---
_id: '12764'
abstract:
- lang: eng
text: We study a new discretization of the Gaussian curvature for polyhedral surfaces.
This discrete Gaussian curvature is defined on each conical singularity of a polyhedral
surface as the quotient of the angle defect and the area of the Voronoi cell corresponding
to the singularity. We divide polyhedral surfaces into discrete conformal classes
using a generalization of discrete conformal equivalence pioneered by Feng Luo.
We subsequently show that, in every discrete conformal class, there exists a polyhedral
surface with constant discrete Gaussian curvature. We also provide explicit examples
to demonstrate that this surface is in general not unique.
acknowledgement: Open access funding provided by the Austrian Science Fund (FWF).
This research was supported by the FWF grant, Project number I4245-N35, and by the
Deutsche Forschungsgemeinschaft (DFG - German Research Foundation) - Project-ID
195170736 - TRR109.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Hana
full_name: Kourimska, Hana
id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E
last_name: Kourimska
orcid: 0000-0001-7841-0091
citation:
ama: Kourimska H. Discrete yamabe problem for polyhedral surfaces. Discrete and
Computational Geometry. 2023;70:123-153. doi:10.1007/s00454-023-00484-2
apa: Kourimska, H. (2023). Discrete yamabe problem for polyhedral surfaces. Discrete
and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00484-2
chicago: Kourimska, Hana. “Discrete Yamabe Problem for Polyhedral Surfaces.” Discrete
and Computational Geometry. Springer Nature, 2023. https://doi.org/10.1007/s00454-023-00484-2.
ieee: H. Kourimska, “Discrete yamabe problem for polyhedral surfaces,” Discrete
and Computational Geometry, vol. 70. Springer Nature, pp. 123–153, 2023.
ista: Kourimska H. 2023. Discrete yamabe problem for polyhedral surfaces. Discrete
and Computational Geometry. 70, 123–153.
mla: Kourimska, Hana. “Discrete Yamabe Problem for Polyhedral Surfaces.” Discrete
and Computational Geometry, vol. 70, Springer Nature, 2023, pp. 123–53, doi:10.1007/s00454-023-00484-2.
short: H. Kourimska, Discrete and Computational Geometry 70 (2023) 123–153.
date_created: 2023-03-26T22:01:09Z
date_published: 2023-07-01T00:00:00Z
date_updated: 2023-10-04T11:46:48Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-023-00484-2
external_id:
isi:
- '000948148000001'
file:
- access_level: open_access
checksum: cdbf90ba4a7ddcb190d37b9e9d4cb9d3
content_type: application/pdf
creator: dernst
date_created: 2023-10-04T11:46:24Z
date_updated: 2023-10-04T11:46:24Z
file_id: '14396'
file_name: 2023_DiscreteGeometry_Kourimska.pdf
file_size: 1026683
relation: main_file
success: 1
file_date_updated: 2023-10-04T11:46:24Z
has_accepted_license: '1'
intvolume: ' 70'
isi: 1
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 123-153
project:
- _id: 26AD5D90-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I04245
name: Algebraic Footprints of Geometric Features in Homology
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Discrete yamabe problem for polyhedral surfaces
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 70
year: '2023'
...
---
_id: '12709'
abstract:
- lang: eng
text: Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within
distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter
family of spaces that grow larger when r increases or k decreases, called the
multicover bifiltration. Motivated by the problem of computing the homology of
this bifiltration, we introduce two closely related combinatorial bifiltrations,
one polyhedral and the other simplicial, which are both topologically equivalent
to the multicover bifiltration and far smaller than a Čech-based model considered
in prior work of Sheehy. Our polyhedral construction is a bifiltration of the
rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using
a variant of an algorithm given by these authors as well. Using an implementation
for dimension 2 and 3, we provide experimental results. Our simplicial construction
is useful for understanding the polyhedral construction and proving its correctness.
