We introduce the Hierarchical Poisson Disk Sampling Multi-Triangulation (HPDS-MT) of surfaces, a novel structure that combines the power of multi-triangulation (MT) with the benefits of Hierarchical Poisson Disk Sampling (HPDS). MT is a general framework for representing surfaces through variable resolution triangle meshes, while HPDS is a well-spaced random distribution with blue noise characteristics. The distinguishing feature of the HPDS-MT is its ability to extract adaptive meshes whose triangles are guaranteed to have good shape quality. The key idea behind the HPDS-MT is a preprocessed hierarchy of points, which is used in the construction of a MT via incremental simplification. In addition to proving theoretical properties on the shape quality of the triangle meshes extracted by the HPDS-MT, we provide an implementation that computes the HPDS-MT with high accuracy. Our results confirm the theoretical guarantees and outperform similar methods. We also prove that the Hausdorff distance between the original surface and any (extracted) adaptive mesh is bounded by the sampling distribution of the radii of Poisson-disks over the surface. Finally, we illustrate the advantages of the HPDS-MT in some typical problems of geometry processing.

We propose a new approach to front propagation algorithms based on a topological variant of well-composedness which contrasts with previous methods based on simple point detection. This provides for a theoretical justification, based on the digital Jordan separation theorem, for digitally "gluing" evolved well-composed objects separated by well-composed curves or surfaces. Additionally, our framework can be extended to more relaxed topologically constrained algorithms based on multisimple points. For both methods this framework has the additional benefit of obviating the requirement for both a user-specified connectivity and a topologically-consistent marching cubes/squares algorithm in meshing the resulting segmentation.

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-regular objects. Based on the trilinear interpolation, we also construct a smooth isosurface from the digital image that has the same topology as the original surface. Finally, we give a surprisingly simple topology preserving reconstruction method by using overlapping balls instead of cubical voxels. This is the first approach of digitizing 3D objects which guarantees topology preservation and gives an upper bound for the geometric distortion. Since the output can be chosen as a pure voxel presentation, a union of balls, a reconstruction by trilinear interpolation, a smooth isosurface, or the piecewise linear marching cubes surface, the results are directly applicable to a huge class of image analysis algorithms. Moreover, we show how one can efficiently estimate the volume and the surface area of 3D objects by looking at their digitizations. Measuring volume and surface area of digital objects are important problems in 3D image analysis. Good estimators should be multigrid convergent, i.e., the error goes to zero with increasing sampling density. We will show that every presented reconstruction method can be used for volume estimation and we will give a solution for the much more difficult problem of multigrid-convergent surface area estimation. Our solution is based on simple counting of voxels and we are the first to be able to give absolute bounds for the surface area.