---
_id: '1930'
abstract:
- lang: eng
text: (Figure Presented) Data acquisition, numerical inaccuracies, and sampling
often introduce noise in measurements and simulations. Removing this noise is
often necessary for efficient analysis and visualization of this data, yet many
denoising techniques change the minima and maxima of a scalar field. For example,
the extrema can appear or disappear, spatially move, and change their value. This
can lead to wrong interpretations of the data, e.g., when the maximum temperature
over an area is falsely reported being a few degrees cooler because the denoising
method is unaware of these features. Recently, a topological denoising technique
based on a global energy optimization was proposed, which allows the topology-controlled
denoising of 2D scalar fields. While this method preserves the minima and maxima,
it is constrained by the size of the data. We extend this work to large 2D data
and medium-sized 3D data by introducing a novel domain decomposition approach.
It allows processing small patches of the domain independently while still avoiding
the introduction of new critical points. Furthermore, we propose an iterative
refinement of the solution, which decreases the optimization energy compared to
the previous approach and therefore gives smoother results that are closer to
the input. We illustrate our technique on synthetic and real-world 2D and 3D data
sets that highlight potential applications.
acknowledgement: RTRA Digiteoproject; ERC grant; SNF award; Intel Doctoral Fellowship;
MPC-VCC
author:
- first_name: David
full_name: Günther, David
last_name: Günther
- first_name: Alec
full_name: Jacobson, Alec
last_name: Jacobson
- first_name: Jan
full_name: Reininghaus, Jan
id: 4505473A-F248-11E8-B48F-1D18A9856A87
last_name: Reininghaus
- first_name: Hans
full_name: Seidel, Hans
last_name: Seidel
- first_name: Olga
full_name: Sorkine Hornung, Olga
last_name: Sorkine Hornung
- first_name: Tino
full_name: Weinkauf, Tino
last_name: Weinkauf
citation:
ama: Günther D, Jacobson A, Reininghaus J, Seidel H, Sorkine Hornung O, Weinkauf
T. Fast and memory-efficient topological denoising of 2D and 3D scalar fields.
IEEE Transactions on Visualization and Computer Graphics. 2014;20(12):2585-2594.
doi:10.1109/TVCG.2014.2346432
apa: Günther, D., Jacobson, A., Reininghaus, J., Seidel, H., Sorkine Hornung, O.,
& Weinkauf, T. (2014). Fast and memory-efficient topological denoising of
2D and 3D scalar fields. IEEE Transactions on Visualization and Computer Graphics.
IEEE. https://doi.org/10.1109/TVCG.2014.2346432
chicago: Günther, David, Alec Jacobson, Jan Reininghaus, Hans Seidel, Olga Sorkine
Hornung, and Tino Weinkauf. “Fast and Memory-Efficient Topological Denoising of
2D and 3D Scalar Fields.” IEEE Transactions on Visualization and Computer Graphics.
IEEE, 2014. https://doi.org/10.1109/TVCG.2014.2346432.
ieee: D. Günther, A. Jacobson, J. Reininghaus, H. Seidel, O. Sorkine Hornung, and
T. Weinkauf, “Fast and memory-efficient topological denoising of 2D and 3D scalar
fields,” IEEE Transactions on Visualization and Computer Graphics, vol.
20, no. 12. IEEE, pp. 2585–2594, 2014.
ista: Günther D, Jacobson A, Reininghaus J, Seidel H, Sorkine Hornung O, Weinkauf
T. 2014. Fast and memory-efficient topological denoising of 2D and 3D scalar fields.
IEEE Transactions on Visualization and Computer Graphics. 20(12), 2585–2594.
mla: Günther, David, et al. “Fast and Memory-Efficient Topological Denoising of
2D and 3D Scalar Fields.” IEEE Transactions on Visualization and Computer Graphics,
vol. 20, no. 12, IEEE, 2014, pp. 2585–94, doi:10.1109/TVCG.2014.2346432.
short: D. Günther, A. Jacobson, J. Reininghaus, H. Seidel, O. Sorkine Hornung, T.
Weinkauf, IEEE Transactions on Visualization and Computer Graphics 20 (2014) 2585–2594.
date_created: 2018-12-11T11:54:46Z
date_published: 2014-12-31T00:00:00Z
date_updated: 2021-01-12T06:54:09Z
day: '31'
department:
- _id: HeEd
doi: 10.1109/TVCG.2014.2346432
intvolume: ' 20'
issue: '12'
language:
- iso: eng
month: '12'
oa_version: None
page: 2585 - 2594
publication: IEEE Transactions on Visualization and Computer Graphics
publication_status: published
publisher: IEEE
publist_id: '5164'
quality_controlled: '1'
scopus_import: 1
status: public
title: Fast and memory-efficient topological denoising of 2D and 3D scalar fields
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 20
year: '2014'
...