acknowledgement: We thank the anonymous reviewers for many helpful comments and suggestions,
which led to substantial improvements of the paper. The first two authors were supported
by the Austrian Science Fund (FWF) grant number P 29984-N35 and W1230. The first
author was partly supported by an Austrian Marshall Plan Scholarship, and by the
Brummer & Partners MathDataLab. A conference version of this paper was presented
at the 37th International Symposium on Computational Geometry (SoCG 2021). Open
access funding provided by the Royal Institute of Technology.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: René
full_name: Corbet, René
last_name: Corbet
- first_name: Michael
full_name: Kerber, Michael
id: 36E4574A-F248-11E8-B48F-1D18A9856A87
last_name: Kerber
orcid: 0000-0002-8030-9299
- first_name: Michael
full_name: Lesnick, Michael
last_name: Lesnick
- first_name: Georg F
full_name: Osang, Georg F
id: 464B40D6-F248-11E8-B48F-1D18A9856A87
last_name: Osang
orcid: 0000-0002-8882-5116
citation:
ama: Corbet R, Kerber M, Lesnick M, Osang GF. Computing the multicover bifiltration.
Discrete and Computational Geometry. 2023;70:376-405. doi:10.1007/s00454-022-00476-8
apa: Corbet, R., Kerber, M., Lesnick, M., & Osang, G. F. (2023). Computing the
multicover bifiltration. Discrete and Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-022-00476-8
chicago: Corbet, René, Michael Kerber, Michael Lesnick, and Georg F Osang. “Computing
the Multicover Bifiltration.” Discrete and Computational Geometry. Springer
Nature, 2023. https://doi.org/10.1007/s00454-022-00476-8.
ieee: R. Corbet, M. Kerber, M. Lesnick, and G. F. Osang, “Computing the multicover
bifiltration,” Discrete and Computational Geometry, vol. 70. Springer Nature,
pp. 376–405, 2023.
ista: Corbet R, Kerber M, Lesnick M, Osang GF. 2023. Computing the multicover bifiltration.
Discrete and Computational Geometry. 70, 376–405.
mla: Corbet, René, et al. “Computing the Multicover Bifiltration.” Discrete and
Computational Geometry, vol. 70, Springer Nature, 2023, pp. 376–405, doi:10.1007/s00454-022-00476-8.
short: R. Corbet, M. Kerber, M. Lesnick, G.F. Osang, Discrete and Computational
Geometry 70 (2023) 376–405.
date_created: 2023-03-05T23:01:06Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2023-10-04T12:03:40Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s00454-022-00476-8
external_id:
arxiv:
- '2103.07823'
isi:
- '000936496800001'
file:
- access_level: open_access
checksum: 71ce7e59f7ee4620acc704fecca620c2
content_type: application/pdf
creator: cchlebak
date_created: 2023-03-07T14:40:14Z
date_updated: 2023-03-07T14:40:14Z
file_id: '12715'
file_name: 2023_DisCompGeo_Corbet.pdf
file_size: 1359323
relation: main_file
success: 1
file_date_updated: 2023-03-07T14:40:14Z
has_accepted_license: '1'
intvolume: ' 70'
isi: 1
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 376-405
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '9605'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Computing the multicover bifiltration
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 70
year: '2023'
...
---
_id: '12763'
abstract:
- lang: eng
text: 'Kleinjohann (Archiv der Mathematik 35(1):574–582, 1980; Mathematische Zeitschrift
176(3), 327–344, 1981) and Bangert (Archiv der Mathematik 38(1):54–57, 1982) extended
the reach rch(S) from subsets S of Euclidean space to the reach rchM(S) of subsets
S of Riemannian manifolds M, where M is smooth (we’ll assume at least C3). Bangert
showed that sets of positive reach in Euclidean space and Riemannian manifolds
are very similar. In this paper we introduce a slight variant of Kleinjohann’s
and Bangert’s extension and quantify the similarity between sets of positive reach
in Euclidean space and Riemannian manifolds in a new way: Given p∈M and q∈S, we
bound the local feature size (a local version of the reach) of its lifting to
the tangent space via the inverse exponential map (exp−1p(S)) at q, assuming that
rchM(S) and the geodesic distance dM(p,q) are bounded. These bounds are motivated
by the importance of the reach and local feature size to manifold learning, topological
inference, and triangulating manifolds and the fact that intrinsic approaches
circumvent the curse of dimensionality.'
