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MULTIRESOLUTION OF SURFACES WITH ARBITRARY TOPOLOGIES USING A REVERSE DOO SUBDIVISION SCHEME

Determination of the reverse of a subdivision using the global least squares fitting leads to a multiresolution structure which can be considered as a semiorthogonal wavelet system. However in the construction of multiresolution surfaces having arbitrary topologies, biorthogonal wavelets are used. For reversing the subdivision of curves and tensor product surfaces, one may employ the local least squares fitting leading to a biorthogonal system. Here, multiresolution surfaces having arbitrary topologies are constructed using a local reverse Doo subdivision scheme. Using a Doo subdivision, a coarse surface is converted to a fine one. This is achieved by the contraction of coarse faces and the addition of new faces. A method is introduced for reversing the Doo subdivision to convert a fine surface into a coarse one along with the associated error. In this process, first those faces of the fine surface, which could have been produced by the contraction of a couse face in a Doo subdivision process, are found. Then these faces are expanded. Since the expanding faces are not necessarily joined properly, several candidates are usually considered for each vertex of the coarse surface. To identity the set of candidates, a graph is constructed in which any set of candidates for a vertex of the coarse surface corresponds to a connected component. To recognize the connected components, the depth first search traversal of the graph is employed. Finally, each vertex of the coarse surface is set to be the average of its corresponding candidates. The results obtained have applications in the compact representation and effective reconstruction of digital images.