---
_id: '2043'
abstract:
- lang: eng
text: Persistent homology is a popular and powerful tool for capturing topological
features of data. Advances in algorithms for computing persistent homology have
reduced the computation time drastically – as long as the algorithm does not exhaust
the available memory. Following up on a recently presented parallel method for
persistence computation on shared memory systems [1], we demonstrate that a simple
adaption of the standard reduction algorithm leads to a variant for distributed
systems. Our algorithmic design ensures that the data is distributed over the
nodes without redundancy; this permits the computation of much larger instances
than on a single machine. Moreover, we observe that the parallelism at least compensates
for the overhead caused by communication between nodes, and often even speeds
up the computation compared to sequential and even parallel shared memory algorithms.
In our experiments, we were able to compute the persistent homology of filtrations
with more than a billion (109) elements within seconds on a cluster with 32 nodes
using less than 6GB of memory per node.
author:
- first_name: Ulrich
full_name: Bauer, Ulrich
id: 2ADD483A-F248-11E8-B48F-1D18A9856A87
last_name: Bauer
orcid: 0000-0002-9683-0724
- first_name: Michael
full_name: Kerber, Michael
last_name: Kerber
orcid: 0000-0002-8030-9299
- first_name: Jan
full_name: Reininghaus, Jan
id: 4505473A-F248-11E8-B48F-1D18A9856A87
last_name: Reininghaus
citation:
ama: 'Bauer U, Kerber M, Reininghaus J. Distributed computation of persistent homology.
In: McGeoch C, Meyer U, eds. Proceedings of the Workshop on Algorithm Engineering
and Experiments. Society of Industrial and Applied Mathematics; 2014:31-38.
doi:10.1137/1.9781611973198.4'
apa: 'Bauer, U., Kerber, M., & Reininghaus, J. (2014). Distributed computation
of persistent homology. In C. McGeoch & U. Meyer (Eds.), Proceedings of
the Workshop on Algorithm Engineering and Experiments (pp. 31–38). Portland,
USA: Society of Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973198.4'
chicago: Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Distributed Computation
of Persistent Homology.” In Proceedings of the Workshop on Algorithm Engineering
and Experiments, edited by Catherine McGeoch and Ulrich Meyer, 31–38. Society
of Industrial and Applied Mathematics, 2014. https://doi.org/10.1137/1.9781611973198.4.
ieee: U. Bauer, M. Kerber, and J. Reininghaus, “Distributed computation of persistent
homology,” in Proceedings of the Workshop on Algorithm Engineering and Experiments,
Portland, USA, 2014, pp. 31–38.
ista: 'Bauer U, Kerber M, Reininghaus J. 2014. Distributed computation of persistent
homology. Proceedings of the Workshop on Algorithm Engineering and Experiments.
ALENEX: Algorithm Engineering and Experiments, 31–38.'
mla: Bauer, Ulrich, et al. “Distributed Computation of Persistent Homology.” Proceedings
of the Workshop on Algorithm Engineering and Experiments, edited by Catherine McGeoch
and Ulrich Meyer, Society of Industrial and Applied Mathematics, 2014, pp. 31–38,
doi:10.1137/1.9781611973198.4.
short: U. Bauer, M. Kerber, J. Reininghaus, in:, C. McGeoch, U. Meyer (Eds.), Proceedings
of the Workshop on Algorithm Engineering and Experiments, Society of Industrial
and Applied Mathematics, 2014, pp. 31–38.
conference:
end_date: 2014-01-05
location: Portland, USA
name: 'ALENEX: Algorithm Engineering and Experiments'
start_date: 2014-01-05
date_created: 2018-12-11T11:55:23Z
date_published: 2014-01-01T00:00:00Z
date_updated: 2021-01-12T06:54:56Z
day: '01'
department:
- _id: HeEd
doi: 10.1137/1.9781611973198.4
ec_funded: 1
editor:
- first_name: Catherine
full_name: ' McGeoch, Catherine'
last_name: ' McGeoch'
- first_name: Ulrich
full_name: Meyer, Ulrich
last_name: Meyer
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1310.0710
month: '01'
oa: 1
oa_version: Submitted Version
page: 31 - 38
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '318493'
name: Topological Complex Systems
publication: Proceedings of the Workshop on Algorithm Engineering and Experiments
publication_status: published
publisher: Society of Industrial and Applied Mathematics
publist_id: '5008'
quality_controlled: '1'
scopus_import: 1
status: public
title: Distributed computation of persistent homology
type: conference
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...
