Marching cubes: A high resolution 3D surface construction algorithm
SIGGRAPH '87 Proceedings of the 14th annual conference on Computer graphics and interactive techniques
Topological considerations in isosurface generation extended abstract
VVS '90 Proceedings of the 1990 workshop on Volume visualization
Graphical Models and Image Processing
Topology-preserving deformations of two-valued digital pictures
Graphical Models and Image Processing
Preserving Topology by a Digitization Process
Journal of Mathematical Imaging and Vision
ACM-SE 37 Proceedings of the 37th annual Southeast regional conference (CD-ROM)
VVS '89 Proceedings of the 1989 Chapel Hill workshop on Volume visualization
Multigrid Convergence of Calculated Features in Image Analysis
Journal of Mathematical Imaging and Vision
Algorithms for Graphics and Imag
Algorithms for Graphics and Imag
Topological properties of Hausdorff discretization, and comparison to other discretization schemes
Theoretical Computer Science
Improving the Robustness and Accuracy of the Marching Cubes Algorithm for Isosurfacing
IEEE Transactions on Visualization and Computer Graphics
On Approximation of Jordan Surfaces in 3D
Digital and Image Geometry, Advanced Lectures [based on a winter school held at Dagstuhl Castle, Germany in December 2000]
Multi-level partition of unity implicits
ACM SIGGRAPH 2003 Papers
The asymptotic decider: resolving the ambiguity in marching cubes
VIS '91 Proceedings of the 2nd conference on Visualization '91
VIS '04 Proceedings of the conference on Visualization '04
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Towards a general sampling theory for shape preservation
Image and Vision Computing
Multigrid convergence and surface area estimation
Proceedings of the 11th international conference on Theoretical foundations of computer vision
An efficient euclidean distance transform
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
Meshless isosurface generation from multiblock data
VISSYM'04 Proceedings of the Sixth Joint Eurographics - IEEE TCVG conference on Visualization
Topological Repairing of 3D Digital Images
Journal of Mathematical Imaging and Vision
Digitization scheme that assures faithful reconstruction of plane figures
Pattern Recognition
Digital Topology on Adaptive Octree Grids
Journal of Mathematical Imaging and Vision
Scaling of plane figures that assures faithful digitization
IWCIA'08 Proceedings of the 12th international conference on Combinatorial image analysis
On the search of optimal reconstruction resolution
Pattern Recognition Letters
Enhancing the reconstruction from non-uniform point sets using persistence information
CTIC'12 Proceedings of the 4th international conference on Computational Topology in Image Context
GPU-based offset surface computation using point samples
Computer-Aided Design
Smoothness of Boundaries of Regular Sets
Journal of Mathematical Imaging and Vision
On Multigrid Convergence of Local Algorithms for Intrinsic Volumes
Journal of Mathematical Imaging and Vision
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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-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.