Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory
Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Effenberger, Felix [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2010 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag 2011 |
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| Übergeordnetes Werk: |
Enthalten in: Computing and visualization in science - Springer-Verlag, 1997, 13(2010), 8 vom: Dez., Seite 377-396 |
|---|---|
| Übergeordnetes Werk: |
volume:13 ; year:2010 ; number:8 ; month:12 ; pages:377-396 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00791-011-0152-x |
|---|
| Katalog-ID: |
OLC2051430179 |
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| 520 | |a Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. | ||
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10.1007/s00791-011-0152-x doi (DE-627)OLC2051430179 (DE-He213)s00791-011-0152-x-p DE-627 ger DE-627 rakwb eng 500 VZ 004 VZ 11 ssgn Effenberger, Felix verfasserin aut Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2011 Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. Vector field topology Interpolation Barycentric interpolation Linear interpolation Bilinear interpolation Level sets Higher-order singularities Computational group theory Colorings Weiskopf, Daniel aut Enthalten in Computing and visualization in science Springer-Verlag, 1997 13(2010), 8 vom: Dez., Seite 377-396 (DE-627)232187533 (DE-600)1391037-1 (DE-576)060776471 1432-9360 nnns volume:13 year:2010 number:8 month:12 pages:377-396 https://doi.org/10.1007/s00791-011-0152-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT GBV_ILN_31 GBV_ILN_40 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2004 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4277 AR 13 2010 8 12 377-396 |
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10.1007/s00791-011-0152-x doi (DE-627)OLC2051430179 (DE-He213)s00791-011-0152-x-p DE-627 ger DE-627 rakwb eng 500 VZ 004 VZ 11 ssgn Effenberger, Felix verfasserin aut Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory 2010 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 2011 Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. Vector field topology Interpolation Barycentric interpolation Linear interpolation Bilinear interpolation Level sets Higher-order singularities Computational group theory Colorings Weiskopf, Daniel aut Enthalten in Computing and visualization in science Springer-Verlag, 1997 13(2010), 8 vom: Dez., Seite 377-396 (DE-627)232187533 (DE-600)1391037-1 (DE-576)060776471 1432-9360 nnns volume:13 year:2010 number:8 month:12 pages:377-396 https://doi.org/10.1007/s00791-011-0152-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT GBV_ILN_31 GBV_ILN_40 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2004 GBV_ILN_2012 GBV_ILN_2018 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4277 AR 13 2010 8 12 377-396 |
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Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory |
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Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory |
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Effenberger, Felix |
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Effenberger, Felix Weiskopf, Daniel |
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finding and classifying critical points of 2d vector fields: a cell-oriented approach using group theory |
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Finding and classifying critical points of 2D vector fields: a cell-oriented approach using group theory |
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Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. © Springer-Verlag 2011 |
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Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. © Springer-Verlag 2011 |
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Abstract We present a novel approach to finding critical points in cell-wise barycentrically or bilinearly interpolated vector fields on surfaces. The Poincaré index of the critical points is determined by investigating the qualitative behavior of 0-level sets of the interpolants of the vector field components in parameter space using precomputed combinatorial results, thus avoiding the computation of the Jacobian of the vector field at the critical points in order to determine its index. The locations of the critical points within a cell are determined analytically to achieve accurate results. This approach leads to a correct treatment of cases with two first-order critical points or one second-order critical point of bilinearly interpolated vector fields within one cell, which would be missed by examining the linearized field only. We show that for the considered interpolation schemes determining the index of a critical point can be seen as a coloring problem of cell edges. A complete classification of all possible colorings in terms of the types and number of critical points yielded by each coloring is given using computational group theory. We present an efficient algorithm that makes use of these precomputed classifications in order to find and classify critical points in a cell-by-cell fashion. Issues of numerical stability, construction of the topological skeleton, topological simplification, and the statistics of the different types of critical points are also discussed. © Springer-Verlag 2011 |
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