---
_id: '2044'
abstract:
- lang: eng
text: We present a parallel algorithm for computing the persistent homology of a
filtered chain complex. Our approach differs from the commonly used reduction
algorithm by first computing persistence pairs within local chunks, then simplifying
the unpaired columns, and finally applying standard reduction on the simplified
matrix. The approach generalizes a technique by Günther et al., which uses discrete
Morse Theory to compute persistence; we derive the same worst-case complexity
bound in a more general context. The algorithm employs several practical optimization
techniques, which are of independent interest. Our sequential implementation of
the algorithm is competitive with state-of-the-art methods, and we further improve
the performance through parallel computation.
author:
- first_name: Ulrich
full_name: Bauer, Ulrich
id: 2ADD483A-F248-11E8-B48F-1D18A9856A87
last_name: Bauer
orcid: 0000-0002-9683-0724
- first_name: Michael
full_name: Kerber, Michael
last_name: Kerber
orcid: 0000-0002-8030-9299
- first_name: Jan
full_name: Reininghaus, Jan
id: 4505473A-F248-11E8-B48F-1D18A9856A87
last_name: Reininghaus
citation:
ama: 'Bauer U, Kerber M, Reininghaus J. Clear and Compress: Computing Persistent
Homology in Chunks. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. Topological
Methods in Data Analysis and Visualization III. Mathematics and Visualization.
Springer; 2014:103-117. doi:10.1007/978-3-319-04099-8_7'
apa: 'Bauer, U., Kerber, M., & Reininghaus, J. (2014). Clear and Compress: Computing
Persistent Homology in Chunks. In P.-T. Bremer, I. Hotz, V. Pascucci, & R.
Peikert (Eds.), Topological Methods in Data Analysis and Visualization III
(pp. 103–117). Springer. https://doi.org/10.1007/978-3-319-04099-8_7'
chicago: 'Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Clear and Compress:
Computing Persistent Homology in Chunks.” In Topological Methods in Data Analysis
and Visualization III, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci,
and Ronald Peikert, 103–17. Mathematics and Visualization. Springer, 2014. https://doi.org/10.1007/978-3-319-04099-8_7.'
ieee: 'U. Bauer, M. Kerber, and J. Reininghaus, “Clear and Compress: Computing Persistent
Homology in Chunks,” in Topological Methods in Data Analysis and Visualization
III, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Springer, 2014,
pp. 103–117.'
ista: 'Bauer U, Kerber M, Reininghaus J. 2014.Clear and Compress: Computing Persistent
Homology in Chunks. In: Topological Methods in Data Analysis and Visualization
III. , 103–117.'
mla: 'Bauer, Ulrich, et al. “Clear and Compress: Computing Persistent Homology in
Chunks.” Topological Methods in Data Analysis and Visualization III, edited
by Peer-Timo Bremer et al., Springer, 2014, pp. 103–17, doi:10.1007/978-3-319-04099-8_7.'
short: U. Bauer, M. Kerber, J. Reininghaus, in:, P.-T. Bremer, I. Hotz, V. Pascucci,
R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III,
Springer, 2014, pp. 103–117.
date_created: 2018-12-11T11:55:23Z
date_published: 2014-03-19T00:00:00Z
date_updated: 2021-01-12T06:54:56Z
day: '19'
department:
- _id: HeEd
doi: 10.1007/978-3-319-04099-8_7
ec_funded: 1
editor:
- first_name: Peer-Timo
full_name: Bremer, Peer-Timo
last_name: Bremer
- first_name: Ingrid
full_name: Hotz, Ingrid
last_name: Hotz
- first_name: Valerio
full_name: Pascucci, Valerio
last_name: Pascucci
- first_name: Ronald
full_name: Peikert, Ronald
last_name: Peikert
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1303.0477
month: '03'
oa: 1
oa_version: Submitted Version
page: 103 - 117
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '318493'
name: Topological Complex Systems
publication: Topological Methods in Data Analysis and Visualization III
publication_status: published
publisher: Springer
publist_id: '5007'
quality_controlled: '1'
scopus_import: 1
series_title: Mathematics and Visualization
status: public
title: 'Clear and Compress: Computing Persistent Homology in Chunks'
type: book_chapter
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...
