# projects

## Multiresolution Surfaces and Surface Processing

Multiresolution surfaces are a natural extension of subdivision
surfaces; in addition to refining a base control mesh using a fixed
subdivision rule, details are added at each refinement step. The
resulting mesh is a *semiregular mesh*, that is, most vertices
of the mesh have fixed valence (same as for a regular grid), and
only few isolated vertices have a different valence. This
representation inherits many attractive features and efficiency of
subdivision, while allowing one to represent high resolution
surface geometry, either constructed from scratch or obtained using
a 3D scanner.

One of the crucial advantages of multiresolution surfaces for
representing most common high-resolutiuon surfaces comes from the
fact that the *connectivity of the mesh need not be explicitly
stored*, except for few extraordinary vertices, just as pixel
adjacencies need not be explicit for an image. In most cases,
connectivity of a fine mesh approximating a surface is an artifact
of a triangulation process, such as marching cubes, and conversion
to a semiregular mesh does not result in the loss of geometric
information. Regular structure of multiresolution surfaces
considerably reduces memory requirements, improves data coherence
and allows one to use much more efficient algorithms.

Multiresolution surfaces are closely related to wavelet-based
and overcomplete image representations, and more generally, makes
surfaces much more similar to images. This makes it possible to
generalize many of the signal processing techniques to handle
geometric data, with applications in modification, compression and
transmission of surface data (**surface processing**).

Our research in this area includes several directions:

**Multiresolution editing.**Our past work on interactive editing of high-complexity surfaces resulted in development of basic tools for multiresolution representations.**Approximate boolean operations on multiresolution surfaces.**Using unions, intersections and differences is a natural way of constructing more complex objects out of primitives. Free-form boundary representations using NURBS or subdivision surfaces are not closed under these operations; the result can be only approximated. Multiresolution surfaces are well suited for computing such approximate results of boolean operations with any desired precision. A more detailed description and examples are available on a separate page.**Surface processing.**We aim to generalize a variety of filters used for images to the case of multiresoltion surfaces and develop new filters for automatic surface modification. A first step in this direction is our work on surface denoising.

### People

### Publications

- Procedural Shape Synthesis on Subdivision Surfaces
- Luiz Velho, Ken Perlin, Lexing Ying, Henning Biermann
*Proc. SIBGRAPI 2001.*

- Sharp features on Multiresolution Subdivision Surfaces
- Henning Biermann, Ioana M. Martin, Denis Zorin, Fausto Bernardini
*Proc. Pacific Graphics 2001.*

- Texture and Shape Synthesis on Surfaces
- Lexing Ying, Aaron Hertzmann, Henning Biermann, Denis Zorin
*Proc. 12th Eurographics Workshop on Rendering, Jun 2001*.

- Boolean Operations on Free-form Solids
- Henning Biermann, Daniel Kristjansson, Denis Zorin
*Proc. SIGGRAPH 2001.*

- Interactive multiresolution mesh editing
- D. Zorin, P. Schröder, W. Sweldens
*Computer Graphics (SIGGRAPH '97 Proceedings), pp. 256-268.*

- Subdivision and Multiresolution Surfaces
- Denis Zorin
*Ph.D. Thesis, Caltech, 1998.*