---
_id: '18390'
abstract:
- lang: eng
  text: Symmetry and self-similarity is the cornerstone of Nature, exhibiting itself
    through the shapes of natural creations and ubiquitous laws of physics. Since
    many natural objects are symmetric, the absence of symmetry can often be an indication
    of some anomaly or abnormal behavior. Therefore, detection of asymmetries is important
    in numerous practical applications, including crystallography, medical imaging,
    and face recognition, to mention a few. Conversely, the assumption of underlying
    shape symmetry can facilitate solutions to many problems in shape reconstruction
    and analysis. Traditionally, symmetries are described as extrinsic geometric properties
    of the shape. While being adequate for rigid shapes, such a description is inappropriate
    for non-rigid ones. Extrinsic symmetry can be broken as a result of shape deformations,
    while its intrinsic symmetry is preserved. In this paper, we pose the problem
    of finding intrinsic symmetries of non-rigid shapes and propose an efficient method
    for their computation.
article_processing_charge: No
author:
- first_name: Dan
  full_name: Raviv, Dan
  last_name: Raviv
- first_name: Alexander
  full_name: Bronstein, Alexander
  id: 58f3726e-7cba-11ef-ad8b-e6e8cb3904e6
  last_name: Bronstein
  orcid: 0000-0001-9699-8730
- first_name: Michael M.
  full_name: Bronstein, Michael M.
  last_name: Bronstein
- first_name: Ron
  full_name: Kimmel, Ron
  last_name: Kimmel
citation:
  ama: 'Raviv D, Bronstein AM, Bronstein MM, Kimmel R. Symmetries of non-rigid shapes.
    In: <i>11th International Conference on Computer Vision</i>. Vol 72. IEEE; 2007.
    doi:<a href="https://doi.org/10.1109/iccv.2007.4409181">10.1109/iccv.2007.4409181</a>'
  apa: 'Raviv, D., Bronstein, A. M., Bronstein, M. M., &#38; Kimmel, R. (2007). Symmetries
    of non-rigid shapes. In <i>11th International Conference on Computer Vision</i>
    (Vol. 72). Rio de Janeiro, Brazil: IEEE. <a href="https://doi.org/10.1109/iccv.2007.4409181">https://doi.org/10.1109/iccv.2007.4409181</a>'
  chicago: Raviv, Dan, Alex M. Bronstein, Michael M. Bronstein, and Ron Kimmel. “Symmetries
    of Non-Rigid Shapes.” In <i>11th International Conference on Computer Vision</i>,
    Vol. 72. IEEE, 2007. <a href="https://doi.org/10.1109/iccv.2007.4409181">https://doi.org/10.1109/iccv.2007.4409181</a>.
  ieee: D. Raviv, A. M. Bronstein, M. M. Bronstein, and R. Kimmel, “Symmetries of
    non-rigid shapes,” in <i>11th International Conference on Computer Vision</i>,
    Rio de Janeiro, Brazil, 2007, vol. 72.
  ista: 'Raviv D, Bronstein AM, Bronstein MM, Kimmel R. 2007. Symmetries of non-rigid
    shapes. 11th International Conference on Computer Vision. ICCV: International
    Conference on Computer Vision vol. 72.'
  mla: Raviv, Dan, et al. “Symmetries of Non-Rigid Shapes.” <i>11th International
    Conference on Computer Vision</i>, vol. 72, IEEE, 2007, doi:<a href="https://doi.org/10.1109/iccv.2007.4409181">10.1109/iccv.2007.4409181</a>.
  short: D. Raviv, A.M. Bronstein, M.M. Bronstein, R. Kimmel, in:, 11th International
    Conference on Computer Vision, IEEE, 2007.
conference:
  end_date: 2007-10-21
  location: Rio de Janeiro, Brazil
  name: 'ICCV: International Conference on Computer Vision'
  start_date: 2007-10-14
date_created: 2024-10-15T11:20:54Z
date_published: 2007-12-26T00:00:00Z
date_updated: 2024-11-18T10:07:58Z
day: '26'
doi: 10.1109/iccv.2007.4409181
extern: '1'
intvolume: '        72'
language:
- iso: eng
month: '12'
oa_version: None
publication: 11th International Conference on Computer Vision
publication_identifier:
  eissn:
  - 2380-7504
  isbn:
  - '9781424416301'
  issn:
  - 1550-5499
publication_status: published
publisher: IEEE
quality_controlled: '1'
scopus_import: '1'
status: public
title: Symmetries of non-rigid shapes
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 72
year: '2007'
...