---
_id: '2153'
abstract:
- lang: eng
text: 'We define a simple, explicit map sending a morphism f : M → N of pointwise
finite dimensional persistence modules to a matching between the barcodes of M
and N. Our main result is that, in a precise sense, the quality of this matching
is tightly controlled by the lengths of the longest intervals in the barcodes
of ker f and coker f . As an immediate corollary, we obtain a new proof of the
algebraic stability theorem for persistence barcodes [5, 9], a fundamental result
in the theory of persistent homology. In contrast to previous proofs, ours shows
explicitly how a δ-interleaving morphism between two persistence modules induces
a δ-matching between the barcodes of the two modules. Our main result also specializes
to a structure theorem for submodules and quotients of persistence modules. Copyright
is held by the owner/author(s).'
author:
- first_name: Ulrich
full_name: Bauer, Ulrich
id: 2ADD483A-F248-11E8-B48F-1D18A9856A87
last_name: Bauer
orcid: 0000-0002-9683-0724
- first_name: Michael
full_name: Lesnick, Michael
last_name: Lesnick
citation:
ama: 'Bauer U, Lesnick M. Induced matchings of barcodes and the algebraic stability
of persistence. In: Proceedings of the Annual Symposium on Computational Geometry.
ACM; 2014:355-364. doi:10.1145/2582112.2582168'
apa: 'Bauer, U., & Lesnick, M. (2014). Induced matchings of barcodes and the
algebraic stability of persistence. In Proceedings of the Annual Symposium
on Computational Geometry (pp. 355–364). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582168'
chicago: Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and
the Algebraic Stability of Persistence.” In Proceedings of the Annual Symposium
on Computational Geometry, 355–64. ACM, 2014. https://doi.org/10.1145/2582112.2582168.
ieee: U. Bauer and M. Lesnick, “Induced matchings of barcodes and the algebraic
stability of persistence,” in Proceedings of the Annual Symposium on Computational
Geometry, Kyoto, Japan, 2014, pp. 355–364.
ista: 'Bauer U, Lesnick M. 2014. Induced matchings of barcodes and the algebraic
stability of persistence. Proceedings of the Annual Symposium on Computational
Geometry. SoCG: Symposium on Computational Geometry, 355–364.'
mla: Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and the
Algebraic Stability of Persistence.” Proceedings of the Annual Symposium on
Computational Geometry, ACM, 2014, pp. 355–64, doi:10.1145/2582112.2582168.
short: U. Bauer, M. Lesnick, in:, Proceedings of the Annual Symposium on Computational
Geometry, ACM, 2014, pp. 355–364.
conference:
end_date: 2014-06-11
location: Kyoto, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2014-06-08
date_created: 2018-12-11T11:56:01Z
date_published: 2014-06-01T00:00:00Z
date_updated: 2021-01-12T06:55:38Z
day: '01'
department:
- _id: HeEd
doi: 10.1145/2582112.2582168
ec_funded: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1311.3681
month: '06'
oa: 1
oa_version: Submitted Version
page: 355 - 364
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '318493'
name: Topological Complex Systems
publication: Proceedings of the Annual Symposium on Computational Geometry
publication_status: published
publisher: ACM
publist_id: '4853'
quality_controlled: '1'
scopus_import: 1
status: public
title: Induced matchings of barcodes and the algebraic stability of persistence
type: conference
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...
---
_id: '2156'
abstract:
- lang: eng
text: We propose a metric for Reeb graphs, called the functional distortion distance.
Under this distance, the Reeb graph is stable against small changes of input functions.
At the same time, it remains discriminative at differentiating input functions.
In particular, the main result is that the functional distortion distance between
two Reeb graphs is bounded from below by the bottleneck distance between both
the ordinary and extended persistence diagrams for appropriate dimensions. As
an application of our results, we analyze a natural simplification scheme for
Reeb graphs, and show that persistent features in Reeb graph remains persistent
under simplification. Understanding the stability of important features of the
Reeb graph under simplification is an interesting problem on its own right, and
critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s).
acknowledgement: National Science Foundation under grants CCF-1319406, CCF-1116258.
