An output-sensitive algorithm for persistent homology
Proceedings of the twenty-seventh annual symposium on Computational geometry
Alpha, betti and the megaparsec universe: on the topology of the cosmic web
Transactions on Computational Science XIV
A point calculus for interlevel set homology
Pattern Recognition Letters
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
Stable morse decompositions for piecewise constant vector fields on surfaces
EuroVis'11 Proceedings of the 13th Eurographics / IEEE - VGTC conference on Visualization
An output-sensitive algorithm for persistent homology
Computational Geometry: Theory and Applications
A study of monodromy in the computation of multidimensional persistence
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
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We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to acontinuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbationneeded to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can bevisualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchicalalgorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, thedual complexes are geometrically realized in $R^3$ and can thus be used to construct level and interlevel sets. We apply these tools tostudy 3-dimensional images of plant root systems.