polytope is homeomorphic to a 2-dimensional sphere that is decomposed in cells. Under reasonable conditions on the cell decomposition the converse is also true:
Every regular cell decomposition of the 2-sphere that obeys an intersection property can be straightened to become the boundary of a 3-polytope. In this sense 3-polytopes and 2-spheres are "the same."
One dimension higher the analogue is not true. There are regular cell decompositions of spheres that are not polytopal.
The f-vector (f_0,f_1,f_2,f_3) of a cell decomposition respectively of a polytope counts the number of vertices, edges, 2-cells and 3-cells. In our context one can ask if there are certain f-vectors that occur as f-vectors of decomposed 3-spheres but not as f-vectors of 4-polytopes.
One way of finding an answer may be to consider the tilings that can be constructed from the boundaries of polytopes...