author:
- first_name: Ulrich
full_name: Bauer, Ulrich
id: 2ADD483A-F248-11E8-B48F-1D18A9856A87
last_name: Bauer
orcid: 0000-0002-9683-0724
- first_name: Xiaoyin
full_name: Ge, Xiaoyin
last_name: Ge
- first_name: Yusu
full_name: Wang, Yusu
last_name: Wang
citation:
ama: 'Bauer U, Ge X, Wang Y. Measuring distance between Reeb graphs. In: Proceedings
of the Annual Symposium on Computational Geometry. ACM; 2014:464-473. doi:10.1145/2582112.2582169'
apa: 'Bauer, U., Ge, X., & Wang, Y. (2014). Measuring distance between Reeb
graphs. In Proceedings of the Annual Symposium on Computational Geometry
(pp. 464–473). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582169'
chicago: Bauer, Ulrich, Xiaoyin Ge, and Yusu Wang. “Measuring Distance between Reeb
Graphs.” In Proceedings of the Annual Symposium on Computational Geometry,
464–73. ACM, 2014. https://doi.org/10.1145/2582112.2582169.
ieee: U. Bauer, X. Ge, and Y. Wang, “Measuring distance between Reeb graphs,” in
Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan,
2014, pp. 464–473.
ista: 'Bauer U, Ge X, Wang Y. 2014. Measuring distance between Reeb graphs. Proceedings
of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational
Geometry, 464–473.'
mla: Bauer, Ulrich, et al. “Measuring Distance between Reeb Graphs.” Proceedings
of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 464–73,
doi:10.1145/2582112.2582169.
short: U. Bauer, X. Ge, Y. Wang, in:, Proceedings of the Annual Symposium on Computational
Geometry, ACM, 2014, pp. 464–473.
conference:
end_date: 2014-06-11
location: Kyoto, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2014-06-08
date_created: 2018-12-11T11:56:02Z
date_published: 2014-06-01T00:00:00Z
date_updated: 2021-01-12T06:55:39Z
day: '01'
department:
- _id: HeEd
doi: 10.1145/2582112.2582169
ec_funded: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1307.2839
month: '06'
oa: 1
oa_version: Submitted Version
page: 464 - 473
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '318493'
name: Topological Complex Systems
publication: Proceedings of the Annual Symposium on Computational Geometry
publication_status: published
publisher: ACM
publist_id: '4850'
quality_controlled: '1'
scopus_import: 1
status: public
title: Measuring distance between Reeb graphs
type: conference
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...
---
_id: '2155'
abstract:
- lang: eng
text: Given a finite set of points in Rn and a positive radius, we study the Čech,
Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete
Morse theory. We prove that the latter three complexes are simple-homotopy equivalent.
Our results have applications in topological data analysis and in the reconstruction
of shapes from sampled data. Copyright is held by the owner/author(s).
acknowledgement: This research is partially supported by ESF under the ACAT Research
Network Programme, and by the Russian Government under mega project 11.G34.31.0053
author:
- first_name: Ulrich
full_name: Bauer, Ulrich
id: 2ADD483A-F248-11E8-B48F-1D18A9856A87
last_name: Bauer
orcid: 0000-0002-9683-0724
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
citation:
ama: 'Bauer U, Edelsbrunner H. The morse theory of Čech and Delaunay filtrations.
In: Proceedings of the Annual Symposium on Computational Geometry. ACM;
2014:484-490. doi:10.1145/2582112.2582167'
apa: 'Bauer, U., & Edelsbrunner, H. (2014). The morse theory of Čech and Delaunay
filtrations. In Proceedings of the Annual Symposium on Computational Geometry
(pp. 484–490). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582167'
chicago: Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and
Delaunay Filtrations.” In Proceedings of the Annual Symposium on Computational
Geometry, 484–90. ACM, 2014. https://doi.org/10.1145/2582112.2582167.
ieee: U. Bauer and H. Edelsbrunner, “The morse theory of Čech and Delaunay filtrations,”
in Proceedings of the Annual Symposium on Computational Geometry, Kyoto,
Japan, 2014, pp. 484–490.
ista: 'Bauer U, Edelsbrunner H. 2014. The morse theory of Čech and Delaunay filtrations.
Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium
on Computational Geometry, 484–490.'
mla: Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay
Filtrations.” Proceedings of the Annual Symposium on Computational Geometry,
ACM, 2014, pp. 484–90, doi:10.1145/2582112.2582167.
short: U. Bauer, H. Edelsbrunner, in:, Proceedings of the Annual Symposium on Computational
Geometry, ACM, 2014, pp. 484–490.
conference:
end_date: 2014-06-11
location: Kyoto, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2014-06-08
date_created: 2018-12-11T11:56:01Z
date_published: 2014-06-01T00:00:00Z
date_updated: 2021-01-12T06:55:38Z
day: '01'
department:
- _id: HeEd
doi: 10.1145/2582112.2582167
ec_funded: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1312.1231
month: '06'
oa: 1
oa_version: Submitted Version
page: 484 - 490
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '318493'
name: Topological Complex Systems
publication: Proceedings of the Annual Symposium on Computational Geometry
publication_status: published
publisher: ACM
publist_id: '4851'
quality_controlled: '1'
scopus_import: 1
status: public
title: The morse theory of Čech and Delaunay filtrations
type: conference
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...
