Topological Equivalence between a 3D Object and the Reconstruction of Its Digital Image

13 years 4 months ago

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— Digitization is not as easy as it looks. If one digitizes a 3D object even with a dense sampling grid, the reconstructed digital object may have topological distortions and in general there exists no upper bound for the Hausdorff distance. This explains why so far no algorithm has been known which guarantees topology preservation. However, as we will show, it is possible to repair the obtained digital image in a locally bounded way so that it is homeomorphic and close to the 3D object. The resulting digital object is always well-composed, which has nice implications for a lot of image analysis problems. Moreover, we will show that the surface of the original object is homeomorphic to the result of the marching cubes algorithm. This is really surprising since it means that the well known topological problems of the marching cubes reconstruction simply do not occur for digital images of r-reglar objects. Based on the trilinear interpolation we also construct a smooth isosurface from ...

Peer Stelldinger, Longin Jan Latecki, Marcelo Siqu

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