acknowledgement: "We thank Eddie Aamari, David Cohen-Steiner, Isa Costantini, Fred
Chazal, Ramsay Dyer, André Lieutier, and Alef Sterk for discussion and Pierre Pansu
for encouragement. We further acknowledge the anonymous reviewers whose comments
helped improve the exposition.\r\nThe research leading to these results has received
funding from the European Research Council (ERC) under the European Union’s Seventh
Framework Programme (FP/2007-2013) / ERC Grant Agreement No. 339025 GUDHI (Algorithmic
Foundations of Geometry Understanding in Higher Dimensions). The first author is
further supported by the French government, through the 3IA Côte d’Azur Investments
in the Future project managed by the National Research Agency (ANR) with the reference
number ANR-19-P3IA-0002. The second author is supported by the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
Grant Agreement No. 754411 and the Austrian science fund (FWF) M-3073."
article_processing_charge: No
article_type: original
author:
- first_name: Jean Daniel
full_name: Boissonnat, Jean Daniel
last_name: Boissonnat
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat JD, Wintraecken M. The reach of subsets of manifolds. Journal
of Applied and Computational Topology. 2023;7:619-641. doi:10.1007/s41468-023-00116-x
apa: Boissonnat, J. D., & Wintraecken, M. (2023). The reach of subsets of manifolds.
Journal of Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00116-x
chicago: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets
of Manifolds.” Journal of Applied and Computational Topology. Springer
Nature, 2023. https://doi.org/10.1007/s41468-023-00116-x.
ieee: J. D. Boissonnat and M. Wintraecken, “The reach of subsets of manifolds,”
Journal of Applied and Computational Topology, vol. 7. Springer Nature,
pp. 619–641, 2023.
ista: Boissonnat JD, Wintraecken M. 2023. The reach of subsets of manifolds. Journal
of Applied and Computational Topology. 7, 619–641.
mla: Boissonnat, Jean Daniel, and Mathijs Wintraecken. “The Reach of Subsets of
Manifolds.” Journal of Applied and Computational Topology, vol. 7, Springer
Nature, 2023, pp. 619–41, doi:10.1007/s41468-023-00116-x.
short: J.D. Boissonnat, M. Wintraecken, Journal of Applied and Computational Topology
7 (2023) 619–641.
date_created: 2023-03-26T22:01:08Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2023-10-04T12:07:18Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00116-x
ec_funded: 1
intvolume: ' 7'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://inserm.hal.science/INRIA-SACLAY/hal-04083524v1
month: '09'
oa: 1
oa_version: Submitted Version
page: 619-641
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: The reach of subsets of manifolds
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2023'
...