---
_id: '2177'
abstract:
- lang: eng
text: We give evidence for the difficulty of computing Betti numbers of simplicial
complexes over a finite field. We do this by reducing the rank computation for
sparse matrices with to non-zero entries to computing Betti numbers of simplicial
complexes consisting of at most a constant times to simplices. Together with the
known reduction in the other direction, this implies that the two problems have
the same computational complexity.
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Salman
full_name: Parsa, Salman
id: 4BDBD4F2-F248-11E8-B48F-1D18A9856A87
last_name: Parsa
citation:
ama: 'Edelsbrunner H, Parsa S. On the computational complexity of betti numbers
reductions from matrix rank. In: Proceedings of the Annual ACM-SIAM Symposium
on Discrete Algorithms. SIAM; 2014:152-160. doi:10.1137/1.9781611973402.11'
apa: 'Edelsbrunner, H., & Parsa, S. (2014). On the computational complexity
of betti numbers reductions from matrix rank. In Proceedings of the Annual
ACM-SIAM Symposium on Discrete Algorithms (pp. 152–160). Portland, USA: SIAM.
https://doi.org/10.1137/1.9781611973402.11'
chicago: Edelsbrunner, Herbert, and Salman Parsa. “On the Computational Complexity
of Betti Numbers Reductions from Matrix Rank.” In Proceedings of the Annual
ACM-SIAM Symposium on Discrete Algorithms, 152–60. SIAM, 2014. https://doi.org/10.1137/1.9781611973402.11.
ieee: H. Edelsbrunner and S. Parsa, “On the computational complexity of betti numbers
reductions from matrix rank,” in Proceedings of the Annual ACM-SIAM Symposium
on Discrete Algorithms, Portland, USA, 2014, pp. 152–160.
ista: 'Edelsbrunner H, Parsa S. 2014. On the computational complexity of betti numbers
reductions from matrix rank. Proceedings of the Annual ACM-SIAM Symposium on Discrete
Algorithms. SODA: Symposium on Discrete Algorithms, 152–160.'
mla: Edelsbrunner, Herbert, and Salman Parsa. “On the Computational Complexity of
Betti Numbers Reductions from Matrix Rank.” Proceedings of the Annual ACM-SIAM
Symposium on Discrete Algorithms, SIAM, 2014, pp. 152–60, doi:10.1137/1.9781611973402.11.
short: H. Edelsbrunner, S. Parsa, in:, Proceedings of the Annual ACM-SIAM Symposium
on Discrete Algorithms, SIAM, 2014, pp. 152–160.
conference:
end_date: 2014-01-07
location: Portland, USA
name: 'SODA: Symposium on Discrete Algorithms'
start_date: 2014-01-05
date_created: 2018-12-11T11:56:09Z
date_published: 2014-01-01T00:00:00Z
date_updated: 2021-01-12T06:55:48Z
day: '01'
department:
- _id: HeEd
doi: 10.1137/1.9781611973402.11
language:
- iso: eng
month: '01'
oa_version: None
page: 152 - 160
publication: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
publication_status: published
publisher: SIAM
publist_id: '4805'
quality_controlled: '1'
scopus_import: 1
status: public
title: On the computational complexity of betti numbers reductions from matrix rank
type: conference
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...
---
_id: '2184'
abstract:
- lang: eng
text: 'Given topological spaces X,Y, a fundamental problem of algebraic topology
is understanding the structure of all continuous maps X→ Y. We consider a computational
version, where X,Y are given as finite simplicial complexes, and the goal is to
compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem
in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected;
in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical
tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and
simplicial sets) with algorithmic tools from effective algebraic topology (locally
effective simplicial sets and objects with effective homology). In contrast, [X,Y]
is known to be uncomputable for general X,Y, since for X = S1 it includes a well
known undecidable problem: testing triviality of the fundamental group of Y. In
follow-up papers, the algorithm is shown to run in polynomial time for d fixed,
and extended to other problems, such as the extension problem, where we are given
a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or
computing the Z2-index-everything in the stable range. Outside the stable range,
the extension problem is undecidable.'
