@article{5800,
  abstract     = {This paper presents a novel study on the functional gradation of coordinate planes in connection with the thinnest and tunnel-free (i.e., naive) discretization of sphere in the integer space. For each of the 48-symmetric quadraginta octants of naive sphere with integer radius and integer center, we show that the corresponding voxel set forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as its functional plane. We use this fundamental property to prove several other theoretical results for naive sphere. First, the quadraginta octants form symmetry groups and subgroups with certain equivalent topological properties. Second, a naive sphere is always unique and consists of fewest voxels. Third, it is efficiently constructible from its functional-plane projection. And finally, a special class of 4-symmetric discrete 3D circles can be constructed on a naive sphere based on back projection from the functional plane.},
  author       = {Biswas, Ranita and Bhowmick, Partha},
  issn         = {09249907},
  journal      = {Journal of Mathematical Imaging and Vision},
  number       = {1},
  pages        = {69--83},
  publisher    = {Springer Nature},
  title        = {{On the functionality and usefulness of Quadraginta octants of naive sphere}},
  doi          = {10.1007/s10851-017-0718-4},
  volume       = {59},
  year         = {2017},
}

@article{2255,
  abstract     = {Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma-Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm.},
  author       = {Edelsbrunner, Herbert and Pausinger, Florian},
  issn         = {09249907},
  journal      = {Journal of Mathematical Imaging and Vision},
  number       = {1},
  pages        = {164 -- 177},
  publisher    = {Springer},
  title        = {{Stable length estimates of tube-like shapes}},
  doi          = {10.1007/s10851-013-0468-x},
  volume       = {50},
  year         = {2014},
}