🍩 Database of Original & NonTheoretical Uses of Topology
(found 3 matches in 0.001654s)


Theory and Algorithms for Constructing Discrete Morse Complexes From Grayscale Digital Images (2011)
V. Robins, P. J. Wood, A. P. SheppardAbstract
We present an algorithm for determining the Morse complex of a two or threedimensional grayscale digital image. Each cell in the Morse complex corresponds to a topological change in the level sets (i.e., a critical point) of the grayscale image. Since more than one critical point may be associated with a single image voxel, we model digital images by cubical complexes. A new homotopic algorithm is used to construct a discrete Morse function on the cubical complex that agrees with the digital image and has exactly the number and type of critical cells necessary to characterize the topological changes in the level sets. We make use of discrete Morse theory and simple homotopy theory to prove correctness of this algorithm. The resulting Morse complex is considerably simpler than the cubical complex originally used to represent the image and may be used to compute persistent homology. 
Computing Robustness and Persistence for Images (2010)
P. Bendich, H. Edelsbrunner, M. KerberAbstract
We are interested in 3dimensional images given as arrays of voxels with intensity values. Extending these values to a continuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbation needed to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can be visualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchical algorithm using the dual complexes of octtree approximations of the function. In addition, we show that for balanced octtrees, the dual complexes are geometrically realized in R3 and can thus be used to construct level and interlevel sets. We apply these tools to study 3dimensional images of plant root systems.