acknowledgement: The research by M. K. was supported by project GAUK 49209. The research
by M. K. was also supported by project 1M0545 by the Ministry of Education of the
Czech Republic and by Center of Excellence { Inst. for Theor. Comput. Sci., Prague
(project P202/12/G061 of GACR). The research by U. W. was supported by the Swiss
National Science Foundation (SNF Projects 200021-125309, 200020-138230, and PP00P2-138948).
article_number: '17 '
author:
- first_name: Martin
full_name: Čadek, Martin
last_name: Čadek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Francis
full_name: Sergeraert, Francis
last_name: Sergeraert
- first_name: Lukáš
full_name: Vokřínek, Lukáš
last_name: Vokřínek
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. Computing
all maps into a sphere. Journal of the ACM. 2014;61(3). doi:10.1145/2597629
apa: Čadek, M., Krcál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., & Wagner,
U. (2014). Computing all maps into a sphere. Journal of the ACM. ACM. https://doi.org/10.1145/2597629
chicago: Čadek, Martin, Marek Krcál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek,
and Uli Wagner. “Computing All Maps into a Sphere.” Journal of the ACM.
ACM, 2014. https://doi.org/10.1145/2597629.
ieee: M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, and U. Wagner,
“Computing all maps into a sphere,” Journal of the ACM, vol. 61, no. 3.
ACM, 2014.
ista: Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. 2014. Computing
all maps into a sphere. Journal of the ACM. 61(3), 17.
mla: Čadek, Martin, et al. “Computing All Maps into a Sphere.” Journal of the
ACM, vol. 61, no. 3, 17, ACM, 2014, doi:10.1145/2597629.
short: M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, U. Wagner, Journal
of the ACM 61 (2014).
date_created: 2018-12-11T11:56:12Z
date_published: 2014-05-01T00:00:00Z
date_updated: 2021-01-12T06:55:50Z
day: '01'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1145/2597629
intvolume: ' 61'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1105.6257
month: '05'
oa: 1
oa_version: Preprint
publication: Journal of the ACM
publication_status: published
publisher: ACM
publist_id: '4797'
quality_controlled: '1'
scopus_import: 1
status: public
title: Computing all maps into a sphere
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 61
year: '2014'
...
---
_id: '2905'
abstract:
- lang: eng
text: "Persistent homology is a recent grandchild of homology that has found use
in\r\nscience and engineering as well as in mathematics. This paper surveys the
method as well\r\nas the applications, neglecting completeness in favor of highlighting
ideas and directions."
acknowledgement: This research is partially supported by NSF under grant DBI-0820624,
by ESF under the Research Networking Programme, and by the Russian Government Project
11.G34.31.0053.
article_processing_charge: No
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Dmitriy
full_name: Morozovy, Dmitriy
last_name: Morozovy
citation:
ama: 'Edelsbrunner H, Morozovy D. Persistent homology: Theory and practice. In:
European Mathematical Society Publishing House; 2014:31-50. doi:10.4171/120-1/3'
apa: 'Edelsbrunner, H., & Morozovy, D. (2014). Persistent homology: Theory and
practice (pp. 31–50). Presented at the ECM: European Congress of Mathematics,
Kraków, Poland: European Mathematical Society Publishing House. https://doi.org/10.4171/120-1/3'
chicago: 'Edelsbrunner, Herbert, and Dmitriy Morozovy. “Persistent Homology: Theory
and Practice,” 31–50. European Mathematical Society Publishing House, 2014. https://doi.org/10.4171/120-1/3.'
ieee: 'H. Edelsbrunner and D. Morozovy, “Persistent homology: Theory and practice,”
presented at the ECM: European Congress of Mathematics, Kraków, Poland, 2014,
pp. 31–50.'
ista: 'Edelsbrunner H, Morozovy D. 2014. Persistent homology: Theory and practice.
ECM: European Congress of Mathematics, 31–50.'
mla: 'Edelsbrunner, Herbert, and Dmitriy Morozovy. Persistent Homology: Theory
and Practice. European Mathematical Society Publishing House, 2014, pp. 31–50,
doi:10.4171/120-1/3.'
short: H. Edelsbrunner, D. Morozovy, in:, European Mathematical Society Publishing
House, 2014, pp. 31–50.
