---
_id: '5800'
abstract:
- lang: eng
  text: 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:
- first_name: Ranita
  full_name: Biswas, Ranita
  id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
  last_name: Biswas
  orcid: 0000-0002-5372-7890
- first_name: Partha
  full_name: Bhowmick, Partha
  last_name: Bhowmick
citation:
  ama: Biswas R, Bhowmick P. On the functionality and usefulness of Quadraginta octants
    of naive sphere. <i>Journal of Mathematical Imaging and Vision</i>. 2017;59(1):69-83.
    doi:<a href="https://doi.org/10.1007/s10851-017-0718-4">10.1007/s10851-017-0718-4</a>
  apa: Biswas, R., &#38; Bhowmick, P. (2017). On the functionality and usefulness
    of Quadraginta octants of naive sphere. <i>Journal of Mathematical Imaging and
    Vision</i>. Springer Nature. <a href="https://doi.org/10.1007/s10851-017-0718-4">https://doi.org/10.1007/s10851-017-0718-4</a>
  chicago: Biswas, Ranita, and Partha Bhowmick. “On the Functionality and Usefulness
    of Quadraginta Octants of Naive Sphere.” <i>Journal of Mathematical Imaging and
    Vision</i>. Springer Nature, 2017. <a href="https://doi.org/10.1007/s10851-017-0718-4">https://doi.org/10.1007/s10851-017-0718-4</a>.
  ieee: R. Biswas and P. Bhowmick, “On the functionality and usefulness of Quadraginta
    octants of naive sphere,” <i>Journal of Mathematical Imaging and Vision</i>, vol.
    59, no. 1. Springer Nature, pp. 69–83, 2017.
  ista: Biswas R, Bhowmick P. 2017. On the functionality and usefulness of Quadraginta
    octants of naive sphere. Journal of Mathematical Imaging and Vision. 59(1), 69–83.
  mla: Biswas, Ranita, and Partha Bhowmick. “On the Functionality and Usefulness of
    Quadraginta Octants of Naive Sphere.” <i>Journal of Mathematical Imaging and Vision</i>,
    vol. 59, no. 1, Springer Nature, 2017, pp. 69–83, doi:<a href="https://doi.org/10.1007/s10851-017-0718-4">10.1007/s10851-017-0718-4</a>.
  short: R. Biswas, P. Bhowmick, Journal of Mathematical Imaging and Vision 59 (2017)
    69–83.
date_created: 2019-01-08T20:42:08Z
date_published: 2017-09-01T00:00:00Z
date_updated: 2021-01-12T08:03:34Z
day: '01'
doi: 10.1007/s10851-017-0718-4
extern: '1'
intvolume: '        59'
issue: '1'
language:
- iso: eng
month: '09'
oa_version: None
page: 69-83
publication: Journal of Mathematical Imaging and Vision
publication_identifier:
  issn:
  - '09249907'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: On the functionality and usefulness of Quadraginta octants of naive sphere
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 59
year: '2017'
...
---
_id: '2255'
abstract:
- lang: eng
  text: 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:
- first_name: Herbert
  full_name: Edelsbrunner, Herbert
  id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
  last_name: Edelsbrunner
  orcid: 0000-0002-9823-6833
- first_name: Florian
  full_name: Pausinger, Florian
  id: 2A77D7A2-F248-11E8-B48F-1D18A9856A87
  last_name: Pausinger
  orcid: 0000-0002-8379-3768
citation:
  ama: Edelsbrunner H, Pausinger F. Stable length estimates of tube-like shapes. <i>Journal
    of Mathematical Imaging and Vision</i>. 2014;50(1):164-177. doi:<a href="https://doi.org/10.1007/s10851-013-0468-x">10.1007/s10851-013-0468-x</a>
  apa: Edelsbrunner, H., &#38; Pausinger, F. (2014). Stable length estimates of tube-like
    shapes. <i>Journal of Mathematical Imaging and Vision</i>. Springer. <a href="https://doi.org/10.1007/s10851-013-0468-x">https://doi.org/10.1007/s10851-013-0468-x</a>
  chicago: Edelsbrunner, Herbert, and Florian Pausinger. “Stable Length Estimates
    of Tube-like Shapes.” <i>Journal of Mathematical Imaging and Vision</i>. Springer,
    2014. <a href="https://doi.org/10.1007/s10851-013-0468-x">https://doi.org/10.1007/s10851-013-0468-x</a>.
  ieee: H. Edelsbrunner and F. Pausinger, “Stable length estimates of tube-like shapes,”
    <i>Journal of Mathematical Imaging and Vision</i>, vol. 50, no. 1. Springer, pp.
    164–177, 2014.
  ista: Edelsbrunner H, Pausinger F. 2014. Stable length estimates of tube-like shapes.
    Journal of Mathematical Imaging and Vision. 50(1), 164–177.
  mla: Edelsbrunner, Herbert, and Florian Pausinger. “Stable Length Estimates of Tube-like
    Shapes.” <i>Journal of Mathematical Imaging and Vision</i>, vol. 50, no. 1, Springer,
    2014, pp. 164–77, doi:<a href="https://doi.org/10.1007/s10851-013-0468-x">10.1007/s10851-013-0468-x</a>.
  short: H. Edelsbrunner, F. Pausinger, Journal of Mathematical Imaging and Vision
    50 (2014) 164–177.
date_created: 2018-12-11T11:56:36Z
date_published: 2014-09-01T00:00:00Z
date_updated: 2023-09-07T11:41:25Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s10851-013-0468-x
ec_funded: 1
file:
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  creator: system
  date_created: 2018-12-12T10:16:18Z
  date_updated: 2020-07-14T12:45:35Z
  file_id: '5204'
  file_name: IST-2016-549-v1+1_2014-J-06-LengthEstimate.pdf
  file_size: 3941391
  relation: main_file
file_date_updated: 2020-07-14T12:45:35Z
has_accepted_license: '1'
intvolume: '        50'
issue: '1'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Submitted Version
page: 164 - 177
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '318493'
  name: Topological Complex Systems
publication: Journal of Mathematical Imaging and Vision
publication_identifier:
  issn:
  - '09249907'
publication_status: published
publisher: Springer
publist_id: '4691'
pubrep_id: '549'
quality_controlled: '1'
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status: public
title: Stable length estimates of tube-like shapes
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 50
year: '2014'
...