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- Guoning Chen, Konstantin Mischaikow, Robert S. Laramee, Pawel Pilarczyk, Eugene Zhang
- IEEE Transactions on Visualization and Computer…
- 2007

Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the… (More)

- Zin Arai, William D. Kalies, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka, Pawel Pilarczyk
- SIAM J. Applied Dynamical Systems
- 2009

A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed… (More)

- Konstantin Mischaikow, Marian Mrozek, Pawel Pilarczyk
- Foundations of Computational Mathematics
- 2005

We introduce an e cient algorithm to compute the homomorphism induced in (relative) homology by a continous map. The algorithm is based ∗Research supported in part by NSF Grant 0107396 †Partially supported by the Polish Committee for Scienti c Research (KBN), grant no. 2 P03A 041 24. ‡Partially supported by the Polish Committee for Scienti c Research (KBN),… (More)

In this note, we segment and topologically classify brain vessel data obtained from magnetic resonance angiography (MRA). The segmentation is done adaptively and the classification by means of cubical homology, i.e. the computation of homology groups. In this way the number of connected components (measured by H0), the tunnels (given by H1) and the voids… (More)

- Stefano Luzzatto, Pawel Pilarczyk
- Foundations of Computational Mathematics
- 2011

- Marian Mrozek, Pawel Pilarczyk, Natalia Zelazna
- Computers & Mathematics with Applications
- 2008

We present a new reduction algorithm for the efficient computation of the homology of a cubical set. The algorithm is based on constructing a possibly large acyclic subspace, and then computing the relative homology instead of the plain homology. We show that the construction of acyclic subspace may be performed in linear time. This significantly reduces… (More)

- Pawel Pilarczyk, Pedro Real Jurado
- Adv. Comput. Math.
- 2015

We introduce algorithms for the computation of homology, cohomology, and related operations on cubical cell complexes, using the technique based on a chain contraction from the original chain complex to a reduced one that represents its homology. This work is based on previous results for simplicial complexes, and uses Serre’s diagonalization for cubical… (More)

The homology of binary 3–dimensional digital images (digital volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities. Homology generators corresponding to these features are represented by nontrivial 0– cycles, 1–cycles and 2–cycles, respectively. In the framework of cubical representation of… (More)

- Pawel Pilarczyk
- Foundations of Computational Mathematics
- 2010

An automated general purpose method is introduced for computing a rigorous estimate of a bounded region in R whose points satisfy a given property. The method is based on calculations conducted in interval arithmetic and the constructed approximation is built of rectangular boxes of variable sizes. An efficient strategy is proposed, which makes use of… (More)

- Mateusz Juda, Marian Mrozek, +6 authors Natalia Zelazna
- 2014

We present an efficient software package for computing homology of sets, maps and filtrations represented as cubical, simplicial and regular CW complexes. The core homology computation is based on classical Smith diagonalization, but the efficiency of our approach comes from applying several geometric and algebraic reduction techniques combined with smart… (More)