---
_id: '12960'
abstract:
- lang: eng
text: "Isomanifolds are the generalization of isosurfaces to arbitrary dimension
and codimension, i.e., submanifolds of Rd defined as the zero set of some multivariate
multivalued smooth function f:Rd→Rd−n, where n is the intrinsic dimension of the
manifold. A natural way to approximate a smooth isomanifold M=f−1(0) is to consider
its piecewise linear (PL) approximation M^\r\n based on a triangulation T of the
ambient space Rd. In this paper, we describe a simple algorithm to trace isomanifolds
from a given starting point. The algorithm works for arbitrary dimensions n and
d, and any precision D. Our main result is that, when f (or M) has bounded complexity,
the complexity of the algorithm is polynomial in d and δ=1/D (and unavoidably
exponential in n). Since it is known that for δ=Ω(d2.5), M^ is O(D2)-close and
isotopic to M\r\n, our algorithm produces a faithful PL-approximation of isomanifolds
of bounded complexity in time polynomial in d. Combining this algorithm with dimensionality
reduction techniques, the dependency on d in the size of M^ can be completely
removed with high probability. We also show that the algorithm can handle isomanifolds
with boundary and, more generally, isostratifolds. The algorithm for isomanifolds
with boundary has been implemented and experimental results are reported, showing
that it is practical and can handle cases that are far ahead of the state-of-the-art. "
acknowledgement: The authors have received funding from the European Research Council
under the European Union's ERC grant greement 339025 GUDHI (Algorithmic Foundations
of Geometric Un-derstanding in Higher Dimensions). The first author was supported by the French government,through
the 3IA C\^ote d'Azur Investments in the Future project managed by the National
ResearchAgency (ANR) with the reference ANR-19-P3IA-0002. The third author was
supported by the Eu-ropean Union's Horizon 2020 research and innovation programme
under the Marie Sk\lodowska-Curiegrant agreement 754411 and the FWF (Austrian Science
Fund) grant M 3073.
article_processing_charge: No
article_type: original
author:
- first_name: Jean Daniel
full_name: Boissonnat, Jean Daniel
last_name: Boissonnat
- first_name: Siargey
full_name: Kachanovich, Siargey
last_name: Kachanovich
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat JD, Kachanovich S, Wintraecken M. Tracing isomanifolds in Rd in
time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations. SIAM Journal
on Computing. 2023;52(2):452-486. doi:10.1137/21M1412918
apa: Boissonnat, J. D., Kachanovich, S., & Wintraecken, M. (2023). Tracing isomanifolds
in Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations. SIAM
Journal on Computing. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/21M1412918
chicago: Boissonnat, Jean Daniel, Siargey Kachanovich, and Mathijs Wintraecken.
“Tracing Isomanifolds in Rd in Time Polynomial in d Using Coxeter–Freudenthal–Kuhn
Triangulations.” SIAM Journal on Computing. Society for Industrial and
Applied Mathematics, 2023. https://doi.org/10.1137/21M1412918.
ieee: J. D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Tracing isomanifolds
in Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations,”
SIAM Journal on Computing, vol. 52, no. 2. Society for Industrial and Applied
Mathematics, pp. 452–486, 2023.
ista: Boissonnat JD, Kachanovich S, Wintraecken M. 2023. Tracing isomanifolds in
Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn triangulations. SIAM
Journal on Computing. 52(2), 452–486.
mla: Boissonnat, Jean Daniel, et al. “Tracing Isomanifolds in Rd in Time Polynomial
in d Using Coxeter–Freudenthal–Kuhn Triangulations.” SIAM Journal on Computing,
vol. 52, no. 2, Society for Industrial and Applied Mathematics, 2023, pp. 452–86,
doi:10.1137/21M1412918.
short: J.D. Boissonnat, S. Kachanovich, M. Wintraecken, SIAM Journal on Computing
52 (2023) 452–486.
date_created: 2023-05-14T22:01:00Z
date_published: 2023-04-30T00:00:00Z
date_updated: 2023-10-10T07:34:35Z
day: '30'
department:
- _id: HeEd
doi: 10.1137/21M1412918
ec_funded: 1
external_id:
isi:
- '001013183000012'
intvolume: ' 52'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://hal-emse.ccsd.cnrs.fr/3IA-COTEDAZUR/hal-04083489v1
month: '04'
oa: 1
oa_version: Submitted Version
page: 452-486
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: SIAM Journal on Computing
publication_identifier:
eissn:
- 1095-7111
issn:
- 0097-5397
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
related_material:
record:
- id: '9441'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Tracing isomanifolds in Rd in time polynomial in d using Coxeter–Freudenthal–Kuhn
triangulations
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 52
year: '2023'
...
