Classically, the concept of multiple resolution was closely associated with the study of wavelets. Wavelets are useful for describing mathematical objects such as functions (or signals) with different resolution levels. For example, an image can be described in different resolution levels. Understanding and characterizing the differences between images (or functions) at different resolution levels is what wavelets are all about. Wavelets are important in 3D configurator projects.

What is Multi-Resolution?

The classical techniques work well in a regular, slightly disinfected environment. They are less useful in the more general environments where engineers and computer scientists meet in practical examples. Graphics people realized some time ago that wavelets are basically a very powerful ally in overcoming many of the computational challenges in graphics. However, wavelet constructions had to be much more flexible than described in classical literature. The researchers quickly made use of the connection between subdivision and wavelets. In particular, there are a number of arbitrary topology subdivision schemes for surfaces that provide an elegant basis for building multiresolutional representations.

Subdivision defines smooth surfaces as the boundary of a sequence of successively refined polyhedra. It was originally developed as a generalization of polynomial patch-based methods to the arbitrary topology environment and has been studied for about 15 years in the mathematical CAGD literature. In recent years, these techniques have attracted more interest in computer graphics literature due to the many advantages of subdivision.

The basic algorithms are extremely simple. A given triangle (or square) is divided by inserting new midpoints along the edges (and a midpoint for squares). The new points are calculated as weighted mean values of the neighboring points. Because of this simplicity, very few assumptions need to be made about the global nature of the objects to be modelled.

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For example, data structures only need to support operations such as neighbor search in a diagram. Special constraints, such as boundary conditions, can be easily taken into account by locally changing the weights of the subdivision schema.

No global constraint systems are required to ensure smooth operation. The domain of the subdivision is an abstract topological complex. No metrics are required, no global (or even local) parameterization, and no embedding is required. Last but not least, important functionalities such as basics for the tangential spaces to the boundary surface and values of the boundary surface can be calculated exactly.

Since the entire process is based on mesh refinement, the subdivision naturally closes the gap between polygonal mesh representations and higher-level modeling paradigms such as patches. Subdivision provides a basic framework for techniques such as level-of-detail rendering, compression, progressive transmission, adaptive meshing, and many other algorithms based on multiresolution. Together with its deep and rich connections to wavelets, it creates a powerful paradigm for precise representation and efficient numerical computation.

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