conference:
end_date: 2012-07-07
location: Kraków, Poland
name: 'ECM: European Congress of Mathematics'
start_date: 2012-07-02
date_created: 2018-12-11T12:00:16Z
date_published: 2014-01-01T00:00:00Z
date_updated: 2021-01-12T07:00:36Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.4171/120-1/3
file:
- access_level: open_access
checksum: 1d4a046f1af945c407c5c4d411d4c5e4
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:16:43Z
date_updated: 2020-07-14T12:45:52Z
file_id: '5232'
file_name: IST-2016-544-v1+1_2012-P-11-PHTheoryPractice.pdf
file_size: 435320
relation: main_file
file_date_updated: 2020-07-14T12:45:52Z
has_accepted_license: '1'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Submitted Version
page: 31 - 50
publication_status: published
publisher: European Mathematical Society Publishing House
publist_id: '3842'
pubrep_id: '544'
quality_controlled: '1'
status: public
title: 'Persistent homology: Theory and practice'
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2014'
...
---
_id: '10892'
abstract:
- lang: eng
text: "In this paper, we introduce planar matchings on directed pseudo-line arrangements,
which yield a planar set of pseudo-line segments such that only matching-partners
are adjacent. By translating the planar matching problem into a corresponding
stable roommates problem we show that such matchings always exist.\r\nUsing our
new framework, we establish, for the first time, a complete, rigorous definition
of weighted straight skeletons, which are based on a so-called wavefront propagation
process. We present a generalized and unified approach to treat structural changes
in the wavefront that focuses on the restoration of weak planarity by finding
planar matchings."
acknowledgement: 'T. Biedl was supported by NSERC and the Ross and Muriel Cheriton
Fellowship. P. Palfrader was supported by Austrian Science Fund (FWF): P25816-N15.'
alternative_title:
- LNCS
article_processing_charge: No
author:
- first_name: Therese
full_name: Biedl, Therese
last_name: Biedl
- first_name: Stefan
full_name: Huber, Stefan
id: 4700A070-F248-11E8-B48F-1D18A9856A87
last_name: Huber
orcid: 0000-0002-8871-5814
- first_name: Peter
full_name: Palfrader, Peter
last_name: Palfrader
citation:
ama: 'Biedl T, Huber S, Palfrader P. Planar matchings for weighted straight skeletons.
In: 25th International Symposium, ISAAC 2014. Vol 8889. Springer Nature;
2014:117-127. doi:10.1007/978-3-319-13075-0_10'
apa: 'Biedl, T., Huber, S., & Palfrader, P. (2014). Planar matchings for weighted
straight skeletons. In 25th International Symposium, ISAAC 2014 (Vol. 8889,
pp. 117–127). Jeonju, Korea: Springer Nature. https://doi.org/10.1007/978-3-319-13075-0_10'
chicago: Biedl, Therese, Stefan Huber, and Peter Palfrader. “Planar Matchings for
Weighted Straight Skeletons.” In 25th International Symposium, ISAAC 2014,
8889:117–27. Springer Nature, 2014. https://doi.org/10.1007/978-3-319-13075-0_10.
ieee: T. Biedl, S. Huber, and P. Palfrader, “Planar matchings for weighted straight
skeletons,” in 25th International Symposium, ISAAC 2014, Jeonju, Korea,
2014, vol. 8889, pp. 117–127.
ista: 'Biedl T, Huber S, Palfrader P. 2014. Planar matchings for weighted straight
skeletons. 25th International Symposium, ISAAC 2014. ISAAC: International Symposium
on Algorithms and Computation, LNCS, vol. 8889, 117–127.'
mla: Biedl, Therese, et al. “Planar Matchings for Weighted Straight Skeletons.”
25th International Symposium, ISAAC 2014, vol. 8889, Springer Nature, 2014,
pp. 117–27, doi:10.1007/978-3-319-13075-0_10.
short: T. Biedl, S. Huber, P. Palfrader, in:, 25th International Symposium, ISAAC
2014, Springer Nature, 2014, pp. 117–127.
conference:
end_date: 2014-12-17
location: Jeonju, Korea
name: 'ISAAC: International Symposium on Algorithms and Computation'
start_date: 2014-12-15
date_created: 2022-03-21T07:09:03Z
date_published: 2014-11-08T00:00:00Z
date_updated: 2023-02-23T12:20:55Z
day: '08'
department:
- _id: HeEd
doi: 10.1007/978-3-319-13075-0_10
intvolume: ' 8889'
language:
- iso: eng
month: '11'
oa_version: None
page: 117-127
publication: 25th International Symposium, ISAAC 2014
publication_identifier:
eisbn:
- '9783319130750'
eissn:
- 1611-3349
isbn:
- '9783319130743'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '481'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Planar matchings for weighted straight skeletons
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8889
year: '2014'
...