---
_id: '13134'
abstract:
- lang: eng
text: We propose a characterization of discrete analytical spheres, planes and lines
in the body-centered cubic (BCC) grid, both in the Cartesian and in the recently
proposed alternative compact coordinate system, in which each integer triplet
addresses some voxel in the grid. We define spheres and planes through double
Diophantine inequalities and investigate their relevant topological features,
such as functionality or the interrelation between the thickness of the objects
and their connectivity and separation properties. We define lines as the intersection
of planes. The number of the planes (up to six) is equal to the number of the
pairs of faces of a BCC voxel that are parallel to the line.
acknowledgement: The first author has been partially supported by the Ministry of
Science, Technological Development and Innovation of the Republic of Serbia through
the project no. 451-03-47/2023-01/200156. The fourth author is funded by the DFG
Collaborative Research Center TRR 109, ‘Discretization in Geometry and Dynamics’,
Austrian Science Fund (FWF), grant no. I 02979-N35.
article_number: '109693'
article_processing_charge: No
article_type: original
author:
- first_name: Lidija
full_name: Čomić, Lidija
last_name: Čomić
- first_name: Gaëlle
full_name: Largeteau-Skapin, Gaëlle
last_name: Largeteau-Skapin
- first_name: Rita
full_name: Zrour, Rita
last_name: Zrour
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Eric
full_name: Andres, Eric
last_name: Andres
citation:
ama: Čomić L, Largeteau-Skapin G, Zrour R, Biswas R, Andres E. Discrete analytical
objects in the body-centered cubic grid. Pattern Recognition. 2023;142(10).
doi:10.1016/j.patcog.2023.109693
apa: Čomić, L., Largeteau-Skapin, G., Zrour, R., Biswas, R., & Andres, E. (2023).
Discrete analytical objects in the body-centered cubic grid. Pattern Recognition.
Elsevier. https://doi.org/10.1016/j.patcog.2023.109693
chicago: Čomić, Lidija, Gaëlle Largeteau-Skapin, Rita Zrour, Ranita Biswas, and
Eric Andres. “Discrete Analytical Objects in the Body-Centered Cubic Grid.” Pattern
Recognition. Elsevier, 2023. https://doi.org/10.1016/j.patcog.2023.109693.
ieee: L. Čomić, G. Largeteau-Skapin, R. Zrour, R. Biswas, and E. Andres, “Discrete
analytical objects in the body-centered cubic grid,” Pattern Recognition,
vol. 142, no. 10. Elsevier, 2023.
ista: Čomić L, Largeteau-Skapin G, Zrour R, Biswas R, Andres E. 2023. Discrete analytical
objects in the body-centered cubic grid. Pattern Recognition. 142(10), 109693.
mla: Čomić, Lidija, et al. “Discrete Analytical Objects in the Body-Centered Cubic
Grid.” Pattern Recognition, vol. 142, no. 10, 109693, Elsevier, 2023, doi:10.1016/j.patcog.2023.109693.
short: L. Čomić, G. Largeteau-Skapin, R. Zrour, R. Biswas, E. Andres, Pattern Recognition
142 (2023).
date_created: 2023-06-18T22:00:45Z
date_published: 2023-10-01T00:00:00Z
date_updated: 2023-10-10T07:37:16Z
day: '01'
department:
- _id: HeEd
doi: 10.1016/j.patcog.2023.109693
external_id:
isi:
- '001013526000001'
intvolume: ' 142'
isi: 1
issue: '10'
language:
- iso: eng
month: '10'
oa_version: None
project:
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
grant_number: I4887
name: Discretization in Geometry and Dynamics
publication: Pattern Recognition
publication_identifier:
issn:
- 0031-3203
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Discrete analytical objects in the body-centered cubic grid
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 142
year: '2023'
...