---
OA_type: closed access
_id: '3182'
abstract:
- lang: eng
  text: In the work of the authors (2003), we showed that graph cuts can find hypersurfaces
    of globally minimal length (or area) under any Riemannian metric. Here we show
    that graph cuts on directed regular grids can approximate a significantly more
    general class of continuous non-symmetric metrics. Using submodularity condition
    (Boros and Hammer, 2002 and Kolmogorov and Zabih, 2004), we obtain a tight characterization
    of graph-representable metrics. Such &quot;submodular&quot; metrics have an elegant
    geometric interpretation via hypersurface functionals combining length/area and
    flux. Practically speaking, we attend 'geo-cuts' algorithm to a wider class of
    geometrically motivated hypersurface functionals and show how to globally optimize
    any combination of length/area and flux of a given vector field. The concept of
    flux was recently introduced into computer vision by Vasilevskiy and Siddiqi (2002)
    but it was mainly studied within variational framework so far. We are first to
    show that flux can be integrated into graph cuts as well. Combining geometric
    concepts of flux and length/area within the global optimization framework of graph
    cuts allows principled discrete segmentation models and advances the slate of
    the art for the graph cuts methods in vision. In particular we address the &quot;shrinking&quot;
    problem of graph cuts, improve segmentation of long thin objects, and introduce
    useful shape constraints.
article_processing_charge: No
author:
- first_name: Vladimir
  full_name: Kolmogorov, Vladimir
  id: 3D50B0BA-F248-11E8-B48F-1D18A9856A87
  last_name: Kolmogorov
- first_name: Yuri
  full_name: Boykov, Yuri
  last_name: Boykov
citation:
  ama: 'Kolmogorov V, Boykov Y. What metrics can be approximated by geo cuts or global
    optimization of length area and flux. In: <i>Proceedings of the Tenth IEEE International
    Conference on Computer Vision Volume 1</i>. Vol 1. IEEE; 2005:564-571. doi:<a
    href="https://doi.org/10.1109/ICCV.2005.252">10.1109/ICCV.2005.252</a>'
  apa: 'Kolmogorov, V., &#38; Boykov, Y. (2005). What metrics can be approximated
    by geo cuts or global optimization of length area and flux. In <i>Proceedings
    of the Tenth IEEE International Conference on Computer Vision Volume 1</i> (Vol.
    1, pp. 564–571). NW Washington, DC, United States: IEEE. <a href="https://doi.org/10.1109/ICCV.2005.252">https://doi.org/10.1109/ICCV.2005.252</a>'
  chicago: Kolmogorov, Vladimir, and Yuri Boykov. “What Metrics Can Be Approximated
    by Geo Cuts or Global Optimization of Length Area and Flux.” In <i>Proceedings
    of the Tenth IEEE International Conference on Computer Vision Volume 1</i>, 1:564–71.
    IEEE, 2005. <a href="https://doi.org/10.1109/ICCV.2005.252">https://doi.org/10.1109/ICCV.2005.252</a>.
  ieee: V. Kolmogorov and Y. Boykov, “What metrics can be approximated by geo cuts
    or global optimization of length area and flux,” in <i>Proceedings of the Tenth
    IEEE International Conference on Computer Vision Volume 1</i>, NW Washington,
    DC, United States, 2005, vol. 1, pp. 564–571.
  ista: 'Kolmogorov V, Boykov Y. 2005. What metrics can be approximated by geo cuts
    or global optimization of length area and flux. Proceedings of the Tenth IEEE
    International Conference on Computer Vision Volume 1. ICCV: International Conference
    on Computer Vision vol. 1, 564–571.'
  mla: Kolmogorov, Vladimir, and Yuri Boykov. “What Metrics Can Be Approximated by
    Geo Cuts or Global Optimization of Length Area and Flux.” <i>Proceedings of the
    Tenth IEEE International Conference on Computer Vision Volume 1</i>, vol. 1, IEEE,
    2005, pp. 564–71, doi:<a href="https://doi.org/10.1109/ICCV.2005.252">10.1109/ICCV.2005.252</a>.
  short: V. Kolmogorov, Y. Boykov, in:, Proceedings of the Tenth IEEE International
    Conference on Computer Vision Volume 1, IEEE, 2005, pp. 564–571.
conference:
  end_date: 2005-10-20
  location: NW Washington, DC, United States
  name: 'ICCV: International Conference on Computer Vision'
  start_date: 2005-10-17
das_tickbox: '1'
date_created: 2018-12-11T12:01:52Z
date_published: 2005-12-05T00:00:00Z
date_updated: 2026-07-02T08:53:55Z
day: '05'
doi: 10.1109/ICCV.2005.252
extern: '1'
intvolume: '         1'
language:
- iso: eng
month: '12'
oa_version: None
page: 564 - 571
publication: Proceedings of the Tenth IEEE International Conference on Computer Vision
  Volume 1
publication_identifier:
  eissn:
  - 2380-7504
  issn:
  - 1550-5499
publication_status: published
publisher: IEEE
publist_id: '3501'
status: public
title: What metrics can be approximated by geo cuts or global optimization of length
  area and flux
type: conference
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 1